Complex Numbers

Complex numbers are numbers which can be put into the form a + bi, where a and b are real numbers and i² = -1.

Typical complex numbers are 3 - i, 1/2 + 7i, and -6 - 2i. If one writes the real number 17 as 17 + 0i and the imaginary number -2.5i as 0 - 2.5i, they too can be considered complex numbers.

Complex numbers are so called because they are made up of two parts which cannot be combined. Even though the parts are joined by a plus sign, the addition cannot be performed. The expression must be left as an indicated sum.

Complex numbers are occasionally represented with ordered pairs, (z,b). Doing so shows the two-component nature of complex numbers but renders the operations with them somewhat obscure and hides the kind of numbers they are.

Inklings of the need for complex numbers were felt as early as the sixteenth century. Cardan, in about 1545, recognized that his method of solving cubic equations often led to solutions with the square root of negative numbers in them. It was not until the seventeenth and early eighteenth centuries that de Moivre, the Bernoullis, Euler, and others gave formal recognition to imaginary and complex numbers as legitimate numbers.