Hero Image
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 1, 2026 Read
Hero Image
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,

September 30, 2026 Read
Hero Image
The number $e$ and compound interest

1 2 3 4 Explore the scene: tap the numbers. With a keyboard, use Tab and Enter or the arrow keys. 1. Customer: initial capital The person in front of the counter represents someone bringing money for a transaction. In this article’s example, we start with 100 units of money: this is the initial capital. 2. Service: checking the accounts The person behind the counter represents service and checking amounts. To follow an investment in our model, we need the capital, the rate and the number of periods. 3. Coins: interest on interest In the mathematical example at 100% a year, one annual payment turns 100 into 200. With two half-year periods at 50%, 100 becomes 150, then 150 becomes 225: the second payment also earns interest on the earlier interest. 4. Cash box: saving is not compounding Stored money does not grow by itself. The interest in this example follows a mathematical payment rule. Explore the chart below to compare annual, monthly and increasingly frequent payments. Try compound interest ↓ Enlarge the engraving Banca, trapeza, mensa: the table of the first bankers, between 1450 and 1650.Illustrative reconstruction with AI based on an engraving from the Euler’s Eyes collection

September 29, 2026 Read
Hero Image
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
Hero Image
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
Hero Image
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
Hero Image
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 1, 2026 Read
Hero Image
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 $k_4 = k$:

September 30, 2026 Read
Hero Image
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. flowchart LR phenomenon((FENÔMENO)) --> equation["EQUAÇÃODIFERENCIAL"] equation --> solution["FUNÇÃOSOLUÇÃO"] solution --> plottedGraph((GRÁFICO)) plottedGraph --> derivative{"DERIVADA"} derivative --> phenomenon derivative -. "localiza" .-> extrema["Pontos de máximoe de mínimo"] classDef phenomenonStyle fill:#e6f1ee,stroke:#397c75,color:#243448,stroke-width:2px classDef equationStyle fill:#eaf0f7,stroke:#58799b,color:#243448,stroke-width:2px classDef solutionStyle fill:#f8efd9,stroke:#b68b3c,color:#243448,stroke-width:2px classDef derivativeStyle fill:#f5e9e3,stroke:#9b6251,color:#243448,stroke-width:2px class phenomenon phenomenonStyle class equation,plottedGraph equationStyle class solution solutionStyle class derivative,extrema derivativeStyle 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 $F = -kx$.

September 29, 2026 Read
Hero Image
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 1, 2026 Read
Hero Image
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.)

September 30, 2026 Read
Hero Image
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.

September 29, 2026 Read