Complex number

The complex numbers arise when we try to solve equations such as $x^2 = -1$ .

Derivation

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.)

Formal Definition

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 $i = \sqrt{-1}$ is the imaginary unit. The set of complex numbers is denoted by $\mathbb{C}$ . The set of complex numbers contains the set $\mathbb{R}$ of the real numbers, since $a = a + 0i$ , but it is much larger.

Parts

Every complex number $z$ has a real part denoted $\Re(z)$ or $\mathrm{Re}(z)$ and an imaginary part denoted $\Im(z)$ or $\mathrm{Im}(z)$ . Note that the imaginary part of a complex number is real: for example, $\Im(3 + 4i) = 4$ . So, if $z\in \mathbb C$ , we can write $z=\mathrm{Re}(z)+i\mathrm{Im}(z)$ . ( $z$ and $w$ are traditionally used in place of $x$ and $y$ as variables when dealing with complex numbers, while $x$ and $y$ (and frequently also $a$ and $b$ ) are used to represent real values such as the real and imaginary parts of complex numbers. This mathematical convention is often broken when it is inconvenient, so be sure that you know what set variables are taken from when dealing with the complex numbers.)

As you can see, complex numbers enable us to remove the restriction of $x\ge 0$ from the domain of the function $f(x)=\sqrt{x}$ (although some additional considerations are necessary).

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.
Conjugation
The argument function $\arg$