## Carl Gauss (1777 - 1855)

Carl Gauss was a mathematical child prodigy. According to mathematical legend he was only 7 when he was given the task of finding the sum of the integers from 1 to 100. While the rest of the class began the laborious task of adding all these numbers up manually, Gauss showed his mathematical ingenuity. He worked like this:

first he wrote the sum out

1 + 2 + 3 + … + 98 + 99 + 100

then he wrote out the sum in reverse order,

100 + 99 + 98 + … + 3 + 2 + 1

now for the stroke of genius. He added both sums together, to get,

101 + 101 + 101 + … + 101 + 101 + 101

But this sum is twice the sum required, and Gauss had his answer, a half of 100.101 = 5050.

By the age of 15 he discovered Bode’s Law (although this was known already he discovered it independently), the binomial theorem, the law of quadratic reciprocity and the prime number theorem. The latter two are so advanced that students even today only come across them during the later parts of a degree course in mathematics. The Binomial Theorem was first discovered by Newton, but Gauss discovered it for himself.

Later in his life, Gauss was to play an important role in helping a female mathematician, Sophie Germain. His help ensured that she received her well deserved doctorate.

In 1801, Gauss accurately predicted the position of a newly discovered ‘small planet’ called Ceres, which had disappeared behind the sun after only 9 degrees of its orbit. Gauss achieved this incredible feat using his least squares approximation method.

His father died in 1808 and in 1809 his wife died giving birth to their second son, who died shortly after. Gauss was devastated. He remarried a year later to his late wife’s best friend, although by all accounts it was not a happy marriage.

He continued to work, publishing a major dissertation on the motion of celestial bodies, discussing differential equations, conic sections, elliptic orbits, as well as how to accurately estimate a planet’s orbit.

At this time Gauss began to take an interest in a branch of mathematics called geodesy. He took charge of a detailed geodesic survey of the state of Hanover, for which he invented the heliotrope. This used a network of mirrors and a telescope to reflect the sun’s rays to gauge positions and angles.

Along with Weber, Gauss studied the theory of terrestrial magnetism, going on to show it is only possible to have two poles on the earth, and actually specified the location for the magnetic South Pole. They went on to discover Kirchoff’s laws, and built a kind of basic telegraph which could send messages over 5000 feet. However, this was merely a diversion for Gauss. He went on to set up a network of magnetic observation points around the world, and published the atlas of geo-magnetism.

Weber left Göttingen in 1837, and from then on Gauss became less active in mathematics, and took on the management of the University’s widow fund. He continued to correspond with various mathematicians, and followed the work of Lobachevsky, and approved Riemann’s thesis. He died in his sleep on 23 February, 1855.

In just about every sphere of mathematics Gauss left a deep mark.

## Gauss’ mathematics

He was constantly discovering and exploring mathematical ideas, and at university he discovered a construction for a regular 17-gon using only a ruler and compasses.

Hitherto, only constructions for the equilateral triangle, the square, the pentagon and the septagon were known. Gauss gave the definitive result that describes such constructions:

a regular polygon with a prime number of sides can be constructed if - and only if - that prime number is of the form, 2^{2^n}+1.

This means, since we can bisect an angle, that regular polygons with the number of sides that is any even multiple of such numbers can also be constructed. For example:

n=2 gives 2^2+1=17 which is prime and so a 17 sided regular polygon may be constructed; so also may be regular polygons with 2.17=34 sides, 3.257=51 \ldots

Here is a construction for the regular pentagon (n=1, 2^{2^1}+1=5)

• Construct two perpendicular diameters of a circle - label the end of one of them, P_1;
• Construct the mid-point of one of the two halves of the other diameter;
• Join this mid -point to P_1, Bisect the angle shown;
• Where this bisector intersects the other diameter, construct a perpendicular;
• Where this perpendicular intersects the circle, mark as P_2.
• P_1P_2 is a side of the required pentagon. Mark off the other sides with a compass.

For his doctoral dissertation, Gauss proved an incredibly important result - the so-called Fundamental Theorem of Algebra. In essence, this says that a linear equation has just one root, a quadratic equation has two, and if the degree of the equation is n, the equation has n roots.

In later life a friend would send his son to study under him - Bolyai pursued a new form of geometry, but unknown to him and the rest of the world, Gauss had already anticipated much of this work, but had not published it.

Another major contribution that Gauss made was the introduction of a theory of congruences - or as it is sometimes called, clock arithmetic.

This is a crucial tool in number theory, and generalises the idea of odd and even numbers. Proofs in number theory rely heavily on it - for example the proof that \sqrt2 is not rational.

Regarding whole numbers as either odd or even splits them up into two classes; but we may split them up into three classes - what is their remainder when divided by 3? It must be either 0 (it divides exactly) 1 or 2.

Similarly we can split them up into 4 classes, 5 classes, and indeed any number of classes.
An example of this in use concerns the squares: no square leaves a remainder of 2 when divided by 3. Here’s the proof by considering whole numbers split up into three classes:

 Remainder when n is divided by 3 0 1 2 Remainder when n^2 is divided by 3 0 1 1

(The last entry is because 2^2=4 and 4 leaves a remainder of 1 when divided by 3.)