What mathematics is

Wrap a rope round the Earth along the equator, tight, flat against the ground. The rope will be about forty thousand kilometres long. Now make it just one metre longer and lift it off the ground evenly, on every side at once. Could anything crawl underneath?

Make a quick estimate. One metre on forty thousand kilometres is like adding a sliver thinner than a hair to a one-kilometre running track. Place your bet before reading on, and see what others answered.

How far will the rope rise above the ground?

By a hair’s breadth, less than a millimetre By about a millimetre By about a centimetre: an ant could crawl under By fifteen centimetres or so: a cat could crawl under By more than a metre: a person could walk under

By about 16 cm: a cat could crawl under such a rope with room to spare. Why it’s so much is explained by one line of calculation below.

A cat under the rope

Sixteen centimetres. Not a hair’s breadth but a gap a cat could crawl through. And that’s not even the strangest part. Replace the Earth with a tennis ball: a rope round the ball, made the same metre longer, stands off it by the same 16 centimetres. Round the Sun, 16 again. Round our whole Galaxy, 16 once more.

Intuition was wrong, and one line is enough to put it right. The length of a circle is its radius times $2\pi \approx 6.28$. If the radius grows by $h$, the length grows by $2\pi h$, whatever the radius was. A metre of extra rope lifts it by $1 \div 6.28 \approx 0.159$ metres.

The height the rope has risen: the gap between it and the ball. How much longer the rope was made. How long it was to begin with doesn’t matter. The factor by which the length of any circle exceeds its radius: $2\pi \approx 6.28$. Example: $\Delta = 1$ m gives $h = \frac{1}{6.28} \approx 0.159$ m, almost 16 cm. Make the rope 10 cm longer and the gap becomes 1.6 cm, for the ball and for the Earth alike.

The statement “the gap doesn’t depend on the size of the ball” is true for all balls at once, and there’s no need to check it on each one in turn. All of mathematics works like this: it doesn’t go through cases one by one, it explains why things can’t be otherwise.

In this course theorems sit in coloured cards, and the proof comes right after the card. Where there’s an honest picture, the proof is shown as a drawing that builds up step by step along with the text: you can step backwards and forwards, and drag the points in the drawing to see for yourself that the argument doesn’t depend on where they are. Here is the first one.

Let’s compare the lengths of two circles: the one the rope lay on, and the one it lies on now.

The tight rope lies on a circle of radius $r$. Its length is $2\pi r$: the length of a circle is its radius times $2\pi$. Where the number $\pi$ comes from, and why it’s the same for every circle, is told in the chapter on the circle. The lifted rope is also a circle with the same centre, only its radius is bigger by the height it rose: $r + h$. Its length is $2\pi(r + h) = 2\pi r + 2\pi h$. The new rope is longer than the old one by exactly the added $\Delta$. Subtract: $2\pi r + 2\pi h - 2\pi r = 2\pi h$. The term $2\pi r$ has cancelled, and the radius has vanished with it. What’s left is $2\pi h = \Delta$. So $h = \frac{\Delta}{2\pi}$, and the radius doesn’t enter the answer at all. With $\Delta = 1$ m the gap is $\frac{1}{2\pi}$ m $\approx 15.9$ cm, for a ball, for the Earth and for the Sun.

What if we wrap a football pitch instead of a ball?

Wrap a rope round a rectangle, a football pitch say, and again make it a metre longer. Move the rope out by $h$ from the edge of the pitch: along the sides it runs parallel to them, and at each corner it goes round the corner along an arc of a circle of radius $h$. The straight pieces add up to the perimeter of the pitch, and the four quarter-circle arcs make one whole circle of length $2\pi h$. So the extra length is again $2\pi h$, and a metre of rope again gives 16 cm.

The same holds for any convex polygon: arcs are added at the corners, and together they turn the rope through exactly one full turn, because the exterior angles of a convex polygon add up to $360^\circ$ (we’ll prove this in the chapter on the triangle). The ball turns out to be just the best-known case of a general rule.

A sunflower that can count

Look at the middle of a sunflower. The seeds aren’t scattered anyhow: they curl into spirals, some clockwise, some anticlockwise. If you count the spirals, you most often get 34 and 55, or 55 and 89. In 1979 Helmut Vogel proposed a simple model of such a head: each new seed appears at the centre, turned by the same angle relative to the one before, and the older seeds move out towards the rim.

With a turn of exactly a third of a revolution, $120^\circ$, the seeds line up in three straight rays; with $144^\circ$, in five. Empty space is left between the rays, and the plant wastes room. This is no accident but a theorem.

Seed number $k$ is turned by $k \cdot \frac pq$ of a revolution. In the drawing the angle is $\frac 25$ of a revolution, $144^\circ$.

Compare seed $k$ with seed $k + q$. The second is turned by $(k + q) \cdot \frac pq = k \cdot \frac pq + p$ revolutions: $p$ full turns more than the first. A full turn brings the direction back where it was. So seed $k + q$ points the same way as seed $k$, only it sits further from the centre. All the seeds split into chains $k,\ k + q,\ k + 2q, \dots$, and each chain lies on one ray. There are $q$ chains: they start with the seeds $0, 1, \dots, q - 1$. It remains to check that these $q$ rays are different. Suppose two seeds $i < j$ among the first $q$ point the same way. Then the difference of their turns, $(j - i) \cdot \frac pq$, is a whole number of revolutions, and $(j - i) \cdot p$ is divisible by $q$. Since $p$ and $q$ have no common divisors, $j - i$ is divisible by $q$ (we’ll prove this property of coprime numbers in chapter 4). But $j - i$ is a number from $1$ to $q - 1$, and it isn’t divisible by $q$. A contradiction. The seeds lie on exactly $q$ rays, with emptiness between them. The smaller $q$, the more empty space.

So a good angle mustn’t be a fraction of a revolution. But it mustn’t be close to a simple fraction either: at $137^\circ$ the seeds almost fall onto the rays of the fraction $\frac{8}{21}$, and near the rim the gaps already show. We need an angle that fractions approximate worse than any other. There is one. It’s the golden angle, $\frac{360^\circ}{\varphi^2} \approx 137.508^\circ$, where $\varphi = \frac{1 + \sqrt 5}{2} \approx 1.618$ is the golden ratio. Its best fractions are $\frac12, \frac13, \frac25, \frac38, \frac{5}{13}, \frac{8}{21}, \frac{13}{34}, \dots$: the numerators and denominators are Fibonacci numbers, and these fractions close in on the golden angle as slowly as is possible at all. The spirals the eye picks out are traces of these approximations: there are 21, 34 and 55 of them, like the denominators.

How many rays will the seeds fall into if each one is turned by $135^\circ$ relative to the one before?

$135^\circ$ is $\frac{135}{360} = \frac38$ of a revolution, a fraction in lowest terms, so there are $8$ rays. Not three: the number of rays is the denominator of the fraction in lowest terms, not the numerator. Check it in the widget with the “135°” button.

Plants don’t count fractions, of course: the golden angle arises by itself when each new bud appears where its neighbours have left the most room. Alan Turing was thinking about why this happens in the 1950s. And in 2016 participants in a British project counted the spirals on hundreds of sunflowers and found that roughly one in five had spiral counts that weren’t Fibonacci numbers, or no clear spirals at all. Nature approximates mathematics well, but it isn’t obliged to obey it. The golden ratio and the Fibonacci numbers will come up again in chapter 6 and in the chapter on sequences.

A dragon made of a paper strip

Take a long strip of paper and fold it in half. Then in half again, the same way, and again. Unfold it and open every crease to exactly a right angle. Looked at edge-on, the bends form a figure invented in the 1960s by the NASA physicists John Heighway, Bruce Banks and William Harter. Martin Gardner wrote about it in Scientific American in 1967, and ever since it has been called the dragon curve.

How many straight segments will the strip consist of after ten folds?

Each fold doubles the number of layers, and with it the number of segments of the strip: $2^{10} = 1024$. There is one fewer bend between them, $1023$. An ordinary sheet can’t be folded in half more than seven or eight times, because the thickness grows too fast (see chapter 12), but in your imagination you can go on for as long as you like.

Look at how the dragon is built: opening the last fold means taking everything already unfolded and turning a copy of it through a right angle about its end. So the dragon consists of two smaller dragons, each of those of two smaller still, and so on down to single segments. Figures whose parts look like the whole are called fractals; they are the subject of chapter 59.

Despite its thousands of bends, the dragon never runs along the same segment twice and never crosses itself, it only touches itself at corners now and then. Chandler Davis and Donald Knuth proved this in 1970. They also showed that four dragons turned about a common start fit together without overlapping, and that the whole plane can be tiled with dragons. The turns along the strip follow a rule too: at the odd positions they simply alternate, right and left, while at the even positions they repeat the whole sequence from the beginning. Michael Crichton opened every part of the novel Jurassic Park with a dragon, more intricate from part to part.

An ant that builds a highway

In 1986 the American artificial-life researcher Christopher Langton invented an ant with two rules. The ant stands on a squared grid, and all the squares are white. On a white square it turns right, on a black one left; then it repaints the square the opposite colour and steps forward one square. That’s all. No memory, no goal.

The rules are childishly simple, but the behaviour isn’t. For the first few hundred steps the ant draws almost symmetric patterns. Then chaos sets in: for thousands of steps a black tangle grows with no visible order. And at around step ten thousand the ant suddenly starts building a “highway”: it repeats the same 104 steps, moving two squares diagonally each time, and goes off to infinity that way.

If you start not from a clean grid but with a few black squares scattered about, the highway still appears, in every experiment anyone has run. But nobody has yet managed to prove that this happens for every starting pattern. It’s an open problem: the statement has been checked on a huge number of examples, it looks obvious, and there’s no proof. Something about the ant can be proved, though, and the proof fits in half a page.

We argue by contradiction: suppose the ant is shut in some bounded region forever, and derive an absurdity from that.

At every step the ant turns through $90^\circ$, so horizontal and vertical steps alternate. Call a square even if the sum of its column and row numbers is even; every step changes the parity of the square. Since both the direction of the step and the parity of the square change at every step, the ant always enters even squares in one way, say horizontally, and odd squares in the other. Each square has its own way in, and the ant always leaves it crosswise. A bounded region has finitely many squares, while there are infinitely many steps, so the ant enters some squares infinitely often. Call the set of such squares $S$. The ant enters each square outside $S$ only finitely many times, so from some moment on it walks only on squares of $S$. Take the top row of $S$, and in it the rightmost square $c$. The squares above $c$ and to the right of $c$ are not in $S$: after that moment the ant never steps there. Suppose the ant enters $c$ horizontally. It can’t come from the right, so it enters from the left, moving right. It leaves vertically, and it can’t go up, so it goes down. Moving right and leaving downwards is a right turn, and that happens on every visit. But the ant turns right only on a white square, and immediately repaints it black: on the next visit it will turn left. A contradiction. If instead the ant enters $c$ vertically, it comes from below and leaves to the left, turning left every time, and the contradiction is the same. So the assumption is false: the region the ant walks over is unbounded. This theorem was proved by Leonid Bunimovich and Serge Troubetzkoy in 1992. Whether the highway always appears, nobody knows to this day.

A single ant holds all of mathematics in miniature: an observation that can’t yet be proved, and a theorem that holds for every starting pattern even though they can’t all be checked. We’ll now look more closely at the difference between “seen many times” and “proved”.

Forty primes in a row

In 1772 Leonhard Euler noticed that the expression $n^2 + n + 41$ gives prime numbers, those divisible only by one and by themselves, for $n = 0, 1, 2, 3, \dots$ Place your bet before starting the machine.

For which $n$ does $n^2 + n + 41$ first give a composite number?

Almost at once, for $n$ below five Somewhere between $n = 5$ and $n = 39$ Not before $n = 40$ Never: it always gives primes

At $n = 40$: $40^2 + 40 + 41 = 40 \cdot 41 + 41 = 41 \cdot 41 = 1681$. Before that come forty primes in a row. And “never” can’t happen: at $n = 41$ all three terms are divisible by 41.

Forty primes in a row: 41, 43, 47, 53, 61, …, 1601. It’s easy to believe it will go on forever. But at $n = 40$ we get $1681 = 41^2$, and the dots form a perfect square. With hindsight the formula was doomed even without a search: at $n = 41$ all three terms are divisible by 41.

Euler is no exception here. In 1640 Pierre de Fermat was sure that all numbers of the form $2^{2^n} + 1$ are prime: 3, 5, 17, 257, 65,537. The next one, $2^{32} + 1 = 4{,}294{,}967{,}297$, went unchecked for nearly a century, until in 1732 the same Euler found a divisor: $641 \cdot 6{,}700{,}417$. Or a very simple game: 31, 331, 3331, 33,331, 333,331, 3,333,331, 33,333,331 are all prime, but the eighth, $333{,}333{,}331$, equals $17 \cdot 19{,}607{,}843$.

Among our examples the record for patience belongs to a conjecture of the Hungarian mathematician George Pólya (1919). Break every number into prime factors and count them with repetition: $12 = 2 \cdot 2 \cdot 3$ has three factors, $10 = 2 \cdot 5$ has two, and 1 has none. Pólya conjectured that for every $N \ge 2$, at least half of the numbers from 1 to $N$ have an odd number of factors. In 1958 the British mathematician Haselgrove proved that the conjecture fails somewhere, without saying where. The smallest such $N$ was found in 1980 by the Japanese mathematician Minoru Tanaka: $906{,}150{,}257$. More than nine hundred million confirmations in a row, and the conjecture is still false.

Continue the sequence

“Continue the sequence” puzzles test your wits, but they have no single answer. A finite beginning doesn’t determine what follows: the same first numbers belong to as many rules as you like. Here are six perfectly natural ones that all start the same way, $1, 2, 4, 8, 16$, and then go their separate ways. One of them is the largest number of pieces that chords cut a circle into: after 16 it gives 31, and we’ll take that trap apart in the chapter on the circle. In the second tab you can choose the sixth number yourself, and there will be a formula that produces it.

The trick in the second tab is an honest one. Through any six points with different $x$-coordinates there passes exactly one polynomial of degree at most five (why is explained in the chapter on polynomials), and it can be written out explicitly. So the answer “the sixth term is 17” is as legitimate as “32”; you just have to show the rule.

Practise catching patterns that break. The trainer offers a statement that holds for the first few $n$ and asks you to find the first $n$ where it fails. Almost all the statements are about primes; they are covered in detail in chapter 3.

Why proof

If examples guarantee nothing, how do mathematicians ever know anything for sure? They prove it. Remember the rope: we didn’t measure the gap for the ball, the Earth and the Sun in turn; one line of calculation answered for every ball there could ever be. The same goes for the rays of the sunflower and for the ant: the argument works for all cases at once, for five and for a million.

Every proof rests on something: on definitions, on theorems already proved, and at the very bottom on axioms, statements accepted without proof as the rules of the game (more on them in chapter 16). Here is the smallest possible model of such a building: one axiom of geometry and one consequence of it. In this course axioms sit in grey cards with a double border; theorems and propositions sit in coloured ones.

We argue by contradiction: suppose the statement is false and derive a contradiction with the axiom.

Suppose the distinct lines $a$ and $b$ have at least two common points; call them $A$ and $B$. The line $a$ passes through $A$ and $B$. The line $b$ also passes through $A$ and $B$. In the drawing it is bent: you can’t draw a different straight line through $A$ and $B$, and that is exactly what the axiom says. Only one line passes through two distinct points, so $b$ coincides with $a$. So $a$ and $b$ are the same line, whereas we assumed they were distinct. A contradiction. So the assumption is false, and two distinct lines have at most one common point.

Three steps, but they contain everything any proof is made of: starting rules, a statement, and a chain in which every link rests on what has already been accepted. The “suppose the opposite” move will come up again and again; its most famous use is the proof that $\sqrt2$ cannot be written as a fraction (chapter 6).

What has been proved stays true forever. Around 300 BC Euclid proved that there are infinitely many primes, and his proof is still read today without a single correction; in chapter 3 it takes half a page. A new experiment can overturn a physical theory. It can’t overturn a theorem, unless the proof contains a mistake.

The caveat “unless the proof contains a mistake” is not there for show. Proofs are written by people, and people make mistakes, so other mathematicians read and check them. In recent decades computers have increasingly checked them too, line by line; more on that in the last chapter.

The story of the rope is a mathematician’s work in miniature. Intuition suggested an answer: a hair’s breadth. Calculation refuted it: sixteen centimetres. A proof explained why the answer is the same for every ball, and a generalisation showed that the ball has nothing to do with it: any convex field will do. The sunflower went the same way: we noticed the spirals, guessed something about the angle, proved why fractions are bad, and found the number that fractions approximate worst of all. When one observation explains many unlike problems at once, we speak of abstraction.

Sometimes a pattern is true, and even then a proof is needed. Add up the first odd numbers: $1 = 1^2$, $1 + 3 = 2^2$, $1 + 3 + 5 = 3^2$, $1 + 3 + 5 + 7 = 4^2$. This time there’s no trap. In the chapter on sequences this is proved with a single picture, and you’ll learn to prove such statements for all $n$ at once.

And sometimes a conjecture survives every test while a proof stays out of reach for centuries. In 1742 Christian Goldbach, in a letter to Euler, made a guess that today is stated like this: every even number greater than two is the sum of two primes. $4 = 2 + 2$, $10 = 3 + 7$, $100 = 3 + 97$.

Goldbach’s conjecture has been checked by computer for all even numbers up to $4 \cdot 10^{18}$. What does that mean?

It is proved: that many checks are enough It holds up to $4 \cdot 10^{18}$, but in general it is still a conjecture It is almost certainly false, since no one has proved it yet It can no longer be refuted

A check is a lot of examples, and examples can’t cover all numbers. Remember Pólya: there the first counterexample hid beyond nine hundred million. If Goldbach’s conjecture has a counterexample, it is larger than $4 \cdot 10^{18}$. So far nobody has a proof, and nobody has a refutation either.

Why mathematics works

So far this all looks like a game: ropes, sunflowers, paper dragons, prime numbers. Yet games like these grew into the calculations behind bridges, orbits and power grids. In 1960 the physicist Eugene Wigner wrote an article about it called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. There is less that’s unreasonable here than it seems, and two stories show why.

In the 1840s astronomers’ calculations didn’t add up: Uranus moved across the sky not quite the way the law of universal gravitation required. Urbain Le Verrier supposed that an unknown planet was pulling on Uranus, and from the deviations he calculated where it had to be. On 23 September 1846 Johann Galle at the Berlin Observatory pointed a telescope at the predicted spot and found Neptune that same night, about a degree from where it was supposed to be. People say the planet was discovered “at the tip of a pen”. Independently of Le Verrier, the Englishman John Couch Adams made a similar calculation.

The second story comes from engineering. Square roots of negative numbers appeared in the sixteenth century as an intermediate trick in the formula for cubic equations, and for a long time they were considered not real, “imaginary” (see chapter 15). Three centuries later, in the 1890s, Charles Steinmetz and other engineers began calculating alternating-current circuits with complex numbers, which turns differential equations into arithmetic. Electrical engineers still work this way, except that they write the imaginary unit as $j$, because $i$ is already taken by the current.

The secret, if there is one, is that mathematics studies structure: what follows from what. The law of gravitation describes the structure of the world. If it is true, then all of its consequences are true too, including those nobody has seen yet. A mathematician derives the consequences without looking through a telescope and tells you where to look.

Seven big questions

Seven questions run through the whole course. They sound simple, some almost childish, yet their answers became landmarks in the history of mathematics. Each question is asked here and answered in the chapter where the tools for the answer are finally at hand: the first answer comes in chapter two, the last in chapter fifty-nine. Pick a question, read the teaser and try a small experiment.

At school the sign rule is learned by heart: minus times minus makes plus. But what is a debt multiplied by a debt? The rule looks like a convention, and it seems we could have agreed on something else. Or could we?

Chapter 2. Zero and minus Chapter 15. Complex numbers

A circle and a square of the same area. Scissors cutting along straight lines can’t turn the circle into the square, and ruler and compass can’t construct such a square: in 1882 Lindemann proved that $\pi$ is transcendental. Yet in 1990 Miklós Laczkovich proved that cutting and reassembling is possible after all, if the pieces are ones no artist could draw, and there are about $10^{50}$ of them.

Chapter 6. The square root of two Chapter 20. Area and volume Chapter 43. Galois theory Chapter 54. Measure

Quadratic equations were solved as far back as Babylon; cubics and quartics fell in sixteenth-century Italy. Then for nearly three hundred years nobody could find a formula for the fifth degree. Not because people searched badly: there is no such formula, and this can be proved.

Chapter 14. Polynomials Chapter 43. Galois theory

There are infinitely many natural numbers, and infinitely many points on a segment, so it seems there is nothing to compare. In 1638 Galileo noticed something odd: there are as many squares as there are numbers, even though squares get rarer and rarer. And in 1874 Georg Cantor proved that infinities come in different sizes.

Chapter 52. Infinities

You agree on a secret key with your bank, and anyone at all can listen to the whole conversation. Yet the eavesdropper still doesn’t learn the key. It all rests on operations that are easy to perform and, as far as anyone knows, practically impossible to undo.

Chapter 41. Remainders and ciphers Chapter 44. Elliptic curves

The equations that govern the air are known, and computers are fast. But in 1961 Edward Lorenz rounded one number in a calculation, $0.506127$, to $0.506$, and got completely different weather. Weather can be predicted about two weeks ahead at most, and weak computers are not the reason.

Chapter 31. Differential equations Chapter 59. Chaos

It seems that everything true in mathematics gets proved sooner or later, if you just try hard enough. In 1931 Kurt Gödel showed that this is not so: for any reasonable system of axioms for arithmetic there are true statements about numbers that it cannot prove, and new axioms won’t fix that.

Chapter 56. Gödel and Turing

How the course works

The course has 61 chapters in ten parts: Introduction, Numbers, Algebra, Geometry, Calculus, Linear algebra, Structures, Chance & data, Foundations and Horizons. On the main page they are drawn as a metro map: each part is a line of its own, chapters are stations, and interchanges connect places where an idea from one part is at work in another.

Every chapter opens with a wall, a problem the tools so far can’t solve, and builds a ladder over it. At the end, in the “Where next” section, a new wall appears, and the next chapter starts from it. You can read in order, or jump straight into any chapter: its header says what it builds on.

Every chapter has a level. “School” is for ages 11–15, “High school” for ages 16–18, “University” for the first two years, and “Beyond” for readers who aren’t put off by abstraction. The course starts at school level and by the end reaches mathematics from the later years of university.

Inside a chapter you’ll meet a few regular residents.

• Formulas taken apart, like the gap formula above: tap a coloured part of the formula and its explanation lights up.
• Widgets: live drawings that answer “what happens if…”. Drag, change, break things: nothing will be damaged.
• “Try it yourself”: problems with an answer box. The checker accepts any correct form: $\frac12$, $0.5$ and $\frac24$ are all the same to it.
• Theorems and proofs: theorems in coloured cards, axioms in grey ones with a double border, and the proof comes right after the theorem, step by step, with the drawing building up along with the text.
• Sidebars marked History, Deeper, For students and Careful can be skipped without losing the thread.
• Terms, like counterexample and proof, are collected in a glossary with a short definition and a link to the chapter where each first appears.

Besides the chapters, the course has two tools. Trainers, like the counterexample hunt above, invent problems endlessly, accept an answer in any correct form and show a worked solution; they are all collected on the practice page.

Solvers take a problem of your own, say a quadratic equation or breaking a number into prime factors, and solve it step by step, explaining why each step is legitimate. And in the topic catalogue you can filter chapters by subject and level, for example everything that comes in handy for school-leaving exams.

Where next

All through this chapter we handled numbers of every size: sixteen centimetres of gap and forty thousand kilometres of rope, 1024 segments of the dragon, ten thousand steps of the ant, nine hundred million checks of Pólya’s conjecture. And not once did we stop to ask where the numbers themselves, and the way we write them, came from. Leopold Kronecker, as Heinrich Weber recalled, used to say that God made the integers and all else is the work of man. Even if so, it was people who had to write them down. A shepherd can mark each sheep with a notch on a stick. But how do you write a number that no stick is long enough for, and how do you add such numbers in columns? The answer took tens of thousands of years, from a notched bone to binary code. That’s chapter 1.