Complex number

The complex numbers arise when we try to solve equations such as $x^2 = -1$ . We know (from the trivial inequality) that the square of a real number cannot be negative, so this equation has no solutions in the real numbers. However, it is possible to define a number, $i$ , such that $i = \sqrt{-1}$ . If we add this new number to the reals, we will have solutions to $x^2 = -1$ . It turns out that in the system that results from this addition, we are not only able to find the solutions of $x^2 = -1$ but we can now find all solutions to every polynomial. (See the Fundamental Theorem of Algebra for more details.)

We are now ready for a more formal definition. A complex number is a number of the form $a + bi$ where $a,b\in \mathbb{R}$ and $i = \sqrt{-1}$ . The set of complex numbers is denoted by $\mathbb{C}$ . The set of complex numbers contains the set $\mathbb{R}$ of the real numbers, but is much larger. Every complex number has a real part, denoted by $\Re$ , or simply $\mathrm{Re}$ , and an imaginary part, denoted by $\Im$ , or simply $\mathrm{Im}$ . So, if $z\in \mathbb C$ , we can write $z=\mathrm{Re}(z)+i\mathrm{Im}(z)$ , where $i$ is the imaginary unit.

As you can see, complex numbers enable us to remove the restriction of $x\ge 0$ for the domain of $f(x)=\sqrt{x}$ .

The letters $z$ and $\omega$ are usually used to denote complex numbers.

Operations

Addition
Subtraction
Multiplication
Division
Absolute value/Modulus/Magnitude (denoted by $|z|$ ). This is the distance from the origin to the complex number in the complex plane.