The complex plane

The Imaginary unit, or $i$i, is the number with the following equivalent properties:

A complex number is any number that can be written as $\greenD{a}+\blueD{b}i$start color #1fab54, a, end color #1fab54, plus, start color #11accd, b, end color #11accd, i, where $i$i is the imaginary unit and $\greenD{a}$start color #1fab54, a, end color #1fab54 and $\blueD{b}$start color #11accd, b, end color #11accd are real numbers.

$\greenD a$start color #1fab54, a, end color #1fab54 is called the $\greenD{\text{real}}$start color #1fab54, start text, r, e, a, l, end text, end color #1fab54 part of the number, and $\blueD b$start color #11accd, b, end color #11accd is called the $\blueD{\text{imaginary}}$start color #11accd, start text, i, m, a, g, i, n, a, r, y, end text, end color #11accd part of the number.

Just like we can use the number line to visualize the set of real numbers, we can use the complex plane to visualize the set of complex numbers.

The complex plane consists of two number lines that intersect in a right angle at the point $(0,0)$left parenthesis, 0, comma, 0, right parenthesis.

The horizontal number line (what we know as the $x$x-axis on a Cartesian plane) is the real axis.

The vertical number line (the $y$y-axis on a Cartesian plane) is the imaginary axis.

Every complex number can be represented by a point in the complex plane.

For example, consider the number $3-5i$3, minus, 5, i. This number, also expressed as $\greenD{3}+(\blueD{-5})i$start color #1fab54, 3, end color #1fab54, plus, left parenthesis, start color #11accd, minus, 5, end color #11accd, right parenthesis, i, has a real part of $\greenD{3}$start color #1fab54, 3, end color #1fab54 and an imaginary part of $\blueD{-5}$start color #11accd, minus, 5, end color #11accd.

The location of this number on the complex plane is the point that corresponds to $\greenD{3}$start color #1fab54, 3, end color #1fab54 on the real axis and $\blueD{-5}$start color #11accd, minus, 5, end color #11accd on the imaginary axis.

So the number $\greenD{3}+(\blueD{-5})i$start color #1fab54, 3, end color #1fab54, plus, left parenthesis, start color #11accd, minus, 5, end color #11accd, right parenthesis, i is associated with the point $(\greenD{3}, \blueD{-5})$left parenthesis, start color #1fab54, 3, end color #1fab54, comma, start color #11accd, minus, 5, end color #11accd, right parenthesis. In general, the complex number $\greenD a+\blueD bi$start color #1fab54, a, end color #1fab54, plus, start color #11accd, b, end color #11accd, i corresponds to the point $(\greenD a,\blueD b)$left parenthesis, start color #1fab54, a, end color #1fab54, comma, start color #11accd, b, end color #11accd, right parenthesis in the complex plane.

In Pythagoras's days, the existence of irrational numbers was a surprising discovery! They wondered how something like $\sqrt{2}$square root of, 2, end square root could exist without an accurate complete decimal expansion.

The real number line, however, helps rectifying this dilemma. Why? Because $\sqrt{2}$square root of, 2, end square root has a specific location on the real number line, showing that it is indeed a real number. (If you take the diagonal of a unit square and place one end on $0$0, the other end corresponds to the number $\sqrt{2}$square root of, 2, end square root.)

Likewise, every complex number does indeed exist because it corresponds to an exact location on the complex plane! Perhaps by being able to visualize these numbers, we can understand that calling these numbers "imaginary" was an unfortunate misnomer.

Complex numbers exist and are very much a part of mathematics. The real number line is simply the real axis on the complex plane, but there is so much beyond that single line!