Copyright © 2013 jsd

Introduction to Higher Math
John Denker

1  Preview

Let’s talk about “higher math” and its applications. For present purposes, higher math means anything beyond arithmetic. The goal here is not to teach you higher math, but merely to offer a few reasons why you might want to go learn it, i.e. why you might find it interesting and useful.

Let’s start with a simple yet practical example.

Suppose Mr. X lends a shovel to Mr. Y, and then Mr. Y lends it to Mr. Z. When it comes time to return the shovel, Z can return it directly to X, without having to go through Y.

This rule applies to any X, Y, and Z.

This is the language of algebra, pure and simple.

In some official documents, one-letter names such as X, Y, and Z are used in exactly this way, and indeed this is how the famous XYZ Affair got its name; see reference 1. Other documents may use somewhat longer names, such as John Doe and Richard Roe, which serve exactly the same purpose. These are sometimes called dummy names or placeholder names. A mathematician would call them algebraic variables.

A great many important ideas are expressed using this sort of language. In many cases it would be next to impossible to express them any other way. It must be emphasized that this is already part of the language, necessary for daily life, not limited to math and science. However, studying math will give you a better understanding of this language, and allow you to use it in more powerful ways.

Here’s another simple yet very practical example that uses mathematical ideas: Consider the hallway floor shown in figure 1. Using a yardstick, we find that the hallway is very nearly rectangular, 10 yards long, and 2 yards wide.
Figure 1: Hallway Floor

We can express the length and width using simple equations:

L = 10 yd     (1a)
W = 2 yd     (1b)

where L represents the length, W represents the width, and yd is an abbreviation for yard (or yards).

In the diagram, the length is shown in red and the width is shown in blue. The black tick-marks show the length and width divided into yards. We can write this as an equation:

L / yd = 10     (2a)
W / yd = 2     (2b)

Equation 2 is another way of formulating the same idea as equation 1. Depending on circumstances, one formulation or the other may be more convenient.

Note that in equation 2a the right-hand side of the equation is a pure number, namely 10. The left-hand side of the equation is also a pure number, because it is one length divided by another length.   This stands in contrast to equation 1a, where both sides of the equation are lengths, not pure numbers.

Equation 2 is one of those all-too-rare situations where the language of English agrees with the language of algebra: The length of the hallway can be divided into ten yards as surely as a pizza can be divided into six slices. This is formalized by the divide-by symbol on the left-hand side of equation 2a. We can then count the subdivisions. This gives us the number on the right-hand side.

This is an interesting lesson already, because it tells you that mathematics is not just arithmetic. It is not limited to numbers. We can write equations involving things like lengths (as in equation 1a) which are emphatically different from pure numbers (as in equation 2a).

Now suppose we want to express the length in terms of in feet (not yards). It’s the same length, just expressed in different units. We could re-measure the length using a one-foot ruler, but it is easier to figure it out mathematically. We can use the following fact:

1 yd = 3 ft

We can always multiply any expression by 1. This leaves the value unchanged. (The rule about multiplying by 1 comes directly from the laws of mathematics; it is a defining property of 1.) Let’s apply this to equation 1a.

L = 
10 yd · 
10 yd · 

3 ft
1 yd

  = 30 ft

This trick of multiplying by 1 as a means of converting from one set of units to another is called the factor label method. It is very widely used. There are tons of pedagogical resources on the topic. See also section 5.4.

This is an example of applied mathematics. This is also an example of physics. That is, by combining some measurements and some mathematics, we build a theoretical model that allows us to ascertain something about the real world that we did not directly measure. We know the length in feet, even though we didn’t measure it directly using a one-foot ruler. You know it’s not pure mathematics, because the result is not exact. It depends on various approximations, notably the assumption that the floor is flat. If the floor had lots of undulations, measuring it with a yardstick and measuring it with a ruler might well give different lengths. For an ordinary floor, however, the calculation in equation 4 is a good-enough approximation for most purposes. See section 2.9 for more about the limitations of mathematics.

Things get even more interesting if we want to know the area of the hallway floor.

There are ways of measuring the area of the floor directly – perhaps by covering it with tiles of known area and counting the tiles – but for a rectangular region it is quicker and more convenient to measure the edges and multiply. For a rectangular region, the area is equal to the length multiplied by the width. We can write this rule as an equation:

A = L · W

Applying this rule to our hallway, we find:

A = L · W               
  = 10 yd · 2 yd     
  = 20 yd2

where A denotes the area, and yd2 is pronounced “yard squared” or equivalently “square yard”. It must be emphasized that when we write a square yard as yd2 that does not mean two yards. It is not a yard plus a yard. It is a yard times a yard. This is part of the notation and terminology of mathematics: The small superscript 2 means to multiply something by itself.

The approach used in equation 6 – combining measurements with theory – is a lot less work than trying to measure the area directly, even in this simple example. (In more complicated situations, the advantage is even more dramatic.)

In equation 6, note the contrast:

Multiplying 10 by 2 to get 20 is just arithmetic. It’s just numbers.   Multiplying a yard by a yard to get a square yard is higher math.

A yard is not a number; it’s something else entirely. It is proverbially improper to compare apples to oranges, and by the same token it is improper to compare oranges to square yards. It’s also improper to compare yards to square yards. If they were numbers you could compare them, but they aren’t and you can’t. Yards and square yards live in a high-dimensional abstract space of their own ... abstract yet very practical and very relevant to the real world.

Now suppose we want to express the area in units of square feet. It’s the same area, just described in different terms. We can use equation 3 to convert equation 6 using the same ideas as before, with a slight twist.

A = 
20 yd2 · 
20 yd2 · 

3 ft
1 yd


3 ft
1 yd

  = 180 ft2                       

Notice that we had to multiply by 3ft/yd twice. That’s because we started with yd2, which is a yard times a yard, and we need to convert both factors. Even though one yard is equal to three feet, a square yard is not equal to three square feet. In fact, a square yard is equal to nine square feet, as you can see in figure 2. That’s a nontrivial fact.

Figure 2: Foot, Yard, Square Foot, and Square Yard

Note that the floor in figure 1 is tiled in one-foot squares. You could determine the area directly by counting tiles, but it is easier to measure the length and width and then ascertain the area by multiplying.

Again our mathematical model is an excellent approximation to the real world, but it is not exact. See section 2.9 and especially example 2-1.

*   Contents

2  Overview

2.1  Higher Math Topics

As mentioned in section 1, higher math means anything beyond arithmetic. It includes the topics listed in table 1, plus a tremendous amount of other stuff. Some of the applications are mentioned in section 2.4 and elsewhere. We aren’t going to explore all of higher math, just the easiest and most useful parts. We assume you have never studied this in any depth before, or have forgotten everything you learned about it.

set theory      logic      probability
algebra      calculus       
topology      geometry      trigonometry
Table 1: A Few Higher Math Topics

The most important thing you need to know is that math doesn’t have to be weird or complicated. Like music or sports or anything else, if you take it to extremes it can get very very complicated, but we aren’t going to take it that far.

Generally speaking, math is a bag of tricks for reasoning about stuff and solving problems. For a more detailed look at reasoning and problem-solving in general, see reference 2 and reference 3.

2.2  Tools and Techniques – Preview

There are a lot of things that you need to learn that aren’t officially listed as part of any ordinary course. These are important, indeed more important than some of the things that are listed.

multi-step reasoning (section 8.2)    
higher reliability, stronger proof    
trusting certain tools    
avoiding certain booby traps    
skimming, reading, re-reading, and pondering a text (section 8.3)    
learning a new language    
generalization, symbolism, and abstraction    
imagination, creativity, artistry, and elegance    
Table 2: Tools and Techniques

These things are related in various ways. For example, multi-step reasoning demands high reliability, as discussed in section 8.2.

None of these things directly requires mathematics, or is restricted to purely mathematical applications. For example, computer programming requires all of those things, as surely as traditional math does. See section 8.1 for more about this.

It must be emphasized that there is more to mathematics than right and wrong. There is also elegance. Continuing down that road, there is yet again more to applied math. There is not just right and wrong, and not just elegance. There is also relevance and practicality.

2.3  Higher Math is Not Arithmetic

It must be emphasized from the outset that higher math is dramatically different from arithmetic. You don’t even need to be good at arithmetic to do higher math.

Professional mathematicians do not sit around adding up long columns of numbers. Really they don’t. They’ve got better things to do. Higher math is an almost-completely different set of skills.

Basic arithmetic is devoid of the things that make math interesting, including elegance and creativity. Arithmetic is relevant to higher math in much the same way that arithmetic is relevant to cooking: It is sometimes useful, but it’s not really the point.

Some mathematicians are really good at arithmetic, but some of them aren’t. The ones in the latter group can certainly figure out arithmetic problems, but they have to stop and think about it. Note that mathematicians are allowed to use spreadsheets, just like everybody else.

2.4  Applications Are Important: The Basket Analogy

Consider the analogy:

A wicker basket, by itself, it’s not good for much. You can’t eat it. It doesn’t make a very good pillow, or a very good hat, or very good underwear. If you’ve got nothing to store and nothing to carry, the basket is not immediately practical. On the other hand, if you have several important things to carry and somewhere to go, a basket might be tremendously helpful.   Mathematical skills, by themselves, are not good for much. However, when an application comes along that calls for those skills, they can be tremendously helpful.

Sometimes you find something that looks like a basket but is purely ornamental. Sometimes a weaver will make a basket just for fun, or just to experiment, to see what’s possible and what’s not.   Sometimes people do math puzzles just for fun. Sometimes mathematicians are motivated by pure esthetics, and sometimes by curiosity, to see what’s possible and what’s not.

The so-called pure mathematicians pay little attention to applications. According to legend, Euclid mocked a beginning student who asked what geometry was good for. However, in my opinion, that’s really bad pedagogy. Most of us are not pure mathematicians. I can appreciate the artistry in an elegant mathematical proof, but that is not the only thing I expect to get from mathematics.

Higher math is needed for:

2.5  Math is about Patterns and Relationships

A great deal of higher mathematics is devoted to patterns, relationships, and generalizations ... finding them, creating them, understanding them, et cetera. This can involve spatial patterns, linguistic patterns, patterns in completely abstract systems, or whatever. Sometimes mathematicians look for patterns in numbers, but that’s not the only focus or even the main focus.

Some non-numerical examples are given in section 1 and section 3. Meanwhile, some numerical examples are given in section 4.

2.6  Math is a Language

Mathematics provides (among other things) a language that helps us express ideas.

The language of algebra is already part of everyday conversations – and the ideas of algebra are already part of everyday thought – whether you realize it or not. It is not restricted to numbers. For example, in example 1-1, the algebraic variables X, Y, and Z refer to persons, not numbers.

Of course math is not just a language. Knowing the English language does not make you a novelist; the primary requirement for writing a novel is having something interesting to say. The language just helps you say it. Math is primarily a bag of tools for reasoning about stuff and solving problems. Mathematical language helps with this, but it isn’t the goal and it isn’t the only tool.

2.7  Math is about Logic and Proofs

There is a tradition going back 2300 years that calls for studying geometry in terms of proofs ... or vice versa. Euclid’s book is about geometry in the same way that Orwell’s book is about animals, i.e. hardly at all. Orwell’s animals are a backdrop and a pretext for talking about politics. Euclid’s geometry is a backdrop and a pretext for talking about proofs.

Some of the proofs that one encounters in high-school geometry are elegant, intricate, and ingenious.

Beware that the mathematical approach has its limitations, as discussed in section 2.9.

2.8  Math is Required for Science

Algebra and geometry are prerequisites for physics, biology, chemistry, engineering, and computing.

Maybe 30 years ago, students who were interested in science but not interested in mathematics might be advised to go into biology. Nowadays, though, that would be very bad advice. The life sciences (including medicine) have become intensely quantitative and mathematical.

A while back I was out in the middle of the desert, helping some graduate students who were studying Gila monsters. Whenever they found one, they measured its height, width, length, tail volume, and temperature. They recorded the location and environmental conditions. For identification, they took a picture and implanted an electronic RFID tag. Last but not least, they took a DNA sample.

All this went into a database. By analyzing the DNA, they were able to construct lineages, showing who was related to whom. This step was highly mathematical, involving reasoning about structures in an abstract high-dimensional space.

On a more down-to-earth level, dealing with the location involved geometry and algebra. They used GPS coordinates (which use one ellipsoid) and then they needed to convert to map coordinates on an out-of-date map (which used a different ellipsoid). If you don’t know what an ellipsoid is, you are on the outside looking in. You can’t even be part of the conversation.

2.9  Limitations of the Mathematical Approach

Although it is useful to learn the methods of mathematical proof (as mentioned in section 2.7 and elsewhere), in practice the mathematical has some serious limitations.

Religion offers certainty; science generally does not. Instead, science teaches you how to survive and get things done in an uncertain world.

This takes up where example 1-5 left off. Suppose we are buying tiles to cover the floor of a hallway that is similar to figure 1 but not identical: The floor is 30.2 feet long by 5.9 feet wide. The area is less than 180 square feet, and each tile covers one square foot ... but we need more than 180 tiles. Calculating the area tells us the number of tiles approximately but not exactly. The abstract mathematical area is the answer to the wrong question. It’s almost the right question, but not quite. In the real world, the objective is not merely to cover the area, but rather to cover the area in a way that looks nice. Covering the area using misshapen scraps of tile does not look nice.

A better calculation is to round up to an integer number of tiles in each direction, and then multiply. That gives us 6*31, which is 186 tiles. An even better calculation take into account the fact that these tiles come in boxes of 20, so we have to round up again. We have to buy 200 tiles, with the expectation that there will be some leftovers.

Bottom line: Just because you have a theoretical model doesn’t mean it is the correct theoretical model. When you calculate something, check that you are calculating the right thing. Don’t grind out an exact answer to the wrong question.

3  Some Simple Non-Numerical Examples

In this section, we present some examples that are highly mathematical but not arithmetical.

3.1  Sudoku

Sudoku puzzles are logic puzzles, not arithmetic puzzles. They are intensely mathematical, but not numerical. They do not require multiplication or even addition.

Even though they are normally written using the digits {123456789}, the digits are not really representing numbers; you could equally well write the puzzle in terms of the nine letters {abcdefghi}. More to the point, you could write the puzzle in terms of letters that aren’t in alphabetical order, such as the nine letters in the word {sunflower}, as in the example below. You could also do it in terms of nine abstract symbols that have no relationship to each other, such as { ∇ ♒ ‡ ∧ © ≡ ∞ ξ ☿ }.

Here is an example. The usual sudoku rules apply: each of the nine symbols must appear once in each row, once in each column, and once in each of the nine different 9×9 blocks. (The blocks are indicated by the shading.) The symbols are the nine letters in the word {sunflower}. Some of the symbols have been filled in, to help you get started. The solution is given in appendix 13.1.


3.2  Straight-Cut Origami

It must be emphasized that higher math includes a lot of things besides algebra. Here’s an example that involves pure geometry, with no algebra, and certainly no arithmetic. It’s a fold-and-cut puzzle:

Each student (or each 3-student team) gets a pair of scissors plus a piece of paper with an arbitrary triangle drawn on it. The mission, should you decide to accept it, is to cut out the triangle using only a single straight cut. Hint: you may fold the paper any way you like before cutting.

This puzzle is intensely mathematical – yet it involves no numbers, no arithmetic, and no algebra. It is an interesting puzzle. It has the advantage that you can pose it to people who don’t know any physics, don’t know any algebra, and couldn’t multiply 44 by 5 without a calculator.

Reference 6 is a news story that features this puzzle, and offers some hope that the much-needed revolution in math education is starting.

The puzzle comes from chapter 6 of reference 7. The book starts out with an uncompromising manifesto of “art for art’s sake” ... but as it goes along it mentions a few bits of math that started out super-abstract but found important applications.

3.3  Functions

Consider the following six statements:

We can express the same information in a less verbose form, as shown in table 3:

fruit    color
lemon    yellow
banana    yellow
cherry    red
McIntosh    red
Granny_Smith    green
lime    green
Table 3: Color as a Function of Fruit

Grid 1 shows yet another way of expressing the same information.

     yellow  red  green
lemon      ·  ·
banana      ·  ·
cherry    ·    ·
McIntosh    ·    ·
Granny_Smith    ·  ·  
lime    ·  ·  
Grid 1: Color as a Function of Fruit

In this example, the grid representation is slightly less compact than the tabular representation, but in other cases it may be more compact. Furthermore, the grid representation is sometimes easier to interpret.

It is relatively easy to verify that there is only one checkmark per row in grid 1. However, not all grids have this property, as we can see in grid 2, where we have lumped together both varieties of apple, and both kinds of citrus. There are only four rows, but still six checkmarks.

     yellow  red  green
citrus      ·  
banana      ·  ·
cherry    ·    ·
apple    ·    
Grid 2: Color versus Fruit

As a point of terminology, in grid 1 we say that the color is a function of the type of fruit. The defining property of a function is that there is only one checkmark per row. This stands in contrast to grid 2, where color is not a function of the category of fruit. There is a relationship between color and the category of fruit, but this relationship does not qualify as a function.

Very commonly non-experts say “function” even when the relationship is not a function, but this is a mistake.

On the other hand, given a well-behaved non-function, it is usually possible to create a function, using the idea of sets. Table 4 shows how this works when applied to our example.

fruit    set of colors
citrus    {yellow, green}
banana    {yellow}
cherry    {red}
apple    {red, green}
Table 4: Set of Colors as a Function of Fruit

Note that color itself is still not a function of the category of fruit. Instead it is a set of colors that exists as a function of the category of fruit in this example.

There is a subtle distinction between “color” and “set of colors” – but the distinction is important. The idea of “set” is completely abstract, but it’s not very complicated.

4  Some Simple Numerical Examples

In this section, the algebraic variables represent numbers.

4.1  The Meaning of Equality versus Recipe

Consider simple arithmetic problems such as the following:

Fill in the blank:
7 + 3 = ____


Fill in the blank:
____ = 7 + 3

Nowadays students are asked to solve problems like this in kindergarten. Note that there are two ways of reading such a problem:

  1. Carry out the operations on the left-hand side (LHS) of of equation 8, and use that as a recipe to find a value for the RHS.
  2. Find a value for “____” that makes the equation true.

These two readings are significantly different.

  1. The “recipe” in equation 8 is strictly left-to-right. It would be nonsense to use the RHS as a recipe for computing the LHS.

    Meanwhile, equation 9 is strictly right-to-left.

  2. Any reasonable notion of equality is symmetrical. For example, if 1 yd = 3 ft  then  3 ft = 1 yd. You can read the equation either way, left-to-right or right-to-left. The “find a value” reading is symmetrical, and works just fine for a very wide class of equations, including equation 10 in addition to equation 8 and equation 9. (In contrast, the “recipe” interpretation fails for equation 10.)

Fill in the blank:
7 + ____ = 10

The existence of problems such as equation 10 is a Big Deal conceptually and pedagogically. Any students who latch onto the idea that every equation is a recipe (as in equation 8) will have to unlearn that before they can cope with equation 10. Unlearning is always hard.

Figure 3 shows one way of solving equation 10, namely a graphical method. Start with a group of ten things. Draw a loop around seven of them. The number remaining outside the loop is a solution to equation 10. This can be seen from the fact that the number inside plus the number outside adds up to 10.

Figure 3: 10 = 7 + 3

Another way to solve the problem is by counting on your fingers, performing a calculation essentially equivalent to figure 3.

Yet another way to solve the problem is by using an addition table, such as the one shown in table 5. Find the 7th column, and run down that column until you find a 10. The corresponding row-number is the solution to the problem.

There also exists a purely mathematical recipe for solving problems of this kind – a recipe called subtraction – but equation 10 does not explicitly depend on subtraction. We do not need any minus signs in order to write equation 10. In fact, the idea behind equation 10 can be used to define what we mean by subtraction.

There’s a name for what we’re doing here: It’s called algebra. Equation 10 is not very fancy algebra, but it is definitely algebra. Note the contrast:

10 − 7 = ____ is not algebra. It’s just arithmetic. You’re doing subtraction because you were told to do subtraction.   7 + ____ = 10 is algebra. You might solve it by doing subtraction, but the equation doesn’t tell you to do that. You have to apply some mathematical reasoning to change the given equation into a subtraction problem.

The distinction between 10 − 7 = ____ and 7 + ____ = 10 is is like crossing from Nogales, Arizona to Nogales, Sonora. You aren’t very far from the border, but you’re definitely in a different country. The equation 7 + ____ = 10 is definitely on the algebra side of the border.

4.2  Two Unknowns

Let’s consider some much fancier than equation 10, namely equation 11. Equations like this show up in school, sometimes even at kindergarten level nowadays:

Fill in both blanks:
____ +  ____ = 10

This equation has the remarkable property that it has more than one solution. For example, 3 + 7 = 10 and also 6 + 4 = 10.

One way of solving this problem uses the method outlined in figure 3. Start with a group of ten things, then draw a circle around any number of them, any number from zero on up, any number from zero to ten inclusive. The number inside and the number outside can be used to fill in the blanks in equation 11.

Another way of solving this problem is to use an addition table, such as table 5. Look through the table until you find a 10 somewhere. Then read off the column number and the row number.

Some people go bonkers when they see a question of this kind. Sometimes for political or cultural reasons they think the most important thing is for every student to get the same answer, and it horrifies them to think that different students might come up with different yet fully-correct answers.

In kindergarten, the students are asked to find some solution to the problem, i.e. some way of filling in the blanks. In contrast, a mathematician looks at equation 11 and wants to find all solutions.

It is quite remarkable that equation 10 has exactly one solution, while equation 11 can have infinitely many solutions. The two equations look somewhat different, but they don’t look infinitely different.

We are definitely doing higher math now. Basic arithmetic does not produce infinities, and cannot deal with infinities. Arithmetic deals with numbers, whereas infinity is not a number. Higher math deals with all sorts of things that aren’t numbers.

We can make equation 11 look fancier by giving names to the unknowns.

Find some x and y to solve the equation:
x + y = 10                       (12a)
x = ____      (12b)
y = ____      (12c)

That may look fancier than equation 11, but it has exactly the same meaning.

Note that in a system of equations like this, the x in equation 12a must have the same value as the x in equation 12b. By the same token, the y in equation 12a must have the same value as the y in equation 12c. We get to choose a value for x, but whatever we choose has to be consistent across the whole problem, across the whole system of equations. See section 4.15.

We can apply that idea in a useful way in equation 13. This is equivalent to equation 11 with the added requirement that the same number must be used to fill in both blanks.

Find some x to solve the equation:
x + x = 10                       (13a)
x = ____      (13b)

In equation 13 and elsewhere, the rule is: In any given system of equations, every time x appears, it has to have the same value. In equation 12, x can be different from y ... but x cannot be different from x.

Note that equation 13 has only one solution, whereas equation 12 has many solutions.

4.3  Addition Table

0  1  2  3  4  5  6  7  8  9  10   
1  2  3  4  5  6  7  8  9  10  11   
2  3  4  5  6  7  8  9  10  11  12   
3  4  5  6  7  8  9  10  11  12  13   
4  5  6  7  8  9  10  11  12  13  14   
5  6  7  8  9  10  11  12  13  14  15   
6  7  8  9  10  11  12  13  14  15  16   
7  8  9  10  11  12  13  14  15  16  17   
8  9  10  11  12  13  14  15  16  17  18   
9  10  11  12  13  14  15  16  17  18  19   
10  11  12  13  14  15  16  17  18  19  20   
Table 5: Addition Table

The addition table has some interesting properties.

Note the contrast:

Constructing the addition table is just arithmetic. Using the table to perform addition is just arithmetic.   Looking for symmetries and patterns in the table is higher math.

4.4  Multiplication Table

1  2  3  4  5  6  7  8  9  10     
2  4  6  8  10  12  14  16  18  20     
3  6  9  12  15  18  21  24  27  30     
4  8  12  16  20  24  28  32  36  40     
5  10  15  20  25  30  35  40  45  50     
6  12  18  24  30  36  42  48  54  60     
7  14  21  28  35  42  49  56  63  70     
8  16  24  32  40  48  56  64  72  80     
9  18  27  36  45  54  63  72  81  90     
10  20  30  40  50  60  70  80  90  100     
Table 6: Multiplication Table

The multiplicationtable has some interesting properties.

4.5  Relationships and Operators

The language of algebra can be used in many ways. Sometimes it is used to set up an equation to be solved. That’s the first thing some people think of when you mention algebra, but it’s by no means the only thing that algebra is good for.

Consider the contrast:

Setting up an equation to be solved.   Asserting a relationship.

In statement 16, the goal is to find a numerical value for x. The equation tells us about a particular number x.   In statement 17, it is not necessary, desirable, or possible to solve for x or y.

Find a value for x such that       (16a)
2x − 7 = 1       (16b)


For all real numbers x and y,       (17a)
(x + y) = (y + x)       (17b)

Statement 17 is not restricted to any particular numbers. It is a powerful generalization. In one sense, it is a general statement about all real numbers. In an even grander sense it is a general statement about the addition operator itself: It says that addition is commutative (when applied to real numbers).

Let’s be clear: The plus sign in equation 17b represents the addition operator. Addition is quite an abstract thing. It’s definitely not a number. Algebra gives us a language that allows us to say useful things about addition itself. Similarly it allows us to talk about other highly abstract things.

Suppose you see just the “equation” part of an algebraic statement by itself, such as equation 16b or equation 17b. The meaning of such a thing by itself would not be clear. You need the full statement. Note the contrast:

Statement 16a instructs us to find a numerical value for x, by solving equation 16b.   Statement 17a is what we call a universal quantifier: it asserts that equation 17b holds for all values of x and y.

4.6  Reaction Time

You can measure human reaction time using little more than a yardstick. People who have never seen a reaction-time measurement tend to be very surprised at how long reaction times really are.

Figure 4: Setup for Simple Ruler Drop

For details, see reference 8.

4.7  Brownies

Suppose you want to make brownies to feed 15 people. All the brownies must all be rectangular, with the same size and shape, one per person. For esthetic reasons, we want the aspect ratio to be no bigger than 1.5 to 1. That is, the length must be no more than 1.5 times the width. The brownie pan is square.

You can’t do it with exactly 15 brownies. You could make 3 rows of 5 but that doesn’t satisfy the aspect-ratio requirement.

You can however make 16 brownies and have one left over. That’s four rows of four.

The same solution works for 16 people, with nothing left over. In fact, the 4×4 solution is optimal for 13, 14, 15, or 16 people.

For 17 people, we need to find a different solution. 17 is a prime number, so that’s definitely not going to work. 18 can be factored as 3 rows of 6 or 2 rows of 9, but neither of those satisfies the aspect-ratio requirement.

19 is a prime number, so that’s not going to work. 20 works, namely 4 rows of 5. For 17 people, that leaves three left over. In fact the 4×5 solution is optimal for 17, 18, 19, or 20 people.

The 4×6 solution is optimal for 21, 22, 23, or 24 people.

The 5×5 solution is optimal for 25 people.

For present purposes, we define optimal to mean satisfying the requirements with minimal leftovers.

The question arises, how do we know that these are the only solutions? Well, we could do it by brute force, just multiplying together all pairs of numbers and seeing what works. However, mathematics gives us an easier way. We can appeal to the unique-factorization theorem. It says that any given integer can be factored using prime numbers in exactly one way (except for trivial re-ordering of the factors).

The goals and requirements can be expressed in mathematical language. For N people we have:

A × B  N            
A ÷ B  2/3            
A ÷ B  3/2            
A × B as small as possible         

4.8  Economical Car

Suppose you are shopping for a car. The question arises, does it make sense economically to get a hybrid car, or to get the corresponding non-hybrid car. The answer depends on how the car is to be used, so let’s consider two different scenarios.

First scenario: used as taxi, 25,000 miles per year, all city driving.

 Car #1 Car #2  
 Camry LE Camry Hybrid LE units
purchase price24000.00 28000.00 $
delta 4000.00   
hwy mileage35.00 39.00 mpg
city mileage25.00 43.00  
travel 25000.00  miles per year
fraction on highway 0.00  dimensionless
gas unit cost 3.00  $ per gallon
gas volume1000.00 581.40 gallons per year
gas cost3000.00 1744.19 $ per year
delta -1255.81  $ per year
payback time 3.19  years

Second scenario: Same two cars, retired person, much less driving, mostly on the highway.

 Car #1 Car #2  
 Camry LE Camry Hybrid LE units
purchase price24000.00 28000.00 $
delta 4000.00   
hwy mileage35.00 39.00 mpg
city mileage25.00 43.00  
travel 5000.00  miles per year
fraction on highway 0.75  dimensionless
gas unit cost 3.00  $ per gallon
gas volume157.14 125.22 gallons per year
gas cost471.43 375.67 $ per year
delta -95.76  $ per year
payback time 41.77  years

We see that in the first scenario, the hybrid is a good deal. The more expensive car quickly pays for itself via improved fuel economy.

In the second scenario, the more expensive car does not pay for itself.

This is a simplified analysis. It is a reasonable first approximation, suitable for cases where the conclusions are clear-cut. For more marginal situations, a more sophisticated calculation is required, taking into account interest rates, inflation, et cetera. One way to formalize this is to calculate the Net Present Value.

Remember, arithmetic is about numbers, whereas higher math is about patterns. So far we have only done a bunch of arithmetic.

This example begins to touch on higher math if you decide that doing the arithmetic by hand is too laborious and too error prone, so you do it using a spreadsheet instead. The language for programming a spreadsheet is essentially the language of algebra.

This example becomes truly higher math when you try to understand the trends:

The spreadsheet used to do these calculations is given in reference 9.

4.9  Squares and Square Roots

Let’s review some basic facts:

If the EdgeLength of a square is 2 inches, then the Area is 4 square inches. You can verify this by counting the little sub-squares in figure 5.

If the EdgeLength of a square is 3 inches, then the Area is 9 square inches.
Figure 5: Squares of Various Sizes

The general rule for finding the Area of a square is to multiply the EdgeLength times itself, in accordance with equation 19. This is a lot quicker and more reliable than counting sub-squares, especially when the area is large.

Area = EdgeLength · EdgeLength

The same procedure works even if the EdgeLength is not an integer, i.e. not a whole number. For example, if the EdgeLength is 1.5 inches then the area is 2.25 square inches.

We can also use graphs to show the relationship between EdgeLength and Area, as in figure 6 or equivalently figure 7.

sq-edge-area   sq-area-edge
Figure 6: Area versus EdgeLength   Figure 7: EdgeLength versus Area

It must be emphasized that these two figures convey exactly the same information. If you prefer one over the other, that is mostly a matter of personal taste. There are four possibilities, all of which work equally well:

There is a lot more that can be done with such graphs, as discussed in section 6.5.

Consider the contrast:

Sometimes we know the EdgeLength and want to calculate the Area.   Sometimes we know the Area and want to calculate the EdgeLength.

We say 9 the square of 3. This comes up so often that there is a special notation for it, using a superscript 2. The expression 32 is usually pronounced “three squared” and the expression 52 is usually pronounced “five squared”.

 9 = 32    (exactly)      (20a)
 16 = 42    (exactly)      (20b)
 2  = 1.4142    (very nearly)      (20c)

  We say 3 is the square root of 9. This comes up so often that there is a standard abbreviation for it (sqrt), and even a standard symbol (√). The expression √3 is pronounced “square root of three”.

3 = sqrt(9)    (exactly)      (21a)
4 = √(16)    (exactly)      (21b)
1.414 = √(2)    (very nearly)      (21c)

You can calculate the square of any number by direct multiplication, in accordance with equation 19. You can also read off the answer from a graph such as figure 6 or figure 7.   There are procedures for calculating the square root of any number. Details can be found in reference 10. For now, you can just read off the answer from a graph such as figure 6 or figure 7. Also, any spreadsheet program and virtually any pocket calculator will calculate square roots for you. Look for the calculator key labeled with the √ symbol.

The EdgeLength verus Area relationship is the inverse of the Area versus EdgeLength relationship (and vice versa). In other words, if we restrict attention to non-negative numbers, the “square” relationship is the inverse of the “square root” relationship.

This is important, because it gives you a way to check your work. If you are not sure that equation 21c is correct, you can check it by calculating the square of 1.414 by direct multiplication, and comparing with equation 20c. We can express the general rule using the language of algebra: For any non-negative number X

 = X                
X · X
 = X

4.10  Some Generalizations

In general, the Area of a rectangle is equal to the Length of edge X multiplied by the Length of side Y (where X and Y are perpendicular). This should be obvious from basic notions of counting squares. If you remember the formula for a rectangle, you don’t need to separately remember the formula for a square, because a square is just a special kind of rectangle, namely the kind where X=Y.

I could have mentioned this in section 4.9 but I didn’t, because it was more than we needed to know at the time.

Let’s talk about scaling. You may be familiar with the term, perhaps in connection with scaling up a recipe, if you want to make twice as many cookies.

The idea expressed in figure 5 and figure 8 is not limited to square-shaped figures.

Figure 8: Small and Large Squares

The same sort of thing happens with triangles, as shown in figure 9. When the edge of the triangle grows by a factor of two, the area of the triangle grows by a factor of two squared, i.e. 22, i.e. 4.

Figure 9: Small and Large Triangles

The general idea here is that the triangle is a two-dimensional figure, while the edge is one-dimensional. When we increase the edge by a factor of 2, we increase both the horizontal and vertical size of the triangle by a factor of 2, so the area goes up by two factors of 2.

If we increase the edge by a factor of 3, then the area goes up by a factor of 32 i.e. three squared i.e. 3×3 i.e. 9.

The same logic applies to any two-dimensional figure, not just triangles and squares. This is called scaling. For more on this, see reference 11.

4.11  Diagonal Distances

If you move 4 units horizontally and 3 units vertically, you wind up 5 units from where you started, as the crow flies. Similarly, if you move 12 units horizontally and 5 units vertically, you wind up 13 units from where you started, as the crow flies. This is shown in figure 10.

Figure 10: Diagonal Distances

Now suppose you move B units straight horizontally and A units straight vertically, and you find yourself C units in a straight line from where you started. The general rule (subject to mild restrictions) is that these distances obey the equation:

A·A + B·B = C·C     (23a)
A2 + B2 = C2     (23b)

Equation 23b is entirely equivalent to equation 23a. In accordance with standard notation, A2 is pronounced “A squared” and means to multiply A by itself.

Equation 23 is a famous result, known as the Pythagorean theorem. It has been known for more than 2500 years. It tells us something important about the structure of the universe. It didn’t have to be that way. In particular, it only works for straight lines in a flat plane; if you measure great-circle distances on the surface of a sphere, the distances do not uphold equation 23 (unless the triangles are very small). Also, equation 23 does not apply to every triangle in the world; it only applies to right triangles, i.e. triangles where the A-side is perpendicular to the B-side.

4.12  Application: Cutting an Octagon from a Square

Here is a completely non-imaginary application.

In high-school wood-shop class I made an elaborately-carved two-foot-tall candlestick, as shown in figure 11. It needed a base. I decided that a multi-tiered octagonal base would look nice. Starting from a square piece of wood, you can make an octagon by cutting off the corners, but the question is, how much to cut? You could solve the problem using purely mechanical geometrical means, but it is just as easy to solve it using algebra.

Figure 11: Candlestick with Octagonal Base

So, suppose we have a square piece of wood, one foot on a side. We wish to make an octagon by cutting off the corners. Suppose we cut off a certain amount from each corner, as shown in figure 12. We don’t yet know the correct amount, but that’s OK, so long as we know x at the end. That’s one of the things (but not the only thing) that algebra is good for: If you don’t know exactly what something is, call it x and move on.

For the octagon, it is a simple matter to solve for x. Algebra gives us a systematic way of finding a value for x that will make all sides of the octagon equal in length.

After drawing the diagram, the next step is to write some algebraic equations that involve x. We then solve the equations to find the desired numerical value.

Figure 12: Cutting An Octagon Out of a Square

We now use two separate lines of reasoning to calculate two different sides in terms of x:

Since we want it to be a regular octagon, the two “different” sides are different only as to orientation; they are equal in length. We can visualize what is going on by making a graph, although this is not necessary. The length of the horizontal side (as given by equation 24) is shown in red, while the length of the sloping side (as given by equation 25) is shown in black. The requirement that the sides must have equal length is represented by the intersection of the two lines. By reading the chart you can see that the x-value must be slightly less than 0.3 and the corresponding length-value must be slightly more than 0.4.

Figure 13: Octagon Side Lengths versus x

Whether or not we have made a graph, we can express the requirement that the sides of the octagon are equal by combining equation 24 and equation 25 in to a single algebraic equation:

length of   length of        
sloping side = horizontal side        
 = 1 ft − 2x

We can solve it using a sequence of algebraic steps. At each step, we show the rationale and method for obtaining the next equation.

Rationale (R) and Method (M)     Equation
M: Restate equation 26.     
=1 ft − 2x      (27a)
R: We want all terms involving x      
    to be on one side.     
M: Add 2x to both sides.     
2x + 
=1 ft      (27b)
R: We want “something” times x.     
M: Distributive law.     
(2 + 
=1 ft      (27c)
R: We want x by itself.     
M: Divide both sides by 2+√2.     
1 ft
2 + 
Convert to decimal numeral.     
     x=0.2929 ft      (27e)

Last but not least, we should always check our work. The two sides of the octagon have the following lengths:

horizontal side = 1 ft − 2x=0.4142 ft      (28a)
sloping side = 
=0.4142 ft      (28b)

We see that the two sides have the same length, as they should, even though they were calculated in very different ways. We can verify after marking and before cutting that the sides have the correct length.

The algebraic technique we have used here is called “solving two linear equations in two unknowns” – but if that doesn’t mean anything to you, don’t worry about it.

4.13  Graphs, Tables, and Functions

Another big part of algebra is the idea of a function.

Unlike variables, which are already part of everyday language and everyday thought, the idea of a function is something that you may have to think about before you fully understand it.

The basic idea is that a function is a recipe. It is a machine that takes certain things as inputs, performs some manipulations, and produces something else as the output.

For details on this, see section 6.

4.14  Example: Medicine Schedule and Dosage

Coming soon.

4.15  Key Concept: Pick Consistent Values

This continues the discussion of consistency from section 4.2. Consider the following:

5·(1001) = 5005       (29a)
5·(1002) = 5010       (29b)
5·(1003) = 5010       (29c)
5·(1000 + x) = 5000 + 5·x   (for all x)     (29d)

Equation 29d uses the language of algebra to summarize the pattern we see in the previous lines. It is a powerful generalization. If you have 1000 plus something, and you multiply the whole thing by five, you multiply the 1000 by five and multiply the other thing by five. This is an example of what we call the distributive law.

It is crucial to choose the same value of x on both sides of the equation; otherwise you get nonsense. This is one of the most fundamental rules of algebra. It is so fundamental that it is often left unstated, but don’t let that fool you.

Don’t change horses in mid-stream.
Don’t change the meaning of x in mid-calculation.

So long as you choose the same value of x on both sides of the equation, you can use any x-value you like. Equation 29d applies for all x. It applies for each and every x that you care to choose.

You are allowed to have more than one horse, so long as you keep track of which is which.

5  More about Algebra

5.1  Algebra Enriches Geometry

The roots of geometry can be traced back more than 3000 years. The roots of algebra can be traced back even farther. For most of that time, until about 300 years ago, they were separate. However, algebra plus geometry together is more interesting than either of them separately. Basic geometry plus algebra gives you trigonometry. More generally, geometry plus algebra gives you the even larger field known as analytic geometry.

Here’s an example of a real-world problem: Suppose you want to make an octagon by cutting the corners off a square piece of wood. Algebra helps you figure out how much to cut off. See section 4.12 for details on this.

We could figure out how to make an octagon using purely geometrical methods, without using equations or even numbers. However, the algebraic solution is so straightforward that it’s hardly worth looking for a non-algebraic solution. More importantly, the algebraic approach generalizes to other situations where classical geometric methods are guaranteed to fail.

Similarly, there are lots of ways of proving the Pythagorean theorem using purely geometrical methods, without using algebra or even numbers. However, algebra can be used to simplify the proof, as discussed in section 10. Analytic geometry opens up yet more proofs – and yet more applications – of the theorem.

5.2  Algebra Enriches Logic

The history of formal logic can be traced back thousands of years. For most of that time, it was separate from algebra. However, the combination of algebra and logic is more interesting than either one separately. For example, consider the following syllogism. It uses the language of algebra to express one of the fundamental ideas of formal logic:

If fact A proves fact B, and fact B proves fact C, then A is sufficient to prove C.

Technology depends on this, broadly and deeply. Computers are based on Boolean logic ... which is also known as Boolean algebra.

5.3  Algebra Can Express Generalizations

Suppose you see a sign that says “Speed Limit 40 MPH”. That tells you a great many things. Among other things, it tells you that 41 MPH is illegal, 42 MPH is illegal, 43.333 MHP is illegal, et cetera. It would be ridiculously impractical to write down a list of all the forbidden speeds, one by one. Instead you would really rather have a rule. We can express the rule in the language of algebra:

For any speed S, 
If S is greater than 40 MPH, 
then S is illegal. 

In general, in the real world, sometimes you want specific numerical values ... but sometimes you’d much rather have a general rule.

Here’s another argument that leads to a similar conclusion:

Figure 14 shows a box wrench. It works very well for a particular size of nut or bolt. However, is doesn’t work at all if the size is different by any significant amount.   Figure 15 shows an adjustable wrench. It can be adjusted to fit a wide range of differently-sized nuts or bolts. However, it is much bulkier and heavier than a comparably-strong box wrench.

box-wrench   adjustable-wrench
Figure 14: Box Wrench   Figure 15: Adjustable Wrench

Once again, the moral of the story is: Sometimes you want something that applies to a specific case ... but sometimes you want something that can be adjusted to cover a wide range of cases.

We can apply the same logic to mathematics. The variable x in equation 29d and the variable S in equation 30 correspond to the worm gear in the adjustable wrench: They allow the equation to be adjusted to cover a wide range of examples.

This idea gets used over and over again, to express all sorts of mathematical principles. Let’s consider a few more examples:

In first grade or maybe in kindergarten, you learned that 2+7 is equal to 7+2. It is also true that 3+7 is equal to 7+3. There are infinitely many examples of this kind.

It would be absurd to try to learn all the examples one by one. The sensible approach is to learn the general rule.

Note that the word “commutative” comes from the same Latin root as the word for “commuting” to and from work. The core meaning is “back and forth”. When we write that X+Y equals Y+X, it means that the addition can be done left-to-right or right-to-left.

Here are yet more examples of rules that can be adjusted to cover a huge number of examples:

For all real numbers X and Y, we have X·Y = Y·X. In other words, multiplication is commutative (when applied to real numbers). For example, 3·7 = 7·3.

Beware that most things in this world are not commutative. Putting on your shoes does not commute with putting on your socks.

Even multiplication is not necessarily commutative. U×V is not generally equal to V×U if U and V are vectors or matrices.

For all real numbers X, Y, and Z, we have

X·(Y+Z) = (X·Y + X·Z)  

In other words, multiplication distributes over addition. For example, 2·(3 + 7) = 2·3 + 2·7. In more detail:

2·(3 + 7) = 2·10   doing the addition first  
  = 20                  
2·(3 + 7) = 2·3 + 2·7   using the distributive law first 
  = 6 + 14              
  = 20

The distributive law (equation 31) is not primarily a statement about the numbers X, Y, and Z. Rather it is a statement about the multiplication operator, the addition operator, and the relationship between them. This is discussed in more detail in section 4.5.

Talking about operators involves some abstraction. It is not, however, a very tricky kind of abstraction. Young children are good at using abstraction, generalization, and symbolism in this way; they do it routinely. Even a toddler playing with a doll is using a great deal of symbolism and abstraction; everybody knows that the doll is not a real baby; it is just a symbol representing a baby.

One reason for studying algebra is to learn more systematic ways of using symbolism, abstraction, and generalization.

5.4  Algebra Can Express Dimensions and Units of Measurement

Let’s continue the discussion of dimensions and units that began with example 1-2, example 1-3, example 1-4, and example 1-5. Here’s another example in the same vein:

Suppose you cover five acres of land with water, to a depth of 0.5 feet. That is 2.5 acre·feet of water. (This is sometimes written as 2.5 acre−feet, but it is safer to write it as acre·feet, with a dot rather than a hyphen, to remind everyone that we are multiplying, not subtracting.)

Obviously, multiplying 5 acres by 0.5 feet requires multiplying 5 by 0.5 ... but it also requires multiplying acres by feet. Units (such as acres and feet) are known quantities, but the rules for multiplying known quantities are exactly the same as the rules for multiplying unknown quantities such as X and Y.

In this way, algebra gives you systematic methods for converting acre·feet to cubic feet, and then converting cubic feet to liters, and so forth, to obtain whatever units of measurement you like. It also tells you that cubic feet are dramatically different from square feet, which is something worth knowing.

The general topic of how to keep track of dimensions and units of measurement is called Quantity Calculus. It might have made more sense to call it Unit Algebra or something like that, but the experts tend to call it Quantity Calculus. A highly condensed overview of the subject can be found in reference 12.

It is entirely possible to measure something using no units at all. On more than a few occasions I have been miles away from the nearest ruler, so I recorded in my notebook that something was |——| long. That’s an analog measurement.

Physical quantities exist whether you measure them or not. In particular, they exist independent of whatever units (if any!) you use to measure them. In figure 1, the length of the hallway is the same, no matter whether you measure it in meters, yards, feet, cubits, or whatever. In particular, the length of the hallway is not «L feet» or «L yards» or anything like that; the length is simply L.

It is important to distinguish the dimensional quantity L from the dimensionless ratio L/ft. Sometimes you want one or the other, depending on circumstances.

Sometimes the penalty for getting the units wrong is on the order of three hundred million dollars, as in the case of the Mars Climate Orbiter (reference 13 and reference 14).

Figure 16: Mars Climate Orbiter mission logo

Note that most calculators and old-school computer languages can represent dimensionless numbers but do not automatically keep track of the units. This creates all sorts of problems and risks. However, with a modest amount of manual labor, it is possible to keep track of the units, even under adverse circumstances, as follows:

Constructive suggestion: When using an old-school computer language, we can use variable names of the form L__ft and W__yd, where the convention is that the double underscore means “measured in units of” and also “divided by”. (Let’s be sure to document this convention.) This allows us to write things like the following. The first line makes use of the fact that an inch is officially defined to be 2.54 centimeters:

        in__m = 0.0254;         /* exactly, by definition */
        ft__m = 12 * in__m;     /* definition of foot */
        yd__m =  3 * ft__m;     /* definition of yard */
        L__m  = 10 * yd__m;     /* length of hallway */
        L__ft = L__m / ft__m;   /* length of hallway, in feet */
Example 1: Units, Using Old-School Computer Language

Another possibility is to use a computer algebra system. That means that instead of the code in example 1, we can write code like the following:

        L : 3 * yd;     /* length of hallway */
        yd : 3 * ft;
        ev(L);          /* result should be:  9 ft */
Example 2: Units, Using Computer-Algebra System

As far as the computer-algebra system is concerned, yd and ft are algebraic abstractions, with no numerical value.

Mutations: Non-experts should skip this section. It discusses how things should not be done. I almost hate to mention this, because discussing misconceptions is as likely to spread them as to dispel them.

It is an all-too-common blunder to write something like

    y = 10   (the «yardage»)        (33a)
    f = 30   (the «footage»)        (33b)
    f = y         (33c)

The contrast between 1 yd = 3 ft and f = 3 y could not be more extreme. That is the contrast between equation 3 and equation 33c.

There are several things wrong with equation 33. The whole approach is misguided. One problem is that if the length of the hallway is y = 10, what is the width? Another problem is that units should be on the RHS of the equation, or in the denominator of the LHS, whereas the expression y = 10 seems to imply that the units are in the numerator on the LHS – which is crazy backwards.

As previously mentioned, you can solve both of those problems by writing things like L/yd = 10 and W/ft = 6. This allows us to think clearly about the distinction between the dimensional quantity L and the dimensionless ratio L/yd.

5.5  Algebra Reduces the Amount of Stuff You Must Learn

The term “equation hunting” usually refers to a bad habit that students sometimes pick up. For any given problem, they run down the list of equations that they know until they find one that seems to fit. They use this to solve the end-of-chapter problems in the textbook. The only reason it appears to work is that there are relatively few equations in the chapter, and all the end-of-chapter problems can be solved using those few equations.

In contrast, this trick is not nearly so useful in the real world, because the number of equations that you would have to consider is ridiculously large.

Instead, for most purposes, the recommended procedure is to learn a relatively small number of equations ... plus the rules of algebra. The remembered equations rarely fit the given problem directly, but can be transformed by algebraic means into something that does fit.

There is a real-world version of equation-hunting that actually works, although it is very inefficient. Sometimes it is possible to guess the exact form of the desired equation. Somewhat more often, it is possible to guess that the desired equation belongs to a certain family, and then systematically find which member of the family does the job. In all cases the rule is that it’s OK to guess, provided you check and confirm that the guess actually works. It’s not guess-and-hope, it’s guess-and-check.

For example, Galileo equation-hunted the equation of motion for a freely-falling object. He did not derive it. There was nothing he could have derived it from. He conducted a long series of meticulous experiments to confirm that his formula was correct, and that the previous “conventional wisdom” was wrong.

Similarly, Newton equation-hunted the law of universal gravitation. He did not derive it. There was nothing he could have derived it from. He checked that it was consistent with Kepler’s laws, which in turn were consistent with Tycho’s meticulous observations.

Similarly, Planck equation-hunted the first quantum mechanical formula, the black-body spectrum. He did not derive it. There was nothing he could have derived it from. He checked that it fit the facts.

It must be emphasized that real-world equation hunting is very much harder that end-of-chapter equation hunting. The number of possibilities is very much larger. The required amount of subject-matter expertise is very much larger. It might take years to hunt up the desired equation.

Bottom line: Equation-hunting is a tool. Like any other tool, it should not be over-valued or under-valued.

5.5.1  Density Formulas

In physics and chemistry, the density (ρ) is defined to be mass (m) per unit volume (V):

ρ = m / V    for all ρ, m, and V

Using the laws of algebra, we can rearrange things in various ways. For all ρ, m, and V we have:

ρ = m / V     (35a)
V = m / ρ     (35b)
m = ρ V     (35c)

The point here is that a person who understands algebra sees equation 35a, equation 35b, and equation 35c as all the same. In contrast, a person who doesn’t understand algebra sees equation 35a, equation 35b, and equation 35c as three different equations, and must learn each of them separately. This is three times as much work. It also means there are three times as many things that could go wrong.

5.5.2  Gas Laws

Here’s another example: In an ideal gas, there is a relationship involving four variables: the pressure (P), volume (V), number of molecules (N), and temperature (T). There is also a constant involved, namely Boltzmann’s constant (k).

P V = N kT    (ideal gas law)

There are numerous possible rearrangements and corollaries to this law. One of the corollaries is called Boyle’s law, but I don’t know which one. Another of the corollaries is called Charles’s law, but I don’t know which one. Some of the other corollaries might have names, but I don’t even know the names. I don’t need to know any of that stuff, because I know equation 36, and I can rederive the corollaries whenever needed, in less time that it takes to tell about it, using simple algebra.

Each of the corollaries is predicated on certain assumptions, and the assumptions are different in each case. So not only do you need to memorize the equation for each corollary, you need to memorize the assumptions. The number of hard-to-learn and easy-to-forget details is astronomical.   The general law (equation 36) easier to learn, harder to forget, and more reliable ... not to mention more powerful.

5.5.3  Electronics Laws

This brings to mind a morbidly amusing story, as recounted by Joseph Bellina:

After graduating from college and ROTC, this fellow chose to go to the Army electronics school. As a pretest he was asked what are the three most important laws of electronics. Well he thought about that a while and chose j=sigma*rho, and Kirchoff’s two laws. As it happened what they expected was V=IR, I = V/R and R = V/I.

The point here is that if you know a little bit of algebra, you see Ohm’s law as one fundamental law, but if you don’t, you have to learn it as three separate not-so-fundamental laws. Actually it’s even worse than that, exponentially worse, as we now explain.

Let’s start from the beginning. In electronics, there is a relationship involving the voltage (V), the the current (I) and the resistance (R):

V = I·R    (Ohm’s law)

Using the laws of algebra, there are three ways of rearranging this:

V = I·R                       
I = V / R                           
R = V / I                           

So you have a choice: You can either remember three things (equation 38) or you can remember just one thing (equation 37) and use algebra to derive the others whenever needed.

If that were the end of the story, the choice wouldn’t matter much. Learning three things is not very much harder than learning one thing.

However, that’s not the end of the story. There is also an equation for the power:

P = I·V    (Joule heating law)

Using the laws of algebra, there are 12 different ways of combining equation 39 with equation 37 and rearranging things. Would you rather learn 12 equations, or just 2 equations?

Note the trend here: The number of variables went up modestly, from 3 to 4. The number of basic concepts went up modestly, from 1 to 2. The number of derived equations went up explosively, from 3 to 12.

Let’s take this one more step: We introduce the notion of conductance. It’s an exceedingly simple concept:

G := 1/R

This allows us to write things like I = G·V, which makes at least as much sense as Ohm’s law in its original form. Now the number of variables goes up from 4 to 5, and the number of basic concepts goes up from 2 to 3. The number of equations continues its explosive growth: It goes up from 12 to 24. Would you rather learn 24 equations, or just 3 equations?

If you count all the rearrangements, there are a huge number equations. You could try to learn them by rote, but I don’t recommend it. Real-world professional electronic engineers don’t know them all by heart. The details are so gory that they are not shown here. You can look at section 13.2 if you dare.  

They actually sell posters for the benefit of people who don’t understand algebra, to help them learn by rote all 12 possible rearrangements of equation 37 in combination with equation 39. Such a poster is shown in figure 17.   If you understand algebra, you don’t need a poster covered in equations. As soon as you learn the basic concepts, you get all the rearrangements for free.

At some point it becomes easier to just learn algebra than to do everything using brute force and rote memory.

Figure 17: Poster for Learning Electronics Equations by Rote

5.5.4  Thermodynamics

In thermodynamics, the description of even a rather simple system might involve a dozen variables and more than a dozen equations. That gives rise to thousands of permutations and combinations – far more than anyone could remember.

5.6  Algebra Can Find Solutions to Equations

Whenever you mention algebra, people think of methods for solving equations. That is, sometimes you will know an equation for X before you know the exact value of X, and then in a later step you solve the equation to find X.

It must be emphasized that solving for the value of a variable is not the only thing algebra can do. This is an important part of algebra, but definitely not the only part. In particular, none of the examples in section 5.3 involve solving for X. The power of those examples comes from the fact that the equation holds for any and all X.

Furthermore, there are lots of situations where you are looking for a solution, but it cannot be found using algebra alone. Sometimes fancier techniques are needed, such as differential equations.

Solving equations has an enormous range of applications. For example:

Suppose you know the stopping distance for a car at 50 miles per hour. You also know the stopping distance at 10 miles per hour. You would like to know the stopping distance at various other speeds. The brute-force solution would be to measure the stopping distance at every possible speed, one by one, and tabulate the results.

The cleverer approach would be to use physics (including algebra) to build a mathematical model. This allows you to interpolate, so you know the stopping distance even at speeds that you didn’t explicitly measure. What’s even better is that subject to mild restrictions, you can extrapolate the model to speeds that you simply could not measure, perhaps because of speed-limit laws, or because of the car’s performance limits, or because they involve weather conditions that you have not yet experienced, or whatever.

If we want to account for other variables that could increase the stopping distance, such as a downhill slope or a tailwind, the brute-force approach becomes even more impractical, and the advantages of the mathematical approach become even more apparent.

Lives depend on getting this right. Note that in all likelihood, the rule of thumb that they taught you in driver’s-ed class is not reliable; it provides excessive margin under some condition and not enough margin under other conditions. Knowing a little bit of algebra allows you to figure this out.

As a more dramatic example, suppose you are in command of a boat or an aircraft. You would be well advised to calculate the amount of fuel required for the journey. You would be wise to calculate it before departure, because you can’t get more fuel in the middle of the ocean. At the very least, you have to calculate it before you reach the point of no return. There are plenty of situations like this, where making a direct measurement is not a viable option.

Farming is tens of thousands of years older than algebra, but modern farming is intensely analytical. Here’s a simplified example: You can use a small test plot to experiment with the amount of water, and another to experiment with the amount of fertilizer, and another to experiment with pesticides, et cetera. Then you need to interpolate and extrapolate all the variables, in order to apply the results to your large main fields. There are so many variables that you could not possibly find the right answer by brute-force experimentation.

Suppose you have a talent for making exceptionally good cookies. You decide to go into business. The question is, how much should you charge for your cookies? You know how much the ingredients cost. You know how much your competitors are charging, but that doesn’t answer the question, because their costs are different and their product isn’t as good. A small-scale experiment tells you that if the price is too high, the sales-volume goes down and you lose money. On the other hand, if the price is too low you lose even more money.

If you are lucky, you may be able to use brute-force trial-and-error methods to find a price that allows you to stay in business ... but you will do better if you use algebra to analyze the data and find the optimal price-point.

A modern high-efficiency outfit such as Walmart makes decisions based on fantastically complex mathematical models.

6  More about Functions

6.1  Tabular and Graphical Approaches

The topic of this section is functions.

As mentioned in section 2.5, a great deal of mathematics (especially higher mathematics) is devoted to patterns, relationships, and generalizations. A function is a particular type of relationship. Functions can be represented in many ways, including graphs, tables, and algebraic expressions.

As a simple example, let’s take a look at table 7. This is what we call a lookup table. The first column (Tc) is the temperature in degrees Celsius. The second column (Tf) is the temperature in degrees Fahrenheit. For the moment, let’s treat the third column as a mere comment and ignore it.

0      32      water: freezing point
5      41     
10      50     
15      59     
20      68     
25      77     
30      86     
35      95     
37      98.6      body temperature
40      104     
45      113     
50      122     
55      131     
60      140     
65      149     
70      158     
75      167     
80      176     
85      185     
90      194     
95      203     
100      212      water: boiling point
Table 7: Temperature Conversion Lookup Table

Given a lookup table such as this, you can convert a temperature reading from one scale to the other. For example, if the temperature is represented as 10 C, you can find that the corresponding representation is 50 F. This is the third row of the table.

This data can also be represented as a graph:

Figure 18: Temperature Conversion Graph

For some types of data, a table is the best representation. For other types, a graphical representation might be helpful. However, for temperature conversion, neither of these is optimal. The problem is, there are lots of different temperatures in the world, and no table can include all of them in any reasonable way. For example:

To solve the problem, you could interpolate and extrapolate. There are various ways of doing this.

If you want to use the function in table 7 in the medical clinic, you should plot the function, but the whole thing as shown in figure 18. You need a more zoomed-in version, such as shown in figure 19.

Figure 19: Zoomed-In Temperature Conversion Graph

You can construct such a graph by hand, as follows: Take a piece of graph paper. Select a suitable region, and label the grid-lines more-or-less as shown in figure 19. Plot two of the points from table 7, namely the points at (35, 95) and (40, 104). Then draw a straight line connecting them and extending beyond them a little ways in both directions. You don’t absolutely need the intermediate point at (37,98.6), but it is a good idea to plot it anyway, as a check. Remember the rule: Check the work.

This figure can be used for interpolation of clinically-relevant temperatures. For the example of 38 C, find the contour labeled 38 C, and follow it. This is a contour of constant Tc. It contour runs vertically in the figure, and is shown by the magenta dotted line. Follow it this contour until you come to the line that represents the temperature-conversion function. Then follow along a contour of constant Tf. This runs horizontally, and is shown by a red dotted line in the figure. Follow it until you run into a label. You can see that it is a little less than halfway between 100 F and 101F. In fact it is exactly 100.4 F, as we can confirm using algebraic methods as discussed in section 6.2.

The spreadsheet that produces these figures is cited in reference 15.

Beware that extrapolation is always riskier than interpolation.

Being able to construct and interpret graphs is an exceedingly valuable skill. Math gets a lot more interesting and a lot more useful as soon as you move beyond arithmetic. It is very hard to see the significance of a pile of numbers just by looking at the numbers. Doing more and more arithmetic with the numbers is not going to help. Graphic the numbers helps a lot.

6.2  Algebraic Approaches

At some point it becomes easier to ignore the table and calculate the conversion from scratch, using an algebraic formula. The formula for converting Celsius to Fahrenheit is nice and simple:

Tf = Tc × 1.8 + 32

That’s an equation. It says the left-hand-side (LHS) and the right-hand-side (RHS) are equal, which is true.

However, there is something more going on here, which we can write as follows:

Tf  Tc × 1.8 + 32

The arrow in recipe 42 means the LHS is calculated from the RHS. This is an algebraic rule, a machine if you will. Given a Tc value, this machine performs some mathematical manipulations and spits out a Tf value.

Figure 20: A Machine with Inputs and Outputs

Note the following contrast:

When writing a lookup table, it is more-or-less traditional (but certainly not necessary) to write the input in the left column and the output in the right column, so that the table can be read left-to-right.   When writing instructions for calculating something, there is a very strong tradition of writing the output on the left and the expression that involves the inputs on the right. You can see this in recipe 42 and also in figure 20. This may seem backwards to you, but there is no point in fighting it.

Note that the entries in a lookup table do not need to be evenly spaced. You can see this in table 7: There are unevenly-spaced entries near 37 C.

6.3  More Examples

Indeed, the entries in a lookup table do not need to be sorted numerically, or even sortable. Indeed, they do not even need to be numerical. For example, the data in table 3 is non-numerical.

Table 8 shows another example of a function.

−5      25
−4      16
−3      9
−2      4
−1      1
0      0
1      1
2      4
3      9
4      16
5      25
Table 8: Square Numbers

Here is the algebraic form of this function:

s  r2

Figure 21 shows the corresponding graph.

Figure 21: Squares

6.4  Inverse Functions

Table 7 can be used in either direction. So far we have treated the first column as the input and the second column as the output, but you can perfectly well use the table in the other direction. For example, if the temperature is represented as 50 F, you can find that the corresponding representation in 10 C. This gives us a new machine, a new function. We can write it algebraically as:

Tc  (Tf − 32) / 1.8

The function in recipe 44 is called the inverse of the function in recipe 42.

We can convert recipe 44 to an equation:

Tc = (Tf − 32) / 1.8

Note the contrast:

Equation 45 means exactly the same thing as equation 41. If one of them is true the other must be true.   Recipe 44 is not the same as Recipe 42. One recipe says to use Tc to calculate Tf, while the other says to use Tf to calculate Tc.

An equation states that the LHS is equal to the RHS and vice versa; it’s all very symmetrical.   In a function, the input is conceptually different from the output. There’s nothing symmetrical about it.

Interestingly enough, it is not quite so easy to form the inverse of the function in table 3. That’s because for any given color, there are multiple kinds of fruit with that color.

For the same reason, it is not entirely simple to form the inverse of the function in table 8 aka figure 21 aka equation 43.

Mathematicians are quite strict about this: For any given input-value, a function has to produce the same output-value every time. A machine that doesn’t obey this rule is not a function.

There are lots of things in this world that aren’t functions. For instance, a clock gives you a different answer every time you look at it. It’s a perfectly good clock, but it’s not a function.

Sometimes when you have a relationship that is not a function, you can turn it into a function by gathering things into sets. Table 9 is a machine that takes in a color and produces a set of fruits that have that color. This is not exactly the inverse of the function in table 3, but it is a perfectly fine function unto itself.

color    set of fruit
yellow    {banana, lemon}
red    {cherry, McIntosh}
green    {lime, Granny Smith}
Table 9: Fruit as a Function of Color

Similarly table 10 is a machine that takes in a s-value and produces a set of r-values that are consistent with that s value, and consistent with the requirement that s = r2. This is not exactly the inverse of the function in table 8, but it is a perfectly respectable function unto itself.

set of r
0      {0}
1      {1, −1}
4      {2, −2}
9      {3, −3}
16      {4, −4}
25      {5, −5}
Table 10: Square Roots

Figure 22 shows the corresponding graph.

Figure 22: Square Roots

It must be emphasized that even though there is an inverse function for the temperature conversion function in table 7, there is no inverse function for the fruit/color conversion in table 3. The inverse function would completely undo the effect of the fruit/color conversion machine, but this is simply not possible. The output of the fruit/color conversion machine contains less information than its input.

Similarly there is no inverse function for the squaring function in table 8. The inverse function would completely undo the effect of the squaring machine, but this is simply not possible. The output of the squaring machine contains less information than its input.

Sometimes it is satisfactory to have a function that spits out a set of numbers (or a set of fruit), but sometimes not. If you are building a machine for use in grocery stores that figures out the type of fruit, it won’t be very useful if it can’t tell the difference between a cherry and a McIntosh. You need a more complicated function, with more inputs. The color information is still useful as part of the overall solution, but it is not a complete solution unto itself.

A set of ordered pairs is called a mapping. Every function is a mapping, but not conversely. A function is required to have a unique output for any given input, but a mapping has no such restriction. Every mapping has an inverse mapping, but not every function has an inverse function. The inverse function is required to completely undo the effects of the original function, but an inverse mapping has no such requirement.

Figure 21 is a function that converts one number to another. It is also a mapping.

Considered as mappings, figure 22 is the inverse of figure 21. Considered as functions that convert one number to another, figure 22 is not a function at all.

6.5  Graphing Inverse Functions

In item 4-5 we encountered the following graphs:

sq-edge-area   sq-area-edge
Figure 23: Area versus EdgeLength   Figure 24: EdgeLength versus Area

It must be emphasized that these two figures convey exactly the same information. If you prefer one over the other, that is mostly a matter of personal taste.

Anything you can do moving horizontally in one figure you can do moving vertically in the other, and vice versa. In fact, you draw figure 23 on a transparent piece of plastic, you don’t need to draw figure 24 at all; you can just flip figure 6 over and look at its back side. You can flip it left-to-right or top-to-bottom and then rotate it into position. You don’t even need the rotation step if you flip it around a 45 diagonal, as shown in figure 25.

Figure 25: Inverted View of Area versus EdgeLength

Equivalently, you can produce the image of figure 25 by viewing figure 6 in a mirror. The labels are hard to read because they are mirror-inverted, but the data itself is plotted correctly.

You can even combine the plot with its mirror image to create symmetrical “butterfly” diagrams. An example is shown in figure 26. There is something fundamentally hokey about this example, because we have two different coordinates (both length and area) in each direction. In real life it is rare to find a mapping where its range is more-or-less the same as its domain ... but it does happen. In such a case, the flipped diagram can be thought of as a representation of the inverse mapping.

Figure 26: Combined View of Area versus EdgeLength
However, this is not necessarily the best way to think about inverses. When the range is not similar to the domain, as in the fruit/color function in table 3, there is no simple geometrical symmetry, and otherwise just causes confusion.

Figure 27 is perhaps a better way of visualizing the symmetry between a function and its inverse. Enter one of the plots and move vertically along the dashed line that represents Area=7. When you come to the curve, move horizontally along the dashed line that represents EdgeLength=√7. Carry this across to the other plot, and keep moving horizontally. When you get to the curve, move vertically along the dashed line that represents EdgeLength=7. The overall result is a graphical computation of (√7)2.

Figure 27: A Function and its Inverse

In figure 27 you may well ask, which plot represents the function and which represents the inverse? Answer: Whichever you choose. The point is, on one plot you enter vertically and read off the answer horizontally, while on the other plot you enter horizontally and read off the answer vertically. Each function is the inverse of the other.

Figure 28 is perhaps an even better way to visualize the relationship between a function and its inverse. This version will appeal to those who like the input of a function to run horizontally and the output to run vertically.

Figure 28: A Function and its Flipped Inverse

Enter the lower-right diagram and run vertically along the dashed line that represents Area=7. When you get to the curve representing the function, pivot and run horizontally along the dashed line that represents EdgeLength=√7.

Continue this onto the “reflector” panel. Pivot and vertically run along the dashed line that – still – represents EdgeLength=√7. When you get to the curve representing the function, pivot and run horizontally along the dashed line that represents Area=7. The overall result is another graphical computation of (√7)2.

The “reflector panel” plays an important role here. It represents the identity function, in a way that lines up the output of one function with the input of the next. I think it helps to show it explicitly.

The “reflector panel” trick can be used to diagram a graphical computation of the composition of any functions (not just inverses), so long as the range of one matches the domain of the next. Lay them down like dominoes.

Sometimes the range and domain are the same. For example, a permutation is guaranteed to have a range identical to its domain. Whenever the range of some function f is a subset of the domain, we can write equations of the form

p = f (f (f (f (q))))

which we call an iterated mapping. This includes the case where the range is an improper subset of the domain, i.e. the whole thing. In the opposite case, where the range is systematically smaller than the domain, we get what is called a contractive mapping.

The spreadsheet that produces these figures is cited in reference 15.

6.6  Equations versus Recipes, Causation, etc.

Note the contrast:

In some sense, there is a profound distinction between an equation and a recipe. An equation is symmetrical, in that the LHS is equal to the RHS and vice versa.   In another sense, the distinction between an equation and a recipe does not matter much, because any recipe can be converted into an equation, and a wide class of equations can be converted into recipes. Many of the same algebraic operations that can be applied to equations can be applied to recipes.

A lot of people, including experts, tend to gloss over the distinction between equations and recipes. An assignment statement in an imperative computer language such as C++ is written with an equals sign, even though it logically should be written with an arrow or with a “:=” symbol. It “looks like” an equation, and is sometimes even called an equation, even though it really isn’t. There is no symmetry between the LHS and RHS of an assignment statement.

For more about symmetry (or the lack thereof) as applied to equations, assignments, and cause-and-effect relationships, see reference 16.

6.7  Conventional Directions and Symbols

As mentioned in reference 10, introductory textbooks tend to fall into the following habits:

  Supposedly, x is always the input and y is always the output.
  Supposedly, on a graph, x is always horizontal and y is always vertical
  Supposedly, on a graph, the input is always horizontal and the output is always vertical.

In mathematics, none of those things is actually required. You should break those habits. (Some spreadsheet apps force on you the idea that the horizontal direction is called x and the vertical direction is called y, but it is still a bad idea.)

For example, you can use figure 18 to convert Tf to Tc just as easily as vice versa. You don’t even need to redraw the graph; just start with a Tf value, move horizontally until you come to the curve, and then move vertically to find the label for the corresponding Tc value.

7  Some Fancier Applications

The goal is to understand some useful applications. We start by describing the applications. We then work backwards, developing the techniques necessary to solve the problem. We then go over everything again tidying up loose ends. This is an example of the spiral approach to learning and problem-solving. It has advantages in terms of motivation as well as realism, as discussed in reference 17.

7.1  Saggy Suspension Bridge (Setup)

Suspension bridges are important in the real world. People have been building them for thousands of years. Unfortunately, if you build them in the most obvious naïve way, they either sag or break.

Not too long ago, the Mythbusters discovered this the hard way. They built a suspension bridge. They discovered that it sagged, as shown in figure 29. The desired result is shown by the horizontal dotted black line, while the actual result is shown by the saggy solid red line. They could reduce the amount of sag by increasing the tension, but too much tension would cause the thing to break. It was a no-win situation: either too much sag or too much tension, or both.

Figure 29: Crash Test Dummy Crossing a Bridge

Just to be silly, they built the bridge out of duct tape, but that is irrelevant to our story. Any other material would have had the same problem. This problem has been around for thousands of years.

The engineering problem can be understood in terms of physics, which can be understood in terms of geometry, which can be understood in terms of algebra.

engineering → physics → geometry → algebra              (47)

The basic physics idea is mechanical advantage. Leverage is a familiar example of mechanical advantage. Screws and wedges also provide mechanical advantage. The load on the bridge has mechanical advantage against the rope. To say the same thing the other way, the rope suffers from a large amount of mechanical disadvantage. That’s because a small amount of stretch produces a disproportionately large amount of sag.

Using the naïve design shown in figure 29 there is no way to solve this problem. In the real world, suspension bridges are built with tall towers supporting catenary cables. If you want a deck that doesn’t sag, you can support the deck using vertical suspenders that come down from the catenaries, as shown in figure 30. Physics demands that you must let the catenaries sag! Something has to sag, because otherwise the load has an infinite mechanical advantage.

Figure 30: Verrazano Narrows Bridge

In accordance with equation 47, before we can understand the physics or the engineering, we need to understand the geometry of the situation.

Rather than attack the engineering problem head-on, let’s start by doing an easier problem that has the same geometry as figure 29. Algebra helps with this, as we shall see in section 7.2.

There is an important strategic principle here: Sometimes when faced with a hard problem it pays to work on an easier problem first. This can be considered a form of reconnaissance.

It pays to do warm-up exercises.

We will finish the saggy-bridge problem in section 7.3. First, though, let’s work the navigation problem in section 7.2. It has the same geometry as the bridge problem, but with fewer distractions.

7.2  As the Crow Flies

7.2.1  Initial Rough Analysis

Suppose you are traveling from point A to point B in figure 31. In particular, suppose you are driving an all-terrain vehicle in flat, open country, so you are not obliged to follow roads. The same logic applies to airplanes and to birds, which are not confined to roads. In all cases we keep things simple by ignoring headwinds and crosswinds.

Figure 31: Distance as the Crow Flies

One option is to travel from A to C, make a 90 left turn, and then proceed from C to B. The total distance for this route is 49 miles.

Another option is to proceed “as the crow flies” from A directly to B, as shown in red in the diagram. Everybody knows that a straight line is the shortest distance between two points, but let’s see if we can figure out how much shorter it is.

This is a legitimate problem unto itself, although not ultra-important. The real significance comes from the fact that the techniques used here are necessary for solving harder and more important problems, such as suspension bridge problem in section 7.1.

At this stage we don’t have a numerical value for the distance, so we call it X. You could just construct a mechanical model or a careful scale diagram and measure X. Or you could travel the route both ways and keep careful records. These are clumsy ways of discovering X, but they work. If you do that, you find that X is approximately 41 miles. People speak of “cutting corners”, and this is why. The short-cut route saves a substantial amount of distance: 41 miles is almost 20% shorter than 49 miles.

If you were in the delivery business, and your costs were 20% higher than the competition, your business would fail very soon.

7.2.2  Mathematical Analysis

So, now that we are properly motivated, let’s see if we can find a more-mathematical, less-clumsy way of finding the value of X. There are lots of scenarios in which the mathematical approach works better than the clumsy approach:

The mathematical approach uses the Pythagorean theorem (equation 23). It tells us that:

X2 = 402 + 92

You could work this out on a calculator, but just to prove a point let me show how you could solve this problem in your head.

Let’s start with the numbers on the right-hand-side (RHS): 40 times 40 is 1600. You should be able to do that one in your head, starting from the fact that 4 times 4 is 16. Similarly you know that 9×9=81. So the number on the RHS of equation 48 is 1681.

So, if X squared is equal to 1681, then X itself is equal to the square root of 1681, which we write as √1681. That is

X2 = 402 + 92           
  = 1681                 
X = 

Now, you could use your calculator to find the square root of 1681, but again, you could also do it in your head. There are actually several ways of doing it.

7.2.3  Taking the Square Root; Using the Binomial Theorem

One way of doing it is a two-step process. The first step is to guess a value for X, and the second step is to prove that the guess is correct. Since we have already guessed that X is approxmately 41, let’s check to see if X=41 is mathematically correct.

As is so often the case, you can check this result by working the problem in reverse. Just as you can check subtraction problems by adding, you can check square-root problems by squaring. That is, if we think X = √1681, we check it by showing that X·X = 1681.

It’s easy to multiply 41×41 in your head. Once again, algebra comes to the rescue. Here’s how: We can write 41 as 40 + 1. Using the language of algebra, we can rewrite that as (a + b) = 41, where a = 40 and b = 1. To repeat:

a = 40              
b = 1               
(a + b) = 41

The next step is easy if you know the following formula:

(a + b)2 = a2 + 2a·b + b2    (for all a and b)

Equation 51 is a special case of the binomial theorem. It is so widely useful that it is worth remembering. If you don’t remember it, you can always re-derive it using basic algebra. The derivation is shown in section 9.1.

Equation 51 is valid for any a and b whatsoever, which means (among other things) that we can apply it to the a and b values in equation 50.

a = 40       (52a)
b = 1       (52b)
(a + b) = 41       (52c)
(a + b)2 = a2 + 2a·b + b2    (for all a and b)     (52d)
 = 402 + 2·40·1 + 12    (in this example)     (52e)
 = 1600 + 80 + 1       (52f)
 = 1681       (52g)

The multiplications called for in equation 52e are super-easy because there is no carrying involved. Avoiding messy carries is a big part of the rationale for what we did, expanding 41 into two terms and applying the binomial theorem.

7.2.4  Taking the Square Root; First-Order Expansion

Here’s another scheme for finding the square root of a number. This scheme does not require guessing.

Here a simple yet powerful trick. It useful for solving this problem and thousands of similar problems. There is a rule of thumb that says if X goes up by one percent, X2 goes up by approximately two percent. This rule is worth remembering, but if you ever forget it, you can rederive it. The derivation and explanation can be found in section 9.2.

We can apply this rule to equation 49 as follows. We just got through calculating that 1600 is the square of 40, so we know that 40 is the square root of 1600. Next we notice that 1681 is about 5% bigger than 1600. In particular, 1616 would be about 1% bigger, 1632 would be 2% bigger, et cetera. Since 80 is 16×5, and 81 is very nearly 80, we know that 1681 is very nearly 5% bigger than 1600. Since X2 is about 5% bigger than 1600, X itself must be 2½% bigger than 40. You can do that one in your head, too, since 2.5% of 40 is just 1. So our estimate is that X=41.

Given an estimate like this, the first thing you should do is check it. As in section 7.2.3, you can check it by calculating X2 and comparing it to 1681.

7.2.5  Taking the Square Root; Newton’s Method

Suppose you didn’t know the trick for expanding things to first order (as discussed in section 7.2.4) – or suppose for some reason it wasn’t particularly convenient.

Here’s another trick, an incredibly powerful trick, invented by Sir Isaac himself.

Suppose we start with X2 and divide by X. The quotient is just X. That’s obvious.

Suppose we start with X2 and divide by something slightly smaller than X. The quotient will be slightly larger than X. What’s more, if the divisor is too small by a small percentage, the quotient will be too big by the same percentage, to a good approximation. Therefore if we split the difference, we obtain a very accurate value for X.

We can say the same thing more precisely using the language of algebra.

P = initial approximation, close to X            
Q = 
R = 
P + Q
  = better approximation                     

If necessary, you can repeat the process, dividing X2 by R and splitting the difference to get an even more accurate approximation. The sequence converges rapidly. The number of correct digits doubles each time. That is to say, if the initial approximation is good to 1%, one turn of the crank gives you an answer that is good to 0.01%.

Let’s apply this to the problem at hand. Suppose we don’t have a very good guess for X. We still have some sort of guess; in particular, we know that X has to be somewhere close to 40. You can divide 1681 by 40 in your head. Write it as (1600 + 80 + 1) and divide each term by 40. The answer is 42 to a good approximation. It’s (42 + 1/40th) exactly, but let’s not bother with the 1/40th for now.

If you split the difference between 40 and 42, you get 41. You should immediately check this. You will discover that it is in fact exact.

7.2.6  Discussion

This whole section has been something of a fugue, making some low-level points and high-level points at the same time. Let’s summarize:

Utilitarian Level   Intellectual Level

We have been discussing much distance you can save by taking a short-cut, traveling as the crow flies. That’s slightly interesting and slightly useful in practical terms.   Just calculating the difference is not the whole point, or even the main point. The more important lesson here is to see the mathematically-savvy way of looking at such problems.

If all we wanted was a utilitarian solution, we could just grind out the numbers on a calculator. There’s nothing wrong with that.   The mathematician looks at this problem and says, not only can we solve it, we can understand it.

If you solve lots of problems using a calculator, you get good at using the calculator. There’s nothing wrong with that.   If you solve the problem in your head, you get good at solving problems in your head. To make it solvable, you need to understand the structure of the problem, so you can rearrange and simplify the calculations. This means you get good at understanding the structure of problems. This is tremendously valuable.

If all you wanted was a solution, once you had a solution, you’d be done.   To the mathematician, the problem is interesting. Once you have a solution, you look for another solution, and then another.

We used about ten different tools to attack this one little problem. If all you wanted was one solution to one problem, ten tools would be overkill. The time spent learning the tools wouldn’t be worth it.   The same ten tools can be used to solve thousands upon thousands of problems, including big problems as well as little problems. Once you know the tools, you begin to see opportunities to use them. No one particular problem makes the tools worthwhile; the payoff comes from using the tools over and over again.

We have demonstrated how things work when a is equal to some number and b is equal to some other number.   The idea of turning a hard problem into a succession of easy problems by repeated use of the distributive law comes up again and again and again. It works just fine when applied to a tremendous variety of things, including rational numbers, complex numbers, vectors, matrices, or whatever – even if you don’t know what a and b stand for.

  Equation equation 57d works on a wide range of things. On the other hand, you have to be a little bit careful not to apply equation 57e to situations where the commutative law doesn’t apply (such as matrices and Clifford algebras).

Calculating the optimal route is a toy problem. Everybody knows that a straight line is the shortest distance between two points.   If there are obstacles, or if you are living in a curved space, it may not be obvious how to define “straight” line. In particular, the surface of the earth has intrinsic curvature. As a consequence, if you are using a map with a Mercator projection, a straight line on the map – i.e. a rhumb line – is not the shortest distance between two points. The tools exhibited here (including the binomial theorem, expansion to first order, and Newton’s method) can be brought to bear on real-world problems, such as the suspension bridge in section 7.1.

7.2.7  Designing the Problem

In case it wasn’t obvious, I chose artful values (40 and 9) for the perpendicular leg distances in section 7.2 to ensure that the diagonal distance would come out to be a round number. I had to do a few lines of algebra in order to know in advance that 40 and 9 would work out nicely. So this in itself is an example of real-world algebra: I had to do some real algebra just to set up the toy problem.

In particular, consider the case where we have a right triangle. We require all the side-lengths to be integers. We require the hypotenuse to be one unit longer than the base. We can use algebra to find all such triangles.

base = N     (54a)
hypotenuse = N+1     (54b)
altitude = m     (54c)
m2 = (N+1)2 − N2     (54d)
 = 2N + 1     (54e)
2N = m2 − 1     (54f)
 = (m+1)·(m−1)     (54g)
N = (m+1)·(m−1)/2     (54h)

Obviously 2N+1 is an odd integer, so equation 54e tells us this only works for odd values of m. Otherwise it works for any odd m larger than 1. Equation 54h is guaranteed to give us an integer value of N, not a half-integer, because (m−1) is guaranteed to be even.

Here are the first few triangles in this set:

altitude      base   hypotenuse
3     4     5
5     12     13
7     24     25
9     40     41
11     60     61
13     84     85
15     112     113
17     144     145
19     180     181
21     220     221

7.3  Saggy Suspension Bridge (Completion)

There is one tiny bit of physics we need to invoke in order to finish the problem. To compute the mechanical advantage, we need to know how the length of the rope changes as the amount of sag changes. This makes sense if you look at it as follows: Suppose you are using a lever, such as a crowbar or wrench. It doesn’t directly matter where the handle starts or ends up; the thing that directly factors into the mechanical advantage is how far the handle moves, i.e. the distance from start to finish.

So let’s calculate what happens during the last part of the sagging process, when the amount of sag goes from 8 feet to 9 feet. That’s a one-foot change. We calculate that the length of each half of the rope goes from 40.792 feet to 41 feet. That’s a 0.208 foot change in each half of the rope, or 0.416 foot total. Taking the ratio of the changes, we find that the mechanical advantage is about 2.4 to 1. Looking at the next foot of sag, going from nine feet to ten feet, the mechanical advantage is a little less.

Using calculus you could figure out that a more accurate answer is 2.22-to-1 (assuming a 9-foot sag). It’s just 40 divided by 9, divided by 2. However, that’s more than we really need to know. You can solve this problem without calculus. It’s more work and somewhat less exact, but it’s entirely doable.

If you tried to reduce the sag to 4 feet, the mechanical advantage would shoot up to five-to-one. This has some dramatic implications. Suppose you wanted the useful load to be a few hundred pounds. Multiply by the mechanical advantage, then multiply by some safety factor, and you find that the strength of the rope has to be many thousands of pounds.

To say the same thing another way, when the sag is 4 feet, the tension is about worse than you might have guessed. The argument goes like this: Given a certain amount of tension in an 80-foot rope, there will be a certain amount of stretch. However, at the middle of the bridge you are in effect being supported by two 40-foot ropes. Each one is shorter (so there will be less stretch, for any given amount of tension) and there are two of them (so there will be only half as much tension in each rope). That’s all true as far as it goes, but it would be a huge mistake to stop the analysis at that point. One must account for the fact that changing the altitude of a triangle changes the hypotenuse very little. All in all, rather than having a 4-to-1 advantage in favor of the rope, there is a 5-to-1 advantage in favor of the load.

A graph of the mechanical advantage is shown in figure 32.

Figure 32: Mechanical Advantage versus Sag

The raw numbers are shown in equation 56.

horiz     sag     diagonal     stretch     advantage
40     0     40.000        
          }  0.012     40.0
40     1     40.012        
          }  0.037     13.3
40     2     40.050        
          }  0.062     8.0
40     3     40.112        
          }  0.087     5.7
40     4     40.200        
          }  0.112     4.5
40     5     40.311        
          }  0.136     3.7
40     6     40.447        
          }  0.160     3.1
40     7     40.608        
          }  0.184     2.7
40     8     40.792        
          }  0.208     2.4
40     9     41.000        
          }  0.231     2.2
40     10     41.231        

8  Tools and Technques

8.1  Side Effects, or Not

Let’s take another look at the reasoning skills mentioned in section 2.2.

multi-step reasoning (section 8.2)    
higher reliability, stronger proof    
trusting certain tools    
avoiding certain booby traps    
skimming, reading, re-reading, and pondering a text (section 8.3)    
learning a new language    
generalization, symbolism, and abstraction    
imagination, creativity, artistry, and elegance    
Table 11: Tools and Techniques

None of these things is intrinsically mathematical. For example, computer programming requires just as much multi-step reasoning, attention to detail, skimming, pondering, tools, symbolism, linguistics, and creativity. On the other hand, traditionally, through most of history, mathematical training has been the primary means of acquiring these skills ... not universally, not necessarily, but typically. The converse is a stronger statement: won’t be able to handle higher math unless you learn these higher-order reasoning skills. So these skills can be considered side-effects of studying math.

These so-called side-effects are important. Most people don’t really benefit from knowing how to derive the quadratic formula or prove the Pythagorean theorem, but they do benefit from the reasoning skills implicit in such exercises. So the tail is wagging the dog: The side-effects of the math course are more important than the nominal subject matter.

These side-effects are so strongly associated with mathematics that some bureaucrats require an algebra course for reasons having precious little to do with algebra. This is bad policy; it would be better to directly require the things you care about. In particular, a computer programming course could be used to impart all of the higher-order reasoning skills listed above, and would be at least as practical.

Einstein said that an education is what remains after you’ve forgotten everything you learned in school. That’s amusing, but in some sense it is a symptom of bad pedagogy: It means the stuff you were supposedly learning wasn’t what you were actually learning; the stuff that was supposedly important wasn’t what was actually important.

8.2  Multi-Step Processes Require High Reliability

Suppose for example that you are given a quiz with 100 simple one-step questions, and you get 94% of them correct. That sounds like a pretty good score. Now in contrast, imagine a quiz with only 5 questions, each of which requires 20 steps. The overall number of steps is the same as before – 100 steps total – but if each step has only 94% reliability, you’d be lucky to get 2 of the 5 questions right.

There are a lot of situations in real life where you have to handle complex multi-step problems, and you do not get partial credit. If you are driving a car on a crowded street and you manage to miss 95% of the pedestrians, that is not considered an acceptable score. You are required to miss all of them, all of the time.

Supoose you build a moon rocket with a million parts, each of which has to work correctly. Then if each part is 99.9999% reliable, there’s less than a 40% chance that the overall system will work. If you want the overall system to be reliable, the failure rate for each individual part has to be very much less than one part in a million ... and/or you need to arrange for some redundancy, so that if one part fails another can take up the slack.

Similar logic applies to building a computer: It contains billions of transistors. If you want the overall system to be reliable, the requirements on the individual transistors are mind-bogglingly strict.

You should be wondering where that 40% number came from. More precisely, it is 1/e = 1/2.71828 = 36.79%. It is an interesting math exercise to work this out.

8.3  Skimming, Reading, Re-Reading, and Pondering

Nobody is smart enough to understand a math text on first reading. Instead you have to start by skimming it. Then go back and read it more carefully. Later, to back and re-read it.

There is nothing intrinsically mathematical about it. The same could be said for physics books, for Russian novels, for sheet music, et cetera. The notes on a sheet of music don’t tell you everything you need to know; you have to interpret them.

Skimming means that when you come to something you don’t understand, don’t panic, and don’t give up. Skip over it, and keep going. Make a mental note of it. Maybe it will become clearer in the light of later information. Maybe it will turn out to be not worth worrying about. Maybe the book is just wrong about this detail.

Intelligent skimming is an utterly nontrivial skill. There are some things that require high reliability and attention to detail, as discussed in section 8.2, but there are other details that can be skipped on first reading ... and it is hard to tell which is which.

Maybe if books were perfect, it would be possible to understand everything on first reading. However, writing such a book would be a near-impossible task ... and the book would be so large that it would be hard to afford, and hard to carry around.

9  Some Derivations

9.1  Expanding (a+b)2

The goal here is to find a simple expression for the square of (a + b)2. The resulting expression gets used again and again. An example can be seen in section 7.2.3.

So, let’s turn the crank:

At each step, we show the rationale for obtaining the next equation.

Rationale and Method     Equation
Use the definition of “squared”:
     (a + b)2=(a + b)·(a + b     (57a)
Use the distributive law:
      =(a + ba + (a + bb     (57b)
Use the distributive law again:
      =(a·a + b·a) + (a + bb     (57c)
Use the distributive law yet again:
      =(a·a + b·a) + (a·b + b·b)     (57d)
Use the commutative law:      
      =(a·a + a·b) + (a·b + b·b)     (57e)
Collect like terms:
      =a2 + 2a·b + b2     (57f)

On the RHS, I color-coded one of the as and one of the bs to make the calculation a bit easier to follow. The colored variables have the same algebraic meaning as the uncolored variables.

This is summarized in Equation 58. This is a famous result. It is a special case of the binomial theorem.

(a + b)2 = a2 + 2a·b + b2    (for all a and b)

The middle term on the RHS – namely the 2a·b term – is called the cross term. The origin of the name can be understood as follows: If you start with all the a terms in one column and all the b terms in another column, the 2a·b term involves crossing from one column to another.

9.2  Expanding Squares and Square Roots to First Order

There is a rule of thumb that says if X goes up by one percent, X2 goes up by slightly more than two percent. This rule gets used again and again. An example can be seen in section 7.2.4.

Consider the numbers in the following table.

1.002 = 1.00     
1.012  1.02     
1.022  1.04     
1.032  1.06     
1.042  1.08     

Where the squiggly ≈ symbol means “approximately equal to”. Notice that there is an easy-to-remember pattern:

When X goes up by one percent, 
X2 goes up by approximately two percent  

You could use a calculator verify the values in equation 59, but it is easier to work out the values by hand. Not only is it easier, it gives more insight into the structure of what’s going on. In particular, with the help of equation 61 we discover that the factor of two that appears in the cross term in the binomial theorem (equation 58) is the same as the “two” that appears in rule 60.

X2    =    (a+  b)2    =    a2+2a·b+   b2    =    X2   
1.002    =    (1+.00)2    =    1+.00+.0000    =    1.0000    
1.012    =    (1+.01)2    =    1+.02+.0001    =    1.0201    
1.022    =    (1+.02)2    =    1+.04+.0004    =    1.0404    
1.032    =    (1+.03)2    =    1+.06+.0009    =    1.0609    
1.042    =    (1+.04)2    =    1+.08+.0016    =    1.0816    

We can understand what’s going on as follows: The binomial theorem is exact, but rule 60 is only an approximation, because it involves neglecting the b2 term in the binomial theorem. Still, though, the rule is quite accurate when the percentage is small. To say the same thing in other words, when b is small, b2 is very small, and can be neglected.

This rule is called expansion to first order. There is a logical reason for the name, which we can discuss some other time. It’s not worth pursuing right now.

Let’s do an example just for fun. Once upon a time, somebody wanted to know the square root of 50, and asked the question in front of a room full of people. Everybody in the room had a calculator, but before anybody had time to poke the “on” button, I solved the problem in my head and blurted out the answer: It’s 7.07, to better than a tenth of a percent.

The reasoning is simple: First step: The square root of 49 is 7. Second step: 50 is 2% bigger than 49, so the square root of 50 has to be 1% bigger than 7. So the answer is 7.07.

For what it’s worth: I even estimated the accuracy, which requires a little bit of additional work. The second step (estimating the square root) is actually more accurate than the first step. The error in the first step is a couple percent of a couple percent, so I estimated that 7.07 was probably off by a few parts in ten thousand. If you grind out the answer on a computer, you find that the error is only 1.5 parts in ten thousand, so 7.07 is slightly more accurate than I would have guessed.

We can have some more fun with this:

The square root of a half is the square root of 50/100. We just figured out the square root of 50, and we certainly know the square root of 100. So we can write the square root of a half as 7.07/10 ... which comes to 0.707, within roundoff error in the third decimal place.

Also, we know that the square root of a half is half of the square root of 2, in accordance with equation 62, as you can easily verify (perhaps by multiplying both sides by S). Therefore the square root of 2 is 1.414, within roundoff error.


Notice the style of reasoning here: Rather than solving a hard problem in one step, we break it apart into a large number of easy steps. By way of analogy, rather than leaping from the ground to the third floor in a single bound, it’s more practical to take the stairs. It’s a longer path, but much easier.

10  The Pythagorean Theorem

10.1  A Proof

We start with a quick outline of the proof. Consider the diagram in figure 33. We start with a right triangle with altitude a and base b. We make four identical copies, and lay them out as shown in the diagram. We lay them out in a square arrangement that is just big enough to allow them to touch, corner-to-corner.

Figure 33: Diagram for Proving the Pythagorean Theorem

Using the standard formula for the area of a triangle, the area of each one of the bluish triangles is:

Area[1 triangle] = ½ a·b

so the area of the four of them together is:

Area[4 triangles] = a·b

Using the standard formula for the area of a square, the area of the entire colored area is:

Area[entire] = (a+b)2       
  = a2 + 2 a·b + b2     

where we have expanded (a+b)2 using the binomial theorem.

We can infer that the area of the yellow square is the area of the whole figure minus the four triangles.

Area[yellow] = a2 + 2 a·b + b2  − 2 a·b    
  = a2 + b2           

Meanwhile, we have another way of calculating the area of the yellow square: It is just a square, with edges of length c:

Area[yellow] = c2

Combining the two previous equations, we find

ab + b2 = c2

which is what we set out to prove.

To convert this outline into a real proof, we would have to go back and fill in a bunch of details. For one thing, we would need to assert that we are talking about figures in the flat, two-dimensional plane. That’s important, because if you start drawing large triangles on the surface of the earth, or any other space with intrinsic curvature, the Pythagorean theorem is not valid. It’s bad luck to prove things that aren’t true.

We also need to prove that the yellow region is in fact a square. Just because we drew the diagram so as to make it “look” like a square doesn’t count. In fact we can prove this, by using two facts from Euclidean geometry: The interior angles of any triangle always add up to 180. Therefore we know

α + β + 90 = 180           

We also know that the three angles α, β, and θ add up to make a straight angle, so

α + β + θ = 180           

Comparing this to the previous equation suffices to prove that θ = 90. We also know that all four sides of the yellow area are the same length. This suffices to prove that it is a square.

10.2  Discussion

The proof given here is similar to the one given by Pythagoras himself in the mid-500s BC, except that ours is simpler. It is simpler because we used algebra as well as geometry. Pythagoras didn’t have algebra, so he needed an analog, graphical way of performing the required subtraction.

Some algebraic ideas can be traced back 2000 years, but for most of that time algebra was not very widely used. People used geometrical constructions instead. Galileo, writing in the early 1600s, never wrote an equals-sign in his entire life, as far as we can tell. Even Newton, several decades later, even after he had invented calculus, wrote books that largely avoided algebra. Often he would discover things using algebra and calculus, and then reformulate them using geometrical arguments alone. Obviously Newton understood algebra, but in those days many people who were reading Newton’s books – even the ones who were “aware” of algebra – did not trust it.

You can prove the Pythagorean theorem without using algebra, but it is more work.

The Pythagorean theorem is so important the people have collected several hundred different proofs.

11  The Square Root of 2 is Irrational

11.1  Preliminary Remarks

Elsewhere in this document, I have offered examples that have clear practical applications. The example in this section – the irrationality of √2 – is in a different category:

I’m not going to pretend that the value is greater than it is.

Mathematicians are fascinated by this sort of thing. It says something fundamental about numbers. It proves that there are more kinds of numbers than you might have guessed.   For the other 99.9% of the population, this topic is not nearly so fascinating. However, it is interesting insofar as it shows you the sort of things that mathematicians do.

Historians are also fascinated by this topic. Discovering that √2 is irrational played a significant role in history, including the history of science, philosophy, and even religion.   If you are in the other 99.9% of the population, you should be very wary whenever anybody tries to argue that something is important for “historical” reasons. That reminds me too much of certain “celebrities” who are “famous for being famous” even though they have never accomplished anything of consequence.

  You should also be very wary of lessons that claim to teach a valuable skill while applying the skill to bogus problems. Scoundrels have been using such claims for centuries, to justify teaching stuff that wasn’t really all that important. The counter to any such claim goes like this: If the skill is so valuable and broadly applicable, why not show us some applications to real-world problems?

On the third hand, you don’t want to go overboard in the direction of real applications, because sometimes the real-world problems are unduly complicated, and some pedagogical simplification is necessary. This is especially true in introductory algebra, because so many of the applications involve using algebra in the service of geometry in the service of physics in the service of engineering, and those other subjects aren’t taught until later.   Even if some simplification is required, there should still be some visible connection to real-world problems.

  You should also be very wary of any claims that the topic has “entertainment” value because the result is weird and surprising. That is an awfully geeky argument, and it’s almost never true. It seems to suggest that if the subject were even more weird and surprising it would be more entertaining, which I don’t think is true. A good teacher should make the result seem natural, relevant, non-weird, and non-paradoxical. Weird is easy. Entertaining is hard. Useful is hard.

For more about the math, history, and value of this topic, see section 11.3.

11.2  A Proof

A rational number, by definition, can be expressed as the ratio of two integers; that is, one integer divided by another.

Now consider the number √2. Let’s call it S for short. We know that S2 = 2. The question is, is S a rational number?

If S is rational, then all the following indented statements are true:

S must be equal to A/B, for some integers A and B. This is required, by the definition of rational number.

As a warm-up exercise, consider the fact that A must be either even or odd, and similarly B must be either even or odd. If they are both even, then A/2 is an integer and B/2 is an integer and

S = 

In fact, we can go even further than that. There are extremely efficient methods of calculating the greatest common divisor of A and B. (Specifically, we could use the Euclidean algorithm.) In any case, there exists some integer G that is the greatest common divisor of A and B. Therefore we can write:

S = 

where either the numerator or the denominator (or perhaps both) is an odd number. So let’s define C and D, which we can use to express S in lowest terms.

C := A/G               
D := B/G               
S = 

We can make progress by squaring the last expression:

S2 = 
2 = 
D2 = C2                

Since the LHS is an even number, the RHS must be an even number. That means C has to be an even number.

We can introduce a new number:

E := C/2  

and we know that E must be an integer, not a half-integer, because we have already proved that C must be even.

We can make further progress by considering the reciprocal of S. Note that

1/S = S/2

as you can verify (perhaps by multiplying both sides by S). We also know the following:

S = 
S/2 = 
  = 1/S                   
S = 
S2 = 
2 = 
E2 = D2          

which tells us that D must be even. This is very similar to the way we proved that C was even.

To summarize, the only way we can have S=C/D and 1/S=D/C is for both C and D to be even.

Now we have a real contradiction, because if both C and D are even, that is inconsistent with the way we constructed them, when we divided out the greatest common divisor of A and B.

To say the same thing more formally, if both C and D are even, then 2G is a divisor of A and also a divisor of B, which is inconsistent with the fact that G is (by construction) the greatest common divisor of A and B.

We can clarify the argument using a bit of Boolean algebra.

The structure of the argument so far is “P implies Q” where P is the hypothesis that √2 is rational, and Q is all of the indented statements collectively. You have to imagine that there is a set of enormous parentheses around the indented stuff. There is a distributive law in Boolean algebra, and if we were to carry out the distribution, “P implies (X, Y, Z)” becomes “(P implies X, P implies Y, P implies Z)”. That is to say, every statement in the indented block must be true if √2 is rational ... but otherwise all bets are off.

The statement “P implies Q” means that whenever P is true then Q must be true. We can check all four possible values of P and Q, to see which of them are consistent with the proposition that P implies Q.

P   Q   P implies Q ?        
0   0   consistent       
0   1   consistent       
1   0   violated       
1   1   consistent       

Looking at the table, we see another way of summarizing what it means to say “P implies Q”. It means that either Q is true or P is false.

Consider the hypothesis that √2 is rational. This hypothesis implies things that cannot be true. Therefore the hypothesis itself must be false. In other words, √2 must be an irrational number.

11.3  Discussion

Irrational numbers are not rare. If you take the square root of every integer from 1 to a million, 99.9% of them will be irrational. In fact, it can be proved that irrational numbers are infinitely more numerous than rational numbers. That makes it somewhat ironic that rational numbers were discovered first.

As mentioned in section 11.1, this proof has large value to a small number of people, plus some small value to a large number of people. I am not going to pretend the value is greater than it is.

Proving that √2 is irrational is not high on the list of practical real-world problems, in the sense that it won’t put food on the table.   Just as an artist takes pride in his craftsmanship, you can take pride in being able to represent √2 exactly.

Your calculator will tell you that √2 = 1.41421356237 or thereabouts, which is approximately true.   The decimal representation cannot possibly be exactly true, because any decimal is (by definition) a rational number.

The decimal approximation is good enough for a wide range of practical purposes.   There are situations where approximating √2 by a thirteen-digit-long decimal is not good enough. Such situations are rather rare, but they do exist. In such a situation the solution is usually not to use more digits; the smart solution is to use algebra to restructure the problem so that you can more easily calculate something that makes sense.

Sometimes people do things for strictly utilitarian purposes.   Sometimes they do things for religious reasons. Sometimes they do things for entertainment.

You could live to a ripe old age without being able to prove that √2 is irrational.   Boolean algebra is tremendously useful. It is worth learning, even if you never apply it to √2. At some level, essentially everything that goes on in a computer involves Boolean algebra. Also, the style of proof used here can be used to prove other things. This style of proof is called proof by contradiction.

The ancient Greeks who discovered that √2 was irrational assigned religious significance to the discovery. They were not expecting it to be irrational. They did not want it to be irrational. They more-or-less worshipped numbers. It was a combination of mysticism and mathematics. To them, integers were perfect. It was a godlike perfection. One could even say that integers are more perfect than the ancient Greek deities, who were powerful but given to all sorts of bickering, thieving, adultery, murder, et cetera.

They also knew about rational numbers. They had noticed the connection between numbers and music, and were mightily impressed by it. According to simple theory, an octave is a factor of 2/1, a perfect fifth is a factor of 3/2, a perfect fourth is a factor of 4/3, et cetera. In the real world, this theory does not entirely fit the facts, but the ancient Greeks didn’t know that, and they overestimated the importance of the theory.

Before √2 came along, the only numbers the ancient Greeks knew about were rational numbers. The idea of something that was evidently a number but outside their number system came as a real shock.

On the other hand, it reinforced their belief in the power of mathematics. The idea that you can prove something to be true, absolutely positively provably true, even though you didn’t expect it to be true or want it to be true – that’s a really powerful idea.

This is proof far beyond the standard of “proof” that lawyers use in court. This is not proof by the preponderance of the evidence. This is not proof beyond a reasonable doubt. This is a mathematical proof, and if you do it right, there is no wiggle room whatsoever.

There is also some value in understanding the proof, even if you don’t place much value on the result. We proved something that you might have thought was difficult or impossible to prove. It is proverbially difficult to prove a negative, but that is exactly what we have done. Also note that there are infinitely many rationals. Indeed there are infinitely many rationals that are “close” to √2. Proving that none of these is exactly equal to √2 requires some seriously powerful tools.

Mathematics is telling you a negative message and also some positive messages. It is telling you there are some things you simply cannot do, such as describing the diagonal of a unit square in terms of an exact rational number. On the positive side, mathematics is giving you a glimpse of a whole new world, a new kind of numbers that you never dreamed of. Another positive message is that the rules of algebra – add subtract multiply divide associative distributive et cetera – apply just fine to irrational numbers, just the same as rational numbers, so you will very soon feel “at home” in this new world.

It could be argued that this proof has value as a signpost at the beginning of a long road. It tells you where to start. It puts you on notice that there are other types of mathematical quantities out there, not just integers and rational numbers.

On the other hand, one should not take the previous paragraph too seriously. There are lots of other things that would serve equally well as signposts, while having more direct practical applications. Vectors, for example.

There are lots of things in this world that are fancier than irrational numbers, such as transcendental numbers, complex numbers, vector algebra, Clifford algebra, probability, calculus, topology, et cetera. These things are tremendously useful. For example, if you tried to do physics without vectors, it would be horrifically clumsy. Doing quantum mechanics without complex numbers (or the equivalent) would be impossible.

12  Tools and Techniques

12.1  Formalism, Notation, and Terminology

Ideas are primary and fundamental. Terminology is tertiary. Terminology is important only insofar as it helps us formulate and communicate the ideas. Conversely, if you find that the terminology is causing problems, change the terminology.

It is almost never worth arguing about terminology. If you don’t like the standard terminology, invent something better.

Einstein said that any theory should be as simple as possible, but not simplier. John Reppy said that an experimentalist should never build any more apparatus than necessary, or any less. The same goes for mathematical formalism: It is usually a waste of time to build any more formalism than necessary, or any less.

Formalism, notation, and terminology are tools. Sometimes they are very helpful or even necessary ... but they are rarely an end unto themselves.

12.2  Shortcuts

Math in particular, and science in general, can be considered a collection of tools for solving problems, for reasoning about stuff, and for avoiding mistakes.

Nobody really cares how much you work or how hard you work; mainly they care how much you get done. By using the proper tools, you can do the job faster and better.

Please do not think that all shortcuts are good, or all shortcuts are bad. There are lots of possibilities:

I am reminded of a story told by the Car Talk guys (reference 18), the story of The story of Delbert Joyner:

... who when asked to carry in a load of firewood by his grandfather, tried to load 20 pieces in his arms and walk from the woodpile to the house and in doing so managed at one time or another to drop each one of the 20 pieces and when he finally reached the house after the half hour odyssey the grandfather said y’know, you could have carried a few pieces at a time and it is the lazy man who works the hardest.

That’s an amusing story, but the conclusion is misstated. A more nuanced view is required.

12.3  Abstraction

Sometimes making something more abstract makes it more useful. Consider for example the number 12. Without being specific as to what the questions are, imagine that the answers are as follows:

The number 12, by itself, is highly abstract. It is not the complete answer to any of the aforementioned questions. However, it is part of the answer to each of those questions, and unimaginably many other questions. The fact that it is abstract makes it more widely useful.

12.4  Symbolism

People are naturally good at symbolism. It’s the nature of the beast. A two-year-old child playing with a baby doll knows it’s not a real baby; it’s just a symbol representing a baby.

Spoken and written words are also symbols. The word “cat” sometime serves to represent an actual cat. The word “felidae” serves to represent something quite abstract, namely the phylogenetic family of cat-like creatures, including lions, tigers, house-cats, et cetera.

We use numerals to represent numbers. For each number, there are many ways of representing it. The following numerals all represent the same number: twelve, 12, 12.000, 0xC, XII, δώδєκα, et cetera.

13  Appendix – Miscellaneous Details

13.1  Sudoku Solution


13.2  Electronics Formulas in Gory Detail

It must be emphasized that writing down all the combinations and permutations like this is exactly what you are NOT supposed to do. The smart thing would be to remember the most fundamental equations and whatever others you use frequently – plus the rules of algebra – and derive everything else if-and-when needed. You could work for years as an electrical engineer without ever writing things out in this level of mind-numbing detail.

We start with the 12 ways of combining and rearranging Ohm’s law and the Joule heating law, using the four variables V, I, R, and P.

V = I·R     (79a)
I = V / R     (79b)
R = V / I     (79c)
P = I·V     (79d)
V = P / I     (79e)
I = P / V     (79f)
P = V2 / R     (79g)
R = V2 / P     (79h)
V = 
P = I2·R     (79j)
R = P / I2     (79k)
I = 
P / R

If we define one more variable, the conductance, we get 12 more equations:

V = I / G     (80a)
I = V·G     (80b)
G = I / V     (80c)
G = 1/R     (80d)
R = 1/G     (80e)
R·G = 1     (80f)
P = V2·G     (80g)
G = P / V2     (80h)
V = 
P / G
P = I2 / G     (80j)
G = I2 / P     (80k)
I = 

Again: writing down all the combinations and permutations like this is exactly what you are NOT supposed to do. The smart thing would be to remember the most fundamental equations and whatever others you use frequently – plus the rules of algebra – and derive everything else if-and-when needed.

14  References

“XYZ Affair”

John Denker,
“Learning, Remembering, and Thinking”

John Denker
“Problem-Solving Checklist”

David E. Joyce,
An annotated exposition of Euclid’s Elements,

An example of a missing postulate is given at:

Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nicht sicher, und insofern sie sicher sint, beziehen sie sich nicht auf die Wirklichkeit.

Albert Einstein,
“Geometrie und Erfahrung”
English Translation: “Geometry and Experience”
Lecture to the Prussian Academy of Science (27 January 1921).
http://quod.lib.umich.edu/u/umhistmath/ABR1192.0001.001/ http://www-history.mcs.st-and.ac.uk/Extras/Einstein_geometry.html

Jessica Lahey
“Teaching Math to People Who Think They Hate It”

Volker Ecke and Christine von Renesse
with Julian F. Fleron and Philip K. Hotchkiss,
Discovering the Art of Mathematics: Games and Puzzles
https://www.artofmathematics.org/sites/default/files/books/games-2013-06-06.pdf (The whole book is available online, free for all.)

John Denker,
“Ruler-Drop Measurement of Reaction Time”

John Denker,
“spreadsheet for calculating hybrid car economy”

John Denker,
“Hints on How to Do Math”

John Denker,
“Scaling Laws”

Joint Committee for Guides in Metrology
“International Vocabulary of Metrology – Basic and General Concepts and Associated Terms (VIM)”

Douglas Isbell, Mary Hardin, Joan Underwood,

“Mars Climate Orbiter Fact Sheet”

John Denker,
“spreadsheet for calculating linear and quadratic examples”

John Denker,
“Cause and Effect”

John Denker,
“The Spiral Approach to Thinking and Learning”

“Car Talk” episode 1514 segment 3
“The Stingy Man Spends the Most”
Copyright © 2013 jsd