History of Mathematics

The generalized binomial: beyond whole-number powers

At school we learn to expand $(1+x)^2=1+2x+x^2$ and $(1+x)^3=1+3x+3x^2+x^3$. The coefficients come from the arithmetic triangle, which Pascal studied and which was known long before him. For positive whole-number exponents, the expansion always stops. In the winter of 1664–1665, still a student, Newton asked a question nobody had asked so boldly: what if the exponent were $\frac12$? Or $-1$? What would $(1+x)^{1/2}$, that is $\sqrt{1+x}$, look like as a sum of powers of $x$?

October 6, 2026 Read

The sum of 1 to 100

Gauss’s most famous story is from when he was about 9 years old. To keep the class busy, the Professor Büttner told the students to add up all the numbers from 1 to 100. In a few seconds, Gauss placed the blackboard on the table with the correct answer: 5050. (The story was told by the first biographer of Gauss, and the exact numbers in the account may have been different, but the idea is the same.)

October 1, 2026 Read

Coordinates: geometry becomes algebra

The big idea of Geometry is to give each point on the plane a pair of numbers. We choose two lines perpendiculars (the axes) and a point is located by the distances $x$ and $y$ to them. Today We call this Cartesian plane, in homage to him (Cartesius was Descartes’ name in Latin). With coordinates, a figure becomes an equation. See three examples. A page from The Geometry (1637). The letters a, b appear for known values and the powers written at the top.René Descartes · public domain The distance between two points. The points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ form, with the point $(x_2, y_1)$, a right triangle of legs $x_2 - x_1$ and $y_2 - y_1$. By the theorem of Pythagoras:

September 30, 2026 Read

The language that Euler left us

Part of Euler’s influence is in things that seem to have always existed. He created or popularized: $f(x)$ to indicate a function of $x$; the letter $e$ for the base of natural logarithms, $e \approx 2{,}71828$; the letter $\pi$ for the ratio between the length of the circumference and the diameter (the symbol already existed, but it was Euler who spread it); $i$ for the imaginary unit, with $i^2 = -1$; $\Sigma$ to indicate a sum. The equals sign and the letter π Symbols have birthdays. The sign $=$ was invented by the Welshman Robert Recorde in 1557, in the book The Whetstone of Witte, so as not to have to write “is equal to” all the time. He chose two parallel strokes of the same length because, as he explained, no two things can be more equal. The letter $\pi$, the initial of the Greek word periphéreia, the circumference, was first used in today’s sense by another Welshman, William Jones, in 1706. Euler began using it in the 1730s, and his books made it standard.

September 29, 2026 Read

The rearrangement theorem

Since school we learn that “the order of the terms does not change the sum”. For finite sums, it is true. For infinite sums, it is not, and the clearest example appears in the same habilitation work of 1854 in which Riemann defined the integral. A series that adds up to ln 2 The alternating harmonic series $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots$$converges, and its sum is $\ln 2 \approx 0.6931$. But there is something fragile about it. If we separate the positive terms from the negative ones, each group on its own is infinite:

October 31, 2026 Read

The nebular hypothesis

The Exposition of the System of the World, of 1796, was a book for readers without mathematical training. In it Laplace told, in prose, everything celestial mechanics had discovered. At the end, in a note that he himself presented “with the distrust that everything not a result of observation or calculation should inspire”, he ventured an explanation for the origin of the Solar System. Title page of the second edition of the Exposition of the System of the World, 1798 (year VII of the republican calendar).Pierre-Simon Laplace · Wikimedia Commons · public domain Too great a coincidence Laplace started from facts everyone knew. The planets then known go around the Sun in the same direction, in orbits almost in the same plane and almost circular. They also spin on themselves in the same direction, and the same holds for the satellites around the planets. Adding up planets, satellites and rotations, Laplace counted 43 motions, all in the same direction.

October 30, 2026 Read

The law of the lever

Levers had been used long before Archimedes, to lift stones, draw water from wells and weigh goods. What Archimedes did that was new was to turn the lever into a theorem. In the treatise On the Equilibrium of Planes, he starts from a few simple axioms, such as “equal weights at equal distances balance”, and deduces the general law in the manner of the Elements of Euclid. The law Magnitudes, whether commensurable or incommensurable, balance at distances inversely proportional to their weights. — Archimedes, On the Equilibrium of Planes, book I, propositions 6 and 7

October 29, 2026 Read

The Instituzioni analitiche

In the middle of the eighteenth century, anyone who wanted to learn calculus had a problem. Newton and Leibniz’s invention was only seventy years old, and its results were scattered across papers in scientific journals, letters and books written in Latin, French and English, in different notations. The only famous textbook, the Marquis de L’Hôpital’s Analysis of the Infinitely Small, of 1696, dealt only with differential calculus. There was no book that led the student, step by step, from algebra to differential equations.

October 28, 2026 Read

Determinants and permutations

On 30 November 1812, the young engineer from Cherbourg read to the Institut de France a memoir with a long title: on functions that can take only two values, equal and of opposite signs, when the variables are interchanged. Behind the title was the first general theory of determinants. Functions that change sign Take three variables $a_1, a_2, a_3$ and the product of the differences $$\Delta = (a_2 - a_1)(a_3 - a_1)(a_3 - a_2).$$If we swap $a_1$ and $a_2$, each factor either stays the same or changes sign, and altogether $\Delta$ becomes $-\Delta$. Any swap of two variables does the same. A permutation of the variables, which is a sequence of swaps, multiplies $\Delta$ by $+1$ or by $-1$: it is even or odd.

October 27, 2026 Read

Complex numbers as couples

By the early nineteenth century complex numbers had been in use for three hundred years, since Cardano and Bombelli, and Euler had worked wonders with them. Wessel, Argand and Gauss had drawn them as points of the plane. But the unease remained: what, after all, is $\sqrt{-1}$? A number whose square is negative seemed a useful fiction, not an object of mathematics. Hamilton settled the question without drawing anything. In the essay Theory of Conjugate Functions, or Algebraic Couples, presented to the Royal Irish Academy in 1833 and 1835 and published in 1837, he proposed that a complex number is an ordered couple of real numbers, and nothing more.

October 27, 2026 Read

The minimax theorem: how to play without being second-guessed

In odds and evens there is no good move. If you always show one finger, your opponent learns and starts winning every time. The only defence is not to be predictable: to play at random, one half of the time, two the other half. Then, whatever your opponent does, you win half the games. In 1928, in a paper called On the theory of parlour games, von Neumann showed that this idea holds for every two-person zero-sum game, in which what one wins the other loses. And that it always leads to an exact number, the value of the game.

October 26, 2026 Read

The halting problem: the question no machine can answer

In 1928 the German David Hilbert set a challenge he called the Entscheidungsproblem, the “decision problem”: find a mechanical procedure that, given any mathematical statement written in the language of logic, would answer in finite time whether it is true or false. If it existed, mathematics would become an assembly line: just write down the question and turn the handle. Hilbert believed the answer was yes. Eight years later, Turing and the American Alonzo Church showed, each by his own route, that it is no.

October 18, 2026 Read