The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.

Imaginary numbers are an important mathematical concept, which extend the real number system to the complex number system , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square.

There are two complex square roots of −1, namely i and i, just as there are two complex square roots of every real number other than zero (which has one double square root).

In the contexts where use of the letter i is ambiguous or problematic, the letter j or the Greek ι is sometimes used instead.[a] For example, in electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current.

For the history of the imaginary unit, see Complex number § History.


The imaginary number i is defined solely by the property that its square is −1: