Gauss earned a scholarship and in college, he independently rediscovered several important theorems; his breakthrough occurred in 1796 when he was able to show that any regular polygon, each of whose odd factors are distinct Fermat primes, can be constructed by ruler and compass alone, thereby adding to work started by classical Greek mathematicians. Gauss was so pleased by this result that he requested that a regular 17-gon be inscribed on his tombstone.
He was the first to prove the fundamental theorem of algebra; in fact, he produced four entirely different proofs for this theorem over his lifetime, clarifying the concept of complex number considerably along the way. He also made important contributions to number theory with his 1801 book Disquisitiones arithmeticae, which contained a clean presentation of modular arithmetic and the first proof of the law of quadratic reciprocity.
He had been supported by a stipend from the Duke of Brunswick, but he did not appreciate the insecurity of this arrangement and also did not believe mathematics to be important enough to deserve support; he therefore aimed for a position in astronomy, and in 1807 he was appointed professor of astronomy and director of the astronomical observatory in Göttingen.
In 1809, Gauss published a major work about the motion of celestial bodies. Among other things, he introduced the gaussian gravitational constant. It also contains an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of measurement error. He was able to prove the correctness of the method under the assumption of normally distributed errors; see Gauss-Markov theorem. The method had been described earlier by Adrien-Marie Legendre in 1805, but Gauss claimed that he had been using it since 1795.
Gauss discovered the possibility of non-Euclidean geometries but never published it. His friend Farkas Wolfgang Bolyai had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry and failed. Bolyai's son, János Bolyai, rediscovered non-Euclidean geometry in the 1820s; his work was published in 1832. Later, Gauss tried to determine whether the physical world is in fact Euclidean by measuring out huge triangles.
In 1818, Gauss started a geodesic survey of the state of Hanover, work which later lead to the development of the normal distribution for describing measurement errors and an interest in differential geometry and his theorema egregrium establishing an important property of the notion of curvature.
In 1831, a fruitful collaboration with the physics professor Wilhelm Weber developed, leading to results about magnetism, the discovery of Kirchhoff's laws in electricity and the construction of a primitive telegraph.
Gauss was deeply religious and conservative. He supported monarchy and opposed Napoleon whom he saw as an outgrowth of revolution. Gauss' personal life was overshadowed by the early death of his beloved first wife, Johanna Osthoff, in 1809, soon followed by the death of one child, Louis. Gauss plunged into a depression from which he never fully recoverd. He married again, to Friederica Wilhelmine Waldeck (Minna), but the second marriage does not seem to have been very happy. When his second wife died in 1831 after long illness, one of his daughters, Therese, took over the household and cared for Gauss until the end of his life. His mother lived in his house from 1812 until her death in 1839. He rarely if ever collaborated with other mathematicians and was considered aloof and austere by many.
Gauss had six children, three by each wife. With Johnanna (1780-1809), his children were Joseph (1806-1873), Wilhelmina (1808-1846) and Louis (1809-1810). Of all of Gauss' children, Wilhelmina was said to have come closest to his talent, but regrettably, she died young. With Minna Waldeck, he had three children: Eugene (1811-1896), Wilhelm (1813-1879) and Therese (1816-1864). Eugene emigrated to the United States about 1832 after a falling out with his father, eventually settling in St. Charles, Missouri, where he became a well respected member of the community. Wilhelm came to settle in Missouri somewhat later, starting as a farmer and later becoming wealthy in the shoe business in St. Louis. Therese kept house for Gauss until his death, after which she married.
He died in Göttingen, Hanover (now Germany) in 1855 and is interred in the cemetery Albanifriedhof there. From 1989 until the end of 2001, his portrait and a normal distribution curve was featured on the German ten-mark banknote.
G. Waldo Dunnington was a life-long student of Gauss. He wrote many articles, and a biography: Carl Frederick Gauss: Titan of Science.