Gauss

The life of Carl Friedrich Gauss

Timeline 1777Born on 30 April, in Braunschweig, son of a gardener and bricklayer. 1786Around 9 years, sum of 1 to 100 in seconds at school. 1791Duke Carl Wilhelm Ferdinand starts paying for his studies. 1795Enter the University of Göttingen. 1796In March 30, discover how to construct the regular polygon with 17 sides. Start your math journal. 1799Doctorate from Helmstedt, with proof of the fundamental theorem of algebra. 1801Publishes the Disquisitiones Arithmeticae and calculates the orbit of the dwarf planet Ceres. 1807He became director of the Göttingen Observatory, where he remained until his death. 1809Publishes the Theoria motus, with the least squares method and the normal curve. 1818Start measuring the Kingdom of Hannover with triangles. In 1821 he invents the heliotrope. 1827Publishes the Theorema Egregium, on the curvature of surfaces. 1833With Wilhelm Weber, he set up one of the first electric telegraphs. 1855Dies on 23 February, in Göttingen. ◀ Previous Follow the journey Next ▶ 1777–1795 BrunswickThe son of a gardener and bricklayer, he adds 1 to 100 in seconds at school. Duke Carl Wilhelm Ferdinand starts paying for his studies. 1795–1798 GöttingenHe studies at the university. On 30 March 1796 he discovers how to construct the regular 17-gon and begins his diary. 1799 HelmstedtHe takes his doctorate with a proof of the fundamental theorem of algebra. 1799–1807 BrunswickHe publishes the Disquisitiones Arithmeticae (1801) and computes the orbit of Ceres. Here he receives the letters of “Monsieur Le Blanc”. 1807–1855 Göttingen ObservatoryHe directs the observatory for almost fifty years. He surveys the Kingdom of Hanover with triangles, publishes the Theorema Egregium and builds a telegraph with Weber. He dies in 1855. Enable JavaScript to see the map. The stages are listed below. Gauss hardly travelled. He spent his life within a hundred kilometres of Braunschweig and Göttingen. Map: © OpenStreetMap contributors

October 10, 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 10, 2026 Read

The 17 sided polygon

Since the Greeks, it was known how to construct regular polygons with 3, 4, 5 and 15 sides with a ruler and compass, and all that come out of them doubling the number of sides (6, 8, 10, 12…). For 2000 years, no one found another. Gauss showed that constructing the $n$ sided polygon is the same as calculating $\cos\frac{2\pi}{n}$ using only integers, the four operations and square roots. And ruler and compass know how to take root square: it is exactly the construction that we saw in the chapter on Descartes. For 17 sides, he reached

October 10, 2026 Read

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 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

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

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

The normal curve, Gauss's bell

If measurement errors are random, how are they distributed? In Theoria motus, Gauss started from the idea that the mean is the best estimate and arrived at the shape of the error curve: $$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2\sigma^2}},$$ where $\mu$ is the mean and $\sigma$ is the standard deviation. It is the normal curve, or Gaussian bell: high in medium, symmetrical, and increasingly lower away from the mean. It appears everywhere, from the heights of the people to the errors of a machine.

October 10, 2026 Read

The Egregium Theorem and curvature

Measuring Hannover, Gauss thought a lot about how to make maps of a curved Earth. In 1827 published the result he called Theorema Egregium, the “remarkable theorem”: the curvature of a surface can be measured by those who live inside it, without going out to look from the outside. A consequence: in a plane, the angles of a triangle add up to 180°. In a sphere, they add up to more. Take the triangle with one vertex at the north pole and the other two at the equator:

October 10, 2026 Read

Systems of equations: the determinant and Gaussian elimination

To calculate orbits using least squares, Gauss needed to solve systems of equations with many unknowns. Two names that every student knows come from this story: the determinant and the Gaussian elimination. The determinant. Take a system with two equations: $$\begin{cases} a\,x + b\,y = e \\ c\,x + d\,y = f \end{cases} \quad\Longrightarrow\quad x = \frac{e\,d - b\,f}{a\,d - b\,c}, \qquad y = \frac{a\,f - e\,c}{a\,d - b\,c}.$$ It all depends on the number $ad - bc$ in the denominator: if it is not zero, the system has a single solution. This number “determines” the system.

October 10, 2026 Read

Gaussian quadrature: integrate with two points

Quadratura circuli: approximating the area of the circle by polygons, the idea that leads to integral calculus.Euler’s Eyes Collection “Squaring” is the old name for calculating an area. The Greeks approximated the area of the circle by polygons with increasingly more sides, and integral calculus was born. The practical problem is: how to calculate the area under a curve doing few math?

October 10, 2026 Read

Other contributions and legacy

Other contributions Physics: Gauss’s law of electromagnetism, the study of Earth’s magnetism and the Gauss unit. Complex numbers: Gaussian integers, numbers $a + bi$ with $a$ and $b$ integers. Astronomy and geodesy: orbits of planets and comets, the heliotrope and the Hannover measurement. Algorithms: a method for calculating the date of Easter and, in a manuscript by 1805, a version of the fast Fourier transform, rediscovered in 1965 and used today in every cell phone. To remember Gauss was born poor in Braunschweig, studied under a duke and spent his life in Göttingen. Added from 1 to 100 by joining pairs: $1 + 2 + \cdots + n = \frac{n(n+1)}{2}$. At 18 years he showed how to construct a polygon with 17 sides, using only square roots. He created the congruence notation, $a \equiv b \pmod n$, the mathematics of the clock and nines-out. He proved that every polynomial of degree $n$ has $n$ complex roots. Estimated the number of primes: $\pi(x) \approx x / \ln x$. He rediscovered Ceres with the least squares method, which shows why the average is the best estimate. He gave the name to the normal curve, the Gaussian bell. He showed that curvature can be measured from within the surface: this is why there is no perfect flat map. He named the determinant and organized the elimination that solves systems of equations. With two well-chosen points, $\pm 1/\sqrt{3}$, he calculated exact integrals for polynomials up to degree 3. All topics by Carl Friedrich Gauss · ← Gaussian quadrature: integrate with two points · Image credits →

October 10, 2026 Read