Early Mathematical Career

Early Mathematical Career

Carl Gauss was attracted to mathematics from an early age. In 1798 (at the age of 21), Gauss finished what would come to be recognized as a masterpiece of number theory, Disquisitiones Arithmeticae. This work set the foundation of modern number theory and consolidated earlier work done by Fermat, Euler, Lagrange, and Legendre and others. This earlier work was supplemented with original, insightful content provided by Gauss. This was the first book on number theory to bring the entire field together into one cohesive framework, as opposed to a collection of isolated observations and theorems. Disquisitiones also set a organizational convention that would later be mimicked by later books (and one that is still largely followed today). First, a theorem was stated formally, then the theorem was proved, then corollaries of the theorem were explored.

 

Despite the considerable amount of time and effort that Gauss invested in Disquisitiones, he still considered choosing a profession other than mathematics while he was at the University of Gottingen. In particular, he was deeply interested in philology (a field of study similar to linguistics). Another important consequence of Disquisitiones was that it introduced new helpful notation, such as the ≡ symbol for congruence. Finally, Gauss showed in Disquisitiones how a regular heptadecagon (a polygon with 17 equal sides) can be constructed with a compass and a straightedge. This problem had been unsolved since ancient Greece and Gauss's proof was the first progress in the field on constructible polygons (shapes that can be drawn using only a compass and a straightedge) in over 2000 years. Gauss was so proud of this accomplishment that he asked for a heptadecagon to be carved on his tombstone. This last wish was not granted because the engraver said it would be indistinguishable from a circle.

 

In 1799, Gauss finished his dissertation by discovering a new proof of the fundamental theorem of algebra (every polynomial of degree n, has n roots, counting multiplicity). Prior proofs of the theorem (done by Lagrange, Euler, Laplace, and others) were mostly correct but were technically not acceptable by today's standards. Their proofs assumed that every polynomial has n roots and showed that the form of these roots was a + bi. Gauss improved upon their work by not assuming the existence of roots. Gauss's proof was mostly geometric, and -- ironically enough -- is also technically incorrect due to topological reasons.

Another important discovery of Gauss's was that every number can be written as a sum of at most three triangular numbers. The proof of this result is beautiful and makes use of the result of a different proof. The proof of the triangular number theorem starts with the fact that every positive integer congruent to 3 mod 8 can be expressed as the sum of at most three squares.

 

This is a neat result, particularly since it is not immediately obvious.

Collectively, these examples really hammer home the magnitude of Gauss's genius. He accomplished all of this by the time he was 22!

Background

Carl Friedrich Gauss was born in Germany in April 1777 and would go on to make significant contributions in mathematics (especially in the areas of algebra, number theory, and statistics) and physics (particularly in astronomy, optics, and electromagnetism). Gauss devoted his life to mathematics even incorporating the symbols for pi and integration into his signature.

Born to a poor family, Gauss learned to solve "real-world" problems using math from an early age. As an example, Gauss's mother was was unable to read or write and as a result could not record when he had been born -- she could only remember that he had been born on a Wednesday, 8 days before the Feast of Ascension. (The Feast of Ascension is a mostly-abandoned Protestant holiday that always takes place on a Thursday, 40 days after Easter). Gauss was able to figure out when he was born by devising a method to calculate the date for Easter in any given year. Not bad for a young boy from a poor, uneducated family!

A more famous example of Gauss’s problem-solving abilities took place when he was an 8-year-old student, and his teacher instructed him to sum up all of the integers between 1 and 100 as a punishment for misbehaving. To their astonishment, Gauss solved the problem within seconds [1]. His reasoning is sketched out below.

Later in life, he was able to make several important contributions, particularly in number theory. Gauss was the first to prove the Triangular Number Theorem (any positive integer can be written as the sum of at most three triangular numbers), was the first to conjecture the Prime Number Theorem, was the first to calculate the orbit of a dwarf planet named Ceres, and provided the foundation for the modern study of statistics, optics, and electromagnetism. Of these, probably the most important mathematical accomplishment was his conjecture of the Prime Number Theorem, which hypothesizes that the prime numbers are distributed roughly as N/log(N). In other words, the probability that a randomly chosen integer greater than N is prime is given approximately by 1/log(N) [2]. This approximation becomes more accurate as N approaches infinity.

Gauss's personal life was tumultuous. His father had a dominating personality leading to a distant relationship between the two. In 1805, he married a woman named Johanna Osthoff who tragically died just a few years later in 1809. They had two children together - one son and one daughter. The son died a year after Johanna. In 1810, Gauss remarried but was a markedly different man as the result of the grief from his first wife's and son's deaths. He began do treat his children aggressively in the same way that his own father had treated him, leading to tension within the family [1].

Gauss was a zealous perfectionist (his personal motto was pauca sed matura: "few, but ripe"), diligent worker, and devout Lutheran. These traits may have been influenced in part by the nationalistic attitudes in Germany at the time (Gauss rarely collaborated with other mathematicians, especially non-Germans) together with the nihilistic philosophies espoused by the German philosophers, Immanuel Kant and Friedrich Nietzsche, who may have influenced some of Gauss’s more unorthodox religious beliefs [3].

In 1855, Gauss passed away from a heart attack and was buried in Gottingen, Germany [1]. After his death, Gauss’s brain was studied by medical experts who found it to be slightly bigger than average. They also found his brain had more pronounced convolutions (the folds on the surface of the brain), which have been speculated by some as a physiological explanation for his genius [4].

References

[1] Bruno, Leonard C. (2003). Math and mathematicians: the history of math discoveries around the world U·X·L, 1999

[2] Hoffman, Paul (1998). The Man Who Loved Only Numbers. New York: Hyperion Books

[3] Bühler, Walter Kaufmann (1987). Gauss: a biographical study. Springer-Verlag

[4] Bardi, Jason (2008). The Fifth Postulate: How Unraveling A Two Thousand Year Old Mystery Unraveled the Universe. John Wiley & Sons