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Complex Numbers

Imagine a conditional.

\[x^2 = -1\]

We cannot find a value for \(x\) such that it satisfies this conditional.
This is where complex numbers come into play.

Complex Planes

Complex numbers do not follow order axioms and can be retreated as 2D points on a plane called the complex plane, Z plane or argand diagram.
This plane is a result of extending the number line1 into 2 dimensions.
m_complex_plane_1

Representing Complex numbers on Complex Plane

The complex numbers has 2 parts, each corresponding to each axis.
There's the real and then there is the imaginary part.
The \(\iota\) is used to represent imaginary parts.
\(\iota\) is defined to be

\[\iota^2 = -1 \implies \iota = \sqrt{-1}\]

We can represent them in 2 ways.

Similarities with Vectors

A complex number \(z\) may be written as

\[z = a(1) + b\iota = a + b \iota\]

Here \(a\) is the real and \(b\) is the imaginary part.

m_complex_plane_2.svg
The \(1\) acts like the unit vector2 \(\hat i\) for \(x\) axis and \(\iota\) acts like the unit vector2 \(\hat j\) for \(y\) axis.
This makes a complex number behave like a position vector2 for a point (the complex number itself).

Ordered Pair

We can use ordered pair to represent the complex numbers as points.

\[(a, b)\]

Here \(a\) is the real and \(b\) is the imaginary part.

Operations

Imagine 2 complex numbers.

\[z_1 = a_1 + b_1\iota\]
\[z_2 = a_2 + b_2\iota\]

Addition

\[z_1 + z_2 = a_1 + b_1 \iota + a_2 + b_2 \iota\]
\[= (a_1 + a_2) + (b_1 + b_2)\iota\]

Subtraction

\[z_1 - z_2 = (a_1 + b_1 \iota) - (a_2 + b_2 \iota)\]
\[= a_1 + b_1 \iota - a_2 - b_2 \iota\]
\[= (a_1 - a_2) + (b_1 - b_2) \iota\]

Multiplication

\[z_1 \times z_2 = (a_1 + b_1 \iota) \times (a_2 + b_2 \iota)\]
\[= a_1(a_2 + b_2 \iota) + b_1\iota(a_2 + b_2\iota)\]
\[= a_1a_2 + a_1b_2\iota + a_2b_1\iota + b_1b_2\iota^2\]
\[= a_1a_2 + b_1b_2(-1) + (a_1b_2 + a_2b_1)\iota \quad \because \iota^2 = -1\]
\[= (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1) \iota\]

Division

\[\frac {z_1}{z_2} = \frac {a_1 + b_1\iota}{a_2 + b_2 \iota}\]
\[= \frac {a_1 + b_1\iota}{a_2 + b_2 \iota} \times \frac {a_2 - b_2\iota}{a_2 - b_2 \iota}\]
\[= \frac{(a_1 + b_1 \iota) \times (a_2 - b_2 \iota)}{(a_2)^2 - (b_2 \iota)^2}\]
\[= \frac{(a_1a_2 + b_1b_2) + (a_1b_1 - a_1b_2) \iota}{a_2^2 + b^2_2}\]

Properties

Additive Identity

\[(0, 0)\]

Additive Inverse

\[(-a, -b)\]

Multiplicative Identity

\[(1, 0)\]

Multiplicative Inverse

\[\left(\frac{a}{a^2 + b^2}, \frac{-b}{a^2 + b^2}\right)\]

References

  1. Read more about number line. ↩

  2. Read more about vectors. ↩↩↩