Carl Friedrich Gauss was a German mathematician, astronomer and physicist with a wide range of contributions; he is considered to be one of the leading mathematicians of all time.
Gauss was the only son of lower class uneducated parents. He impressed his teachers early on and earned a scholarship. In college, he independently rediscovered several important theorems; his breakthrough occurred in 1796 when he correctly characterized all the regular polygons that can be constructed by ruler and compass alone, thereby completing work started by classical Greek mathematicians.
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.
At the same time, Gauss discovered the immensely important method of least squares which is used in all sciences to this day to minimize the impact of measurement error. He employed this method to correctly predict the position of the asteroid Ceres. The work was later published in 1809 in a major work about the motion of celestial bodies.
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 that mathematics is important enough to deserve to be supported; 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.
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 1820's; 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 devoloped, leading to results about magnetism, the discovery of Kirchhoff's laws in electricity and the construction of a primitive telegraph.
Even though Gauss never worked as a professor of mathematics and disliked teaching, several of his students turned out to be influential mathematicians, among them Richard Dedekind and Bernhard Riemann.
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 in 1809, soon followed by the death of one child. Gauss plunged into a depression from which he never fully recoverd. He married again, 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 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.