Biography

Clock arithmetic

If it is 15h now, what time will it be in 10 hours? It’s not “25 h”, but 1 am: on the clock, after 24 the count starts again. Gauss gave a name and symbol to this idea in Disquisitiones: $$25 \equiv 1 \pmod{24}$$ It reads “25 is congruous to 1, modulo 24”: both leave the same remainder when divided by 24. The symbol $\equiv$ works almost like an equals sign: you can add and multiply on both sides.

October 10, 2026 Read

The rule of signs

How many positive roots can a polynomial have? Descartes gave a simple rule: the number of roots positive is at most the number of sign changes between consecutive coefficients (and, as proved then differs from it by an even number). See with $x^3 - 6x^2 + 11x - 6$. The signs of the coefficients are $+\ -\ +\ -$: three changes. So there is 3 or 1 positive roots. In fact,

October 10, 2026 Read

The number $e$ and compound interest

Banca, trapeza, mensa: the table of the first bankers, between 1450 and 1650. Hence the words bank and banking come from.Euler’s Eyes Collection Before reaching Euler, the number $e$ passed through the banks. The vocabulary of money comes from Latin: depositum (what was put, from ponere), creditum (what was trusted, from credere, to believe) and juro (from jus, juris, the right).

October 10, 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 10, 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.

October 10, 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 $i$:

October 10, 2026 Read

Counting prime numbers

Euler proved that the primes never end (see his chapter). But how many are there? At 15 years old, Gauss He spent his free time counting cousins in tables, one thousand at a time. Did you notice that, close to a number $x$, more or less 1 in every $\ln x$ number is prime. Hence the estimate: the number of primes up to $x$, which we call $\pi(x)$, is similar to

October 10, 2026 Read

The kissing circles theorem

In 1643’s letter to Princess Elisabeth, Descartes dealt with four tangent circles two by two. Call the curvature of a circle the inverse of the radius, $k = \frac{1}{r}$. The theorem says that the curvatures satisfy $$(k_1 + k_2 + k_3 + k_4)^2 = 2\left(k_1^2 + k_2^2 + k_3^2 + k_4^2\right).$$ Application: three equal coins of radius 1 touch each other two by two. What is the radius of the circle that fits in the hole between them? With $k_1 = k_2 = k_3 = 1$ and $k_4 = k$:

October 10, 2026 Read

The spring and differential equations

A differential equation is an equation in which the unknown is a function, and the equation speaks of its derivatives. Euler organized this path into a scheme that we still use today: observe a phenomenon, write the equation, find the function that solves it and draw the graph. Euler’s scheme: phenomenon → differential equation → solution function → graph. The derivative also points to the maximum and minimum points.Euler’s Eyes Collection The phenomenon. In 1678, Robert Hooke summarized the behavior of a spring in Latin: ut tensio, sic vis, “as extension, so strength.” A spring of constant $k$, stretched by $x$, pulls back with strength $F = -kx$.

October 10, 2026 Read

Ceres and the least squares method

On January 1, 1801, astronomer Giuseppe Piazzi, in Palermo, discovered a tiny dot that moved among the stars: it was Ceres, today classified as a dwarf planet. Piazzi accompanied him for 41 days, until it disappeared in the Sun’s glare. With so few observations, no one knew how to calculate the orbit, and Ceres seemed lost forever. Gauss, at 24 years old, created a new method and calculated the orbit in a few weeks. On the night of 31 December 1801, astronomer Franz von Zach pointed the telescope where Gauss indicated, and there it was Ceres.

October 10, 2026 Read

Why does the rainbow appear at 42°

In The Meteors, Descartes explained the rainbow using the law of refraction, which he published in The Dioptric. When light passes from air to water, it bends $$\sin i = n \cdot \sin r,$$ where $i$ is the entry angle, $r$ the angle into the water and $n \approx \frac{4}{3}$ for the water. (The Dutchman Willebrord Snell had found the same law before, but did not publish it. That is why it is called Snell-Descartes law.)

October 10, 2026 Read

The Königsberg bridges and the birth of graphs

Königsberg at 1652, with the river Pregel and its islands. You can count the bridges.Merian-Erben · public domain The city of Königsberg (today Kaliningrad, Russia) was crossed by the Pregel River, which formed two islands. Seven bridges connected the four regions of the city. Residents wondered if it was possible take a walk across each bridge exactly once.

October 10, 2026 Read