Elements

The strangest rule

Here is perhaps the strangest rule in school mathematics. i squared equals minus one. A number whose square is negative. You learn it, you apply it, and you never quite trust it. What if it were just a quarter turn? Articulated Course in Mathematics. Volume one: from number to form.

Who it is for

The first volume makes a simple promise. Calculate with confidence. Draw a figure. Understand what makes a proof hold. It is addressed, the foreword says, to anyone entering formal mathematics, perhaps with curiosity, perhaps with apprehension. To anyone rebuilding the foundations, or preparing for university. It assumes nothing beyond elementary arithmetic. It covers the core of secondary-school mathematics, and adds a few glimpses of university mathematics, where they shed light on an idea. It leads to the threshold of university, which is where volume two begins.

The map of the volume

Here is the map. Seven parts. Twenty-two chapters. One hundred and sixty articles, some seven hundred pages. First, the language: calculation, inequalities, proof, arithmetic, quadratics. Then plane geometry, and geometry in space. Complex numbers. The longest part: functions, sequences, limits, derivatives, integrals. Probability and statistics. And, to finish, the conics.

Four ideas

Four ideas hold the whole together. The first: calculation becomes a language. Two plus three times four is not a sequence of arbitrary gestures, but a hierarchy. And a language is for saying true things, and then proving them. An example, in thirty seconds. The square root of two is not a fraction. Suppose it is: root two equals p over q, in lowest terms. Then p squared equals two q squared. So p squared is even, and p is even. Substitute, and q squared is even, so q is too. Both even: the fraction was not in lowest terms. Contradiction. The second idea: the same object is seen twice, as a calculation and as a figure. Pythagoras’ theorem links form to number. Coordinates, then vectors, turn every figure into a calculation, and every calculation into a figure. The third: measuring change. The limit, the derivative and the integral tell how a quantity varies, and what it accumulates. The fourth: measuring chance, by counting first, then by calculating. And beneath all four runs a thread: the chain of numbers. The natural numbers, the integers, the fractions, the reals. Each time an equation refused to be solved, the numbers were enlarged. The last to hold out fits on one line: x squared plus one equals zero.

Multiplying by i is turning

No real number will do: a square is never negative. So we decree a number, written i, whose square is minus one. The move may seem arbitrary. The book readily admits it. But let’s write each complex number as a plus b i, and place it in the plane: a on the horizontal axis, b on the vertical one. Three plus two i becomes a point. Now multiply it by i. Expand, and replace i squared with minus one: three plus two i becomes minus two plus three i. Look at the point. It has turned a quarter turn about the origin. It holds for every point: a plus b i becomes minus b plus a i. Start from one. A quarter turn: i. Another: minus one. There is i squared equals minus one: two quarter turns, a half turn. Two more, and you are back where you started. The strange rule was no quirk of calculation. It was geometry, written in algebra. In chapter thirteen, multiplying by e to the i theta turns the plane through any angle. One object, seen twice: the whole volume is in that gesture.

Working with the volume

How do you work with this volume? Slowly, pencil in hand, the foreword advises. With three kinds of exercises, all with solutions, printed upside down. A practice exercise, right after the article: compute i to the power one, two, three, four, five. What pattern do you see? A cycle of length four. Four quarter turns, and round again. A study, at the end of the chapter, teaches a method: writing a rotation whose centre is not the origin. Translate, turn, translate back. About one plus i, three plus i goes to one plus three i, at the same distance from the centre. And an expedition takes you further than you’d expect: the roots of unity draw a regular triangle, pentagon and hexagon on the circle, and they add up to zero. Alongside the volume, three companions. The abridged course, to learn or relearn. The compendium, to review. The formula handbook, to check. At one article a day, the volume takes about five months.

What makes the difference

What makes the difference comes down to a few rules. One article, one idea, from its motivation to its interpretation. The figure before the theorem. Every proof has a visible status: complete, sketched or admitted. Ten objects come back from volume to volume; the square root of two, which you just met, accompanies the whole collection. And the volume exists in French and in English. “This book is not an examination,” the foreword ends, “it is a companion for the road.”

To remember

To remember. Volume one: from secondary school to the threshold of university. Seven parts, one hundred and sixty articles, every exercise with its solution. And i: a quarter turn. Volume one opens for free: the opening pages, the table of contents and all of chapter one, exercises included. The link is in the description.