The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation




x

2


+
1
=
0


{\displaystyle x^{2}+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
+
3
i


{\displaystyle 2+3i}
.
Imaginary numbers are an important mathematical concept; they extend the real number system




R



{\displaystyle \mathbb {R} }
to the complex number system




C



{\displaystyle \mathbb {C} }
, 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: i and



−
i


{\displaystyle -i}
, just as there are two complex square roots of every real number other than zero (which has one double square root).
In contexts in which use of the letter i is ambiguous or problematic, the letter j is sometimes used instead. 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.

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