In mathematics, an imaginary number is a number whose square is negative. The term was coined by René Descartes in the seventeenth century and was meant to be derogatory: obviously such numbers don't exist. Nowadays we find the imaginary numbers on the vertical axis of the complex number plane. Every imaginary number can be written as where is a real number and the imaginary unit with the property that

(In electrical engineering and related fields, the imaginary unit is often written as j to avoid confusion with a changing current, traditionally denoted by ''i'\'.) Every complex number can be written uniquely as a sum of a real number and an imaginary number (its imaginary part).

Despite their name, imaginary numbers are just as "real" as real numbers (or just about as real as a number, which is an abstract concept, can be). Imaginary numbers have concrete applications in a variety of sciences and related areas such as electromagnetism, quantum mechanics, and cartography.

The powers of i repeat in a cycle: