Read the first part of this series here.
Although Gauss made contributions in many fields of mathematics, number theory was his favourite. He said that
“mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics.”
A way in which Gauss revolutionised number theory was his work with complex numbers.
Gauss gave the first clear exposition of complex numbers and of the investigation of functions of complex variables. Although imaginary numbers had been used since the 16th century to solve equations that couldn’t be solved any other way, and although Euler made huge progress in this field in the 18th century, there was still no clear idea as to how imaginary numbers were connected with real numbers until early 19th century. Gauss was not the first to picture complex numbers graphically (Robert Argand produced the Argand diagram in 1806). However, Gauss was the one who popularised this idea and introduced the standard notation a + bi. Hence, the study of complex numbers received a great expansion allowing its full potential to be unleashed.
Furthermore, at the age of 22 he proved the Fundamental Theorem of Algebra which states:
Every non-constant single-variable polynomial over the complex numbers has at least one root.
This shows that the field of complex numbers is algebraically closed, unlike the real numbers.
Gauss also had a strong interest in astronomy, and was the Director of the astronomical observatory in Göttingen. When Ceres was in the process of being identifies in the late 17th century, Gauss made a prediction of its position. This prediction was very different from those of other astronomers, but when Ceres was discovered in 1801, it was almost exactly where Gauss had predicted. This was one of the first applications of the least squares approximation method, and Gauss claimed to have done the logarithmic calculations in his head.
Part 3 coming next week!