I decided to start a new series on influential mathematicians, starting with Gauss, one of my personal favourites. Carl Friedrich Gauss (1777-1855) is considered to be one of the greatest mathematicians in the 19th century, and is sometimes referred to as the “Prince of Mathematics”.
His discoveries influenced and left a lasting mark in a variety of different areas, including number theory, astronomy, geodesy, and physics, particularly the study of electromagnetism.
Born in Brunswick, Germany to poor, working class parents, he was discouraged from attending school from his father, a gardner and brick-layer, who expected Gauss would follow one of the family trades. However, Gauss’ mother and uncle recognised Gauss’ early genius and knew he must develop this gift with a proper education.
In arithmetic class, at the age of 10, Gauss showed his skills as a maths prodigy. A well known anecdote about Gauss and his early school education is about when the strict schoolmaster gave the following assignment:
“Write down all the whole numbers from 1 to 100 and add up their sum.”
They expected this assignment to take a while to complete but after a few seconds, to the teacher’s surprise, Carl placed his slate on the desk in front of the teacher, showing he was done with the question. His other classmates took a much longer time to complete the assignment. At the end of class time, although most other students answers were wrong, Gauss’ was correct: 5050. Carl then explained to the teacher that he found the result as he could see that 1+100 = 101, 2+99=101, etc. So he could find 50 pairs of numbers that each add up to 101, and so 50*101 = 5050. I don’t know about you but I definitely could not come up with this sort of argument at the age of 10…
Although his family was poor, Gauss’ intellectual abilities drew the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum at the age of 15, and then to the University of Göttingen – a very prestigious university – where he stayed from 1795 to 1798. During this period, Gauss discovered many important theorems.
No pattern had previously been found in the occurrence of prime numbers until Gauss. Although the occurrence of the primes seems to be completely random, by approaching the problem from a different angle and graphing the incidence of primes as the numbers increased, he noticed a rough trend: as numbers increased by 10, the probability of the numbers reduced by a factor of around 2. However, as his method only gave him an approximation, and as he could not definitively prove his findings, he kept them a secret until much later in his life.
1796 is known as Gauss’ “annus mirabilis” (means “wonderful year” and is used to refer to several years during which events of major importance are remembered). In 1796:
- Gauss constructed a regular 17-sided heptadecagon, which had previously been unknown, using only a ruler and a compass. This was a major advance in geometry since the time of the Greeks.
- Gauss formulated this prime number theorem on the distribution of prime numbers among the integers, which states that Here is the number of primes less than or equal to n. We can also write .
- Gauss proved that every positive integer can be represented as the sum of at most 3 triangular number
More about Gauss in the next post!