The big ideas of mathematics

In 2026 the 2 on the left means two thousand and the 2 in the tens place means twenty. Same symbol, different value, decided by position. This is place value, and it is why ten symbols are enough to write any whole number.

Position only works if you can mark an empty place. Without zero, 26, 206 and 2006 look alike. Once zero exists, the algorithms you learned in school for adding, multiplying and dividing are simple column-by-column rules. With Roman numerals they are a nightmare: try multiplying MCMXLIV by XXIX.

There is nothing special about ten. Each place is worth the base times the place to its right. Computers use base 2, the Babylonians used base 60 (which survives in 60 minutes and 360 degrees), and the Maya used base 20.

A prime is a whole number greater than 1 whose only divisors are 1 and itself: 2, 3, 5, 7, 11, 13, … Every other number breaks into primes, like 60 = 2 × 2 × 3 × 5. The fundamental theorem of arithmetic says that this breakdown is always possible and always unique, apart from the order of the factors. Primes are the atoms of multiplication.

Do they run out? Euclid's proof that they don't is one of the most famous arguments ever written. Suppose you had a complete list of primes. Multiply them all together and add 1. The result leaves remainder 1 when divided by any prime on your list. So its prime factors are missing from the list, and the list wasn't complete.

The primes thin out as numbers grow, but slowly and with a definite pattern. Near a number N, roughly one number in ln N is prime. This is the prime number theorem, proved in 1896.

The Pythagoreans believed everything was ratios of whole numbers. Then someone found that the diagonal of a square with side 1, which has length √2, is not one.

The proof is short. Suppose √2 = p/q with the fraction in lowest terms. Squaring gives p² = 2q², so p² is even, so p is even. Write p = 2k. Then 4k² = 2q², so q² = 2k², and q is even too. Both are even, so the fraction wasn't in lowest terms after all. That is a contradiction, so no such fraction exists.

Irrational numbers are everywhere: √2, π, e, the golden ratio φ. Most real numbers are irrational, in a sense idea 12 makes precise. What fractions can do is approximate them. The lab finds the best fraction for every denominator. Some numbers, like π, have unusually good approximations (355/113 is correct to six decimal places). The golden ratio has the worst ones of any number, so in a precise sense it is the "most irrational."

Science believes things because evidence piles up. Mathematics asks for more: a chain of steps, each one following from the last, that starts from agreed assumptions and ends at the claim. Once a theorem is proved it stays proved. Euclid's theorems are as true today as in 300 BCE.

The Pythagorean theorem says that in a right triangle with legs a and b and hypotenuse c, a² + b² = c². Measuring a thousand triangles would make you confident. The proof in the lab makes you certain. Put four copies of the triangle in a square of side a + b. Arranged one way, they leave a hole of area c². Slide them around and they leave two holes of areas a² and b². The big square and the four triangles didn't change, so neither did the total area of the holes.

Draw two perpendicular number lines. Every point in the plane now has an address (x, y). A curve is just the set of points whose addresses satisfy an equation. The equation y = x² is a parabola, and x² + y² = 1 is a circle.

Geometric questions become algebra. Where does the parabola cross the x-axis? Set y = 0 and solve. How many crossings are there? The discriminant b² − 4ac tells you: positive means two, zero means the curve just touches, negative means it misses entirely. You can see the answer and compute it, and the two always agree.

This bridge works in both directions. Algebra gives geometry its calculating power, and pictures give algebra its intuition. Calculus, physics and every graph in this guide are built on it.

A square has eight symmetries: four rotations (by 0°, 90°, 180°, 270°) and four flips. Do one symmetry and then another, and the result is again one of the eight. Every move can be undone, and "do nothing" counts as a move. Any collection of moves with these properties is called a group.

The surprise is that order matters. Rotate then flip, and you usually get a different result from flip then rotate. Group theory is the algebra of things that don't commute, and it shows up wherever there is symmetry: crystals, molecules, the Rubik's cube, particle physics, and the question of which equations can be solved by a formula (see the graduate edition).

The lab lets you apply moves to a labelled shape. Whatever sequence you try, the result is always one of the 2n symmetries shown below the shape.

Linear growth adds the same amount each step: 100, 110, 120, 130. Exponential growth multiplies by the same factor: 100, 110, 121, 133.1. At first they look alike. Then the exponential pulls away and never comes back. Compound interest, unchecked epidemics, bacteria and computer chips have all grown this way for a while.

Something growing at r% per step doubles roughly every 72/r steps, whatever its starting size. That steady doubling is the signature of exponential growth.

The logarithm is the inverse: it asks "how many doublings (or tenfoldings) does it take to get here?" Plot exponential growth on a logarithmic scale and it becomes a straight line whose slope is the growth rate. Logarithms turn multiplication into addition, which is why scientists used them to do arithmetic for 350 years before calculators.

Walk halfway to a wall, then half the remaining distance, then half again. You take infinitely many steps, yet the total distance is exactly 1. In symbols, ½ + ¼ + ⅛ + … = 1. The sum converges: its running totals get as close to 1 as you want and stay there. That target is called the limit.

Shrinking pieces aren't enough. The harmonic series 1 + ½ + ⅓ + ¼ + … has pieces that shrink to zero, but its running total grows without bound. It grows very slowly: you need about 12,000 terms to pass 10, and more terms than there are atoms in the universe to pass 200.

Some infinite sums land on surprising targets. The squares of the reciprocals add up to π²/6. The alternating odd fractions give π/4. Circles appear in sums that have nothing to do with circles.

A car's average speed is distance divided by time. But what does the speedometer mean when it says 60 km/h at one instant, when zero time has passed and the car has covered zero distance? 0/0 is meaningless.

The answer is a limit. Take the average rate of change over a short stretch h, then let h shrink toward zero. On a graph the average rate is the slope of a secant line through two points of the curve. As the points merge, the secant turns into the tangent, and its slope is the derivative.

The derivative turns "how fast is this changing?" into something you can calculate. For x² it is 2x, for sin x it is cos x, and for e^x it is e^x itself. Velocity, marginal cost, the rate of a reaction and the gradient a neural network follows during training are all derivatives.

How much area lies under a curve? Chop it into thin vertical strips, approximate each by a rectangle, and add them up. More strips give a better answer, and the limit as the strips get infinitely thin is the integral. The same idea computes distance from speed, volume from cross-sections and total probability from a density.

The fundamental theorem of calculus connects this to idea 9. Let A(x) be the area from 0 up to x. Nudge x a little and the area grows by a thin strip of height f(x). So the rate at which the area grows is f(x) itself: A ′ = f. To find an area, you don't have to add infinitely many strips. You only need a function whose derivative is f.

The city of Königsberg had seven bridges joining two islands and two riverbanks. Residents wondered whether they could walk through the city crossing every bridge exactly once. Nobody could find a route, but nobody could prove there wasn't one.

Leonhard Euler saw that the map didn't matter. Shrink each land mass to a dot (a vertex) and each bridge to a line (an edge). The result is a graph. Now think about any vertex that is neither the start nor the end of the walk: every time you arrive you must leave, so its edges pair up and it needs an even number of them. A walk can therefore have at most two vertices with an odd number of edges. In Königsberg all four were odd, so no such walk exists.

Euler's argument began graph theory, and also topology, the study of properties that survive stretching. Graphs now model road maps, the internet, social networks, molecules and the links between web pages that search engines rank.

How do you know two sets have the same size without counting? Pair them off. If every chair has exactly one person and every person has a chair, there are as many people as chairs. Georg Cantor applied this test to infinite sets, with strange results. The even numbers pair up with all whole numbers (2↔1, 4↔2, 6↔3, …), so there are "as many" of each. So do the fractions, using a clever zigzag listing.

Then Cantor showed that the real numbers can't be listed. His diagonal argument works on infinite sequences of 0s and 1s, which are essentially real numbers written in binary. Suppose someone hands you a list that supposedly contains every sequence. Build a new sequence by flipping the 1st digit of row 1, the 2nd digit of row 2, the 3rd digit of row 3, and so on. Your sequence differs from every row in at least one place, so it isn't on the list. Every list misses something.

So there are at least two sizes of infinity: countable (the size of the whole numbers) and the strictly larger size of the real numbers. The same diagonal trick shows there's no largest infinity at all, and in other forms it powers Gödel's incompleteness theorem and Turing's proof that some problems can't be solved by any computer.