From counting to the real line

Mathematics is usually taught as though the numbers were simply there, waiting, and the only question was what to do with them. That is backwards: every kind of number was invented under protest, because someone wrote down a problem that the numbers already available could not answer.

This lesson follows that chain of protests from one end to the other. It assumes nothing beyond arithmetic, and what it produces is the object the rest of the course lives on, the real line, along with an honest account of what that line still cannot do.

Counting, and the first thing it cannot do

Start with what a child starts with: $1, 2, 3, 4, dots$ , the natural numbers, written $N$ . They answer one question, "how many", and they are closed under two operations. Add two naturals and you get a natural. Multiply two naturals and you get a natural. Closure is worth naming, because it is exactly what fails next.

Now ask a question of the same shape as one they can answer. The equation $x + 3 = 7$ has the answer $4$ , and nothing is wrong. The equation $x + 7 = 3$ has no answer at all in $N$ . It is not that the answer is hard to find; there is no natural number that works, so within this system the question is malformed.

There are two honest responses. One is to declare the question illegal and keep the numbers small and safe, which is roughly what Greek mathematics did. The other is to invent whatever object the equation demands and then check that the new object does not wreck the arithmetic that already worked. Every step in this lesson is that second response, taken again and again.

Zero and the negatives

The demand of $x + 7 = 3$ is for a number that undoes addition. Grant it: for every natural $n$ introduce an object $- n$ with the defining property $n + (- n) = 0$ , and introduce $0$ itself as the number that changes nothing when added. The result is the integers, $Z$ , running $dots, - 2, - 1, 0, 1, 2, dots$ and closed under addition, subtraction and multiplication.

The first systematic account of how these behave is Brahmagupta's, in the Brahmasphutasiddhanta of 628 CE, which states rules for zero and for what he calls fortunes and debts: a debt subtracted from zero is a fortune, and the product of two debts is a fortune. That second rule is the one every student finds arbitrary, and it is not arbitrary at all. It is forced, and the next lesson shows exactly what forces it.

Resistance lasted a very long time. Cardano, solving cubics in the Ars Magna of 1545, still called negative roots fictae, fictitious, and European mathematicians were arguing about whether negative quantities were legitimate into the eighteenth century. Two facts sit behind this. A negative number answers no question of the form "how many", so it has no direct counterpart in a heap of stones. And a debt, a temperature below freezing, or a displacement to the left are all perfectly concrete, so the objection was never about the world. It was about what a number was allowed to be.

Fractions, and a system that looks complete

Multiplication now has the same problem addition had. The equation $3 x = 12$ is fine. The equation $3 x = 7$ has no solution among the integers, and the fix is the same trick: for every non-zero integer $n$ introduce an object $1 / n$ with the defining property $n \times (1 / n) = 1$ . What comes out is the rational numbers, $Q$ , every number expressible as $p / q$ with $p$ and $q$ integers and $q$ not zero.

The rationals are closed under all four operations, division by zero excepted, and they are also dense: between any two distinct rationals there is another, since their average is rational and sits strictly between them. Take $1 / 3$ and $2 / 5$ ; their average is $11 / 30$ , which lies between the two, and the same move repeats forever. There is no such thing as the next rational after a given one.

Density makes the system feel finished. Pick any point on a ruler and you can name a rational as close to it as you please, so it is natural to assume the rationals fill the line completely. That assumption is false, and the discovery that it is false is the most important event in this lesson.

Every rational has a decimal expansion that either terminates or eventually repeats, and every terminating or repeating decimal is a rational. The forward direction is long division: dividing by $q$ can only produce $q$ possible remainders, so a remainder must recur, and once it does the digits cycle. The reverse direction is a trick worth having.

Example. Write $0.272727 dots$ as a fraction.

Call the number $x$ . The repeating block is two digits long, so multiply by $100$ to shift the expansion by exactly one block: $100 x = 27.272727 dots$ . Subtracting the original, the infinite tails cancel exactly, since they are identical: $100 x - x = 27$ , so $99 x = 27$ and $x = 27 / 99$ . Dividing top and bottom by $9$ gives $x = 3 / 11$ . Checking by division, $3$ divided by $11$ is $0.2727 dots$ , as claimed.

Now you. Write $0.135135 dots$ as a fraction in lowest terms.

Answer

The block is three digits, so $1000 x - x = 135$ , giving $x = 135 / 999$ . Both are divisible by $27$ , since $135 = 27 \times 5$ and $999 = 27 \times 37$ , so $x = 5 / 37$ .

The diagonal that is not a fraction

Draw a square of side $1$ and ask for the length $d$ of its diagonal. Pythagoras' theorem gives $d^{2} = 1^{2} + 1^{2} = 2$ , so $d$ is a number whose square is $2$ . The Pythagoreans, working in the fifth century BCE, could prove that no fraction has this property. Aristotle refers to the argument as a familiar one, and a version of it survives in Book X of Euclid's Elements.

The full proof is the business of the last lesson of this course, where the technique it uses is the point. What matters here is the consequence, and it is severe. The diagonal of a perfectly ordinary square has a length, you can draw it, and that length is not any ratio of whole numbers. The rationals, dense as they are, have a hole in them exactly where the diagonal lands.

The tradition that the discovery was a scandal, and that Hippasus was drowned at sea for divulging it, is a late story and probably not history. The mathematical damage, however, was real. A school whose slogan was that all is number had assumed that any two lengths share a common unit small enough to measure both, and the diagonal of a square shows they need not.

You can watch the failure numerically. Fractions approximate $\sqrt{2}$ beautifully, and never hit it.

Example. How close does $99 / 70$ come to having a square of $2$ ?

Square it: $99^{2} = 9801$ and $70^{2} = 4900$ , so $(99 / 70)^{2} = 9801 / 4900$ . Now $2 \times 4900 = 9800$ , so the square is $9801 / 4900 = 2 + 1 / 4900$ , missing $2$ by exactly one part in $4900$ , about $0.0002$ . Close, and not equal, and the argument of the last lesson says the miss can never be zero however clever the fraction.

Now you. By how much does the square of $577 / 408$ miss $2$ ?

Answer

$577^{2} = 332929$ and $408^{2} = 166464$ , and $2 \times 166464 = 332928$ , so the square is $332929 / 166464 = 2 + 1 / 166464$ . It misses by exactly one part in $166464$ , which is roughly six millionths, and still misses.

The real line

Patch the holes and you get the real numbers, $R$ : every point on a continuous line has a number, and every number a point. Informally, a real number is a decimal expansion, allowed to run forever without repeating. The rationals are the expansions that terminate or repeat; the irrationals are all the rest, and $\sqrt{2} = 1.41421356 dots$ is one of them.

Making that respectable took until the 1870s, when Dedekind and Cantor gave constructions of the reals in terms of the rationals alone, so that the line stopped being a picture and became a definition. The property their constructions deliver is completeness: there are no gaps left, and any quantity you can squeeze arbitrarily tightly between rationals is itself a real number. Completeness is what makes limits work, so it is the foundation the whole of calculus is built on, and it is why the course you are reading stops here and calculus starts here.

Two facts about the irrationals are worth carrying. First, they come in two kinds. $\sqrt{2}$ is algebraic, meaning it solves a polynomial equation with whole-number coefficients, here $x^{2} - 2 = 0$ . Numbers that solve no such equation at all are transcendental, and both of the famous constants are: Hermite proved it for $e$ in 1873 and Lindemann for $π$ in 1882. Lindemann's result settled squaring the circle, an open problem of two thousand years, in the negative.

Second, the irrationals are not rare. Cantor showed in 1874 that the rationals can be listed in an infinite sequence while the reals cannot, so the two infinities differ in size, and in a precise sense almost every real number is irrational. The numbers we can name are the exceptions.

Decimals, precision and error

In practice, every real number that is not a whole number gets replaced by a decimal that stops, so it is worth being able to say how much that costs. The absolute error is the difference between the approximation and the true value; the relative error is that difference divided by the true value, and it is the one that usually matters, because being wrong by a metre is trivial for a road and fatal for a doorframe.

Example. $π = 3.14159265 dots$ and the schoolroom approximation is $22 / 7 = 3.14285714 dots$ . What are the absolute and relative errors?

Subtracting, the absolute error is $3.14285714 - 3.14159265 = 0.00126449$ , so $22 / 7$ is a little too large. Dividing by $π$ gives a relative error of $0.00126449 / 3.14159265 = 0.000402$ , about four parts in ten thousand, or $0.04 %$ . On a circle the size of a dinner plate that is an error of about a tenth of a millimetre in the circumference.

Now you. The better approximation $355 / 113 = 3.14159292 dots$ was known to Zu Chongzhi in the fifth century. What is its absolute error, and roughly its relative error?

Answer

The absolute error is $3.14159292 - 3.14159265 = 0.00000027$ , near enough $2.7 \times 10^{- 7}$ . Dividing by $π$ gives a relative error of about $8.5 \times 10^{- 8}$ , under one part in ten million, which is why the approximation stood as the best known for nearly a thousand years.

The equation the line cannot solve

The chain of this lesson has a shape: name an equation with no solution, invent the solution, check that nothing already working breaks. Run it once more and it does not terminate.

The equation is $x^{2} = - 1$ . On the real line it has no solution, and for a reason stronger than mere absence: any real number, positive or negative, has a square that is zero or positive, so no arrangement of real numbers will ever produce $- 1$ . The same demand can be made anyway, and it was, by Bombelli in 1572, who found that solving cubic equations sometimes required carrying square roots of negatives through the middle of a calculation even when the final answers were ordinary whole numbers. Grant the demand and you get the complex numbers, and everything that worked before still works.

This course does not follow that road, and the reason is worth saying plainly rather than hiding. The material ahead, functions, graphs, growth, decay and the geometry of angles, is about quantities you can measure and plot: a length, a temperature, a balance, a height. Those live on the line. The complex numbers are the natural home of a different set of questions, and taking them on here would double the machinery without touching the goal.

So the working set is $R$ , with $N$ , $Z$ and $Q$ sitting inside it, and one honest gap left at $x^{2} = - 1$ .

There is something conspicuously unfinished about all of this. Solving $x + 7 = 3$ and $3 x = 7$ meant subtracting from both sides and dividing both sides, and no reason was ever given for why those moves are allowed. They were used because they are familiar. The rules that licence them are few, they can be written down completely, and once they are, results such as Brahmagupta's rule about two debts stop being conventions to memorise and become consequences. That is the next lesson.