Complex Numbers

Abstract

Complex numbers are objects of the form $a + b\sqrt { - 1,}$ where a and b are real numbers and $\sqrt { - 1}$ is. . . what? Mathematicians worried about this question for several centuries and did not come up with a good answer until the 19th century, by which time complex numbers had become indispensable in virtually all fields of mathematics. Their story is perhaps the supreme illustration of a saying of Hilbert’s: “In mathematics, existence means freedom from contradiction.”1 Mathematicians came to believe in complex numbers because they worked, not because they could define them, and finding a definition was not a high priority until all concepts of number came under scrutiny.