History of Mathematics

Algebra done with squares and rectangles

If a straight line is cut at random, the square on the whole equals the squares on the segments plus twice the rectangle contained by the segments. — Euclid, Elements, Proposition II.4 The Greeks had no letters for unknowns, no plus or minus signs, no equals sign. When they wanted to speak of $a\cdot b$, they spoke of a rectangle with sides $a$ and $b$; when they wanted $a^2$, they spoke of a square with side $a$. The words “square” and “cube” that we use for powers come from this. Book II of the Elements is made entirely in this way: it consists of identities of algebra, stated and proved as facts about areas. Historians call it geometric algebra.

October 12, 2026 Read

E = mc²: the energy hidden in mass

One more consequence of the paper on electrodynamics has occurred to me. The principle of relativity, together with Maxwell’s equations, requires that mass be a direct measure of the energy contained in a body. The idea is amusing and enticing; but for all I know, the Lord may be laughing over the whole thing and leading me around by the nose. — Albert Einstein, letter to Conrad Habicht (summer 1905) In September 1905, three months after the relativity paper, Einstein sent the Annalen der Physik a supplement of just three pages, with a title in the form of a question: “Does the inertia of a body depend upon its energy content?”. The answer was yes, and the conclusion, written in words and with other letters, is the most famous formula in physics:

October 12, 2026 Read

The art of conjecturing: counting the chances

To conjecture about something is to measure its probability. Therefore we define the art of conjecturing, or stochastics, as the art of measuring the probabilities of things as exactly as possible, so that in our judgements and actions we may always choose or follow what seems to us better, more satisfactory, safer or more carefully considered. — Jakob Bernoulli, Ars conjectandi (1713), part IV In 1654, Pascal and Fermat exchanged letters on how to divide the stakes of an interrupted game. In 1657, the Dutchman Christiaan Huygens published a fifteen-page booklet on the calculations of dice games. These were parlour problems: dice, cards, bets.

October 11, 2026 Read

The Pascaline: a sum made of gears

In 1640 Étienne Pascal was appointed tax commissioner for Normandy. The job meant adding up column after column of figures late into the night, and his son Blaise, then 17, helped. To make matters worse, French money was not decimal: $$1\ \text{livre}=20\ \text{sols},\qquad 1\ \text{sol}=12\ \text{deniers}.$$Adding $3$ livres, $15$ sols and $8$ deniers to $2$ livres, $7$ sols and $9$ deniers needs three kinds of carry: deniers: $8+9=17$, which is $1$ sol and $5$ deniers; sols: $15+7+1=23$, which is $1$ livre and $3$ sols; livres: $3+2+1=6$. Result: $6$ livres, $3$ sols and $5$ deniers. Do that a thousand times a night and you will make mistakes. In 1642 Blaise began building a machine that would not.

October 8, 2026 Read

Fermat numbers: a conjecture and a counterexample

Mathematicians have always dreamed of a formula that produces only prime numbers. In 1640 Fermat thought he had found one. Five primes in a row Fermat studied numbers of the form $$F_n=2^{2^n}+1.$$The exponent is a power of 2: $1,2,4,8,16,\ldots$ Computing the first few: $$F_0=3,\quad F_1=5,\quad F_2=17,\quad F_3=257,\quad F_4=65{,}537.$$All five are prime. Checking $65{,}537$ by hand, trying divisors up to its square root, was already hard work. Fermat wrote to Frénicle de Bessy that he was convinced all numbers of this form were prime, although he admitted he had no proof.

October 7, 2026 Read

Binary arithmetic: writing numbers with 0 and 1

Why do we count in tens? Probably because we have ten fingers. There is nothing special about the number 10: any number can be written in any base. Leibniz took this idea to the extreme and asked: what is the smallest number of digits that can write everything? The answer is two: 0 and 1. Each place is worth double The numbers 0 to 32 written in binary, in the table from the paper Explanation of Binary Arithmetic, which Leibniz published in 1703 in the memoirs of the Paris Academy of Sciences.G. W. Leibniz (1703) · Wikimedia Commons · public domain In the decimal system the places are worth 1, 10, 100, 1000… and each digit says how many times we use that value. In the binary system the places are worth 1, 2, 4, 8, 16, 32… and each digit can only say yes (1) or no (0).

October 6, 2026 Read

Newton's method: approximating roots with tangents

Some equations have no formula. There is no quadratic formula for $e^x+x-3=0$, and the formula for cubic equations is so laborious that hardly anyone uses it. In these cases the way forward is to approximate: guess a value, see how far off we are, and correct the guess. The question is how to correct it well. Newton’s example In De analysi, of 1669, Newton chose the equation $$y^3-2y-5=0.$$By trial he saw that the root is near $2$, since $2^3-2\cdot2-5=-1$. So he wrote $y=2+p$, where $p$ is the missing correction, and substituted:

October 6, 2026 Read

The fundamental theorem of algebra

In his doctoral thesis, in 1799, Gauss proved that every polynomial of degree $n$ has $n$ roots, as long as complex numbers are accepted. For example, $x^2 + 1 = 0$ has no real solution, but has two complex ones: $$x^2 + 1 = (x - i)(x + i).$$ Gauss also popularized the way of drawing complex numbers as points on a plane, the plane complex: $a + bi$ is the point $(a, b)$. Norwegian Caspar Wessel (1797) and Swiss Jean-Robert Argand (1806) had the same idea, but almost no one read their work; was the fame of Gauss, from 1831, who spread the drawing. That’s why it is also called Argand-Gauss plane. And then a nice connection appears with the polygon with 17 sides. The three roots of $x^3 = 1$ are

October 1, 2026 Read

The angles of polyhedra and Euler's formula

At one vertex of a cube there are three squares, and their angles add up. $3 \times 90° = 270°$. There are $90°$ left to complete one turn: this is the defect of the vertex. By Around 1630, Descartes discovered that, in a convex polyhedron, the sum of the defects of all vertices is always 720°. Polyhedron Angles at a vertex Defect Vertices Sum Tetrahedron 3 × 60° = 180° 180° 4 720° Cube 3 × 90° = 270° 90° 8 720° Octahedron 4 × 60° = 240° 120° 6 720° Dodecahedron 3 × 108° = 324° 36° 20 720° Icosahedron 5 × 60° = 300° 60° 12 720° The faces around a vertex, open in the plane. The red slice is the defect, the angle left to complete 360°. Ative o JavaScript para ver esta figura interativa.

September 30, 2026 Read

Euler's formula and the "most beautiful equation"

The number $e$ can be written as a series: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots$$ Euler had the courage to put an imaginary number in the exponent. Changing $x$ to $i\theta$ and using $i^2 = -1$, $i^3 = -i$, $i^4 = 1$: $$e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2!} - i\frac{\theta^3}{3!} + \frac{\theta^4}{4!} + i\frac{\theta^5}{5!} - \cdots$$ Separating terms without $i$ from terms with $i$:

September 29, 2026 Read

Cauchy's inequality

The 1821 Cours d’analyse ends with nine notes, a kind of appendix with results Cauchy did not want to put in the main text. The second, on inequalities, contains two gems that every olympiad student knows. The inequality Cauchy’s theorem says that, for any real numbers $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$, $$(a_1 b_1 + a_2 b_2 + \cdots + a_n b_n)^2 \le (a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + b_2^2 + \cdots + b_n^2),$$with equality only when the two lists are proportional.

October 31, 2026 Read

The zeta function

The function that bears Riemann’s name was studied earlier by Euler. For a real number $s$ greater than 1, the series $$\zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \cdots = \sum_{n=1}^{\infty}\frac{1}{n^s}$$converges. Euler computed several of its values, such as $\zeta(2) = \frac{\pi^2}{6}$, the solution of the Basel problem. Euler’s product Euler’s deepest discovery, of 1737, was that the same function can be written as a product over the prime numbers:

October 31, 2026 Read