# Complex Number Plane

## Key Questions

• The complex plane is a Cartesian-like plane where each complex number is a point, the x coordinate being the real part of the complex number and the y coordinate the imaginary part. In other words, $z = a + b i$ in the complex plane is the point $\left(a , b\right)$ in the Cartesian plane.

• In simple terms the modulus of a complex number is its size.

If you picture a complex number as a point on the complex plane, it is the distance of that point from the origin.

If a complex number is expressed in polar coordinates (i.e. as $r \left(\cos \theta + i \sin \theta\right)$), then it's just the radius ($r$).

If a complex number is expressed in rectangular coordinates - i.e. in the form $a + i b$ - then it's the length of the hypotenuse of a right angled triangle whose other sides are $a$ and $b$.

From Pythagoras theorem we get: $| a + i b | = \sqrt{{a}^{2} + {b}^{2}}$.

$1 = \left(1 , 0\right)$ and $i = \left(0 , 1\right)$

#### Explanation:

The complex number plane is usually considered as a two dimensional vector space over the reals. The two coordinates represent the real and imaginary parts of the complex numbers.

As such, the standard orthonormal basis consists of the number $1$ and $i$, $1$ being the real unit and $i$ the imaginary unit.

We can consider these as vectors $\left(1 , 0\right)$ and $\left(0 , 1\right)$ in ${\mathbb{R}}^{2}$.

In fact, if you start from a knowledge of the real numbers $\mathbb{R}$ and want to describe the complex numbers $\mathbb{C}$, then you can define them in terms of pairs of real numbers with arithmetic operations:

$\left(a , b\right) + \left(c , d\right) = \left(a + c , b + d\right) \text{ }$ (this is just addition of vectors)

$\left(a , b\right) \cdot \left(c , d\right) = \left(a c - b d , a d + b c\right)$

The mapping $a \to \left(a , 0\right)$ embeds the real numbers in the complex numbers, allowing us to consider real numbers as just complex numbers with a zero imaginary part.

Note that:

$\left(a , 0\right) \cdot \left(c , d\right) = \left(a c , a d\right)$

which is effectively scalar multiplication.