There are no real numbers that solve the equation $x^2=-1$. No matter what you do, neither a positive number multiplied by itself, nor a negative number multiplied by itself, will give you a negative number as the product. So in order to take square roots of negative numbers, mathematicians have introduced something called complex numbers. Complex numbers are based on the imaginary unit $i$.

A complex number consists of two parts: A real part and an imaginary part. The real part of a complex number is a real number on the real number line—just an ordinary number like you’re used to. The imaginary part of a complex number is the real number in front of the imaginary unit $i$. When you write complex numbers, you can choose to split the number into its real part and imaginary part. The number is then written in Cartesian form, which is in contrast to polar form.

The real part of a complex number $z$ is often denoted by $\mathrm{Re}\left(z\right)$. The imaginary part of a complex number $z$ is often denoted by $\mathrm{Im}\left(z\right)$. The set of all complex numbers is denoted by $ℂ$.

Both imaginary and real numbers are contained in the set of complex numbers. Imaginary numbers can be thought of as complex numbers without a real part, and real numbers can be thought of as complex numbers without an imaginary part. The set of real numbers is therefore contained in the set of complex numbers: