Advanced Mathematics

4.3Complex Numbers

Purpose of Section

To introduce the field (ℂ, + , ×) of complex numbers and their Cartesian and polar representations in the complex plane. We also show how Euler's theorem connects the complex exponential to real trigonometric functions.

4.3.1 An Introductory Tale

The study of numbers generally begins with children and counting numbers 1, 2, 3, … then progressing to negative numbers, and then to fractions, and finally to the real numbers. To most students of mathematics, the complex numbers come last, if at all. Throughout history, every enlargement of the meaning of number had practical motivation. Negative numbers were required to solve x + 3 = 1, rational numbers were required to solve 3x = 5, and real numbers were a response to the equation x² = 2. Finally, complex numbers came about when people wanted to solve equations like x² + 1 = 0.

Square roots of negative numbers first appeared in Ars Magna (1545) by the Italian mathematician Gerolamo Cardano (1501–1576) in his solution of the simultaneous equations

getting the solution

Cardano did not give any interpretation of the square root of negative numbers, although he did say that they obeyed the usual rules of algebra and that solutions containing them could be verified. Cardano and other mathematicians ...