Complex Number

The Complex Numbers are a Number Set denoted by $\mathbb{C}$ that extends the Real Numbers via the Imaginary Unit $i=\sqrt{-1}$

• $\mathbb{C}$ is a Vector Space Isomorphism of ${\mathbb{R}}^2$
• Like ${\mathbb{R}}^2$, the complex numbers can be represented within the Cartesian Plane.
• $\mathbb{C}$ is a closed Algebraic Field

Complex numbers are of the form

$$a+bi$$

• where $a,b\in \mathbb{R}$
• where $i$ is the Imaginary Unit

Euler's Formula

$$z=|z|e^{i\varphi }=|z|(\cos (\varphi )+i\sin (\varphi ))$$ $$z_1{z}_2=(|z_1|e^{i{\varphi }_1})(|z_2|e^{i{\varphi }_2})=(|z_1||z_2|e^{i({\varphi }_1+{\varphi }_2)})$$

• Consequence of Euler’s Identity
• Suppose $z_1$ has an angle ${\varphi }_1$ and $z_2$ has ${\varphi }_2$
• The product $z_1{z}_2$ has angle ${\varphi }_1+{\varphi }_2$, and modulus $|z_1||z_2|$

Link to original

Operations

Binary

Addition

Like adding Vectors in ${\mathbb{R}}^2$

$$\begin{array}{rl}{z}_1+z_2& =\\ (a+bi)+(c+di)& =\\ (a+c)+(b+d)i& \end{array}$$

Subtraction

Like subtracting Vectors in ${\mathbb{R}}^2$

$$\begin{array}{rl}{z}_1-z_2& =\\ (a+bi)-(c+di)& =\\ (a-c)+(b-d)i& \end{array}$$

Multiplication

$$z_1\cdot z_2=(a+bi)\cdot (c+di)=ac-bd+(ad+bc)i$$

Division

$$\frac{z_1}{z_2}=\frac{z_1\overline{z_2}}{|z_2{|}^2}=\frac{(a+bi)(c-di)}{c^2+d^2}=\frac{ac+bd+(bc-ad)i}{c^2+d^2}$$

Unary

Real Component Function

Imaginary Component Function

Argument

$$z=|z|e^{i\varphi }⟹\text{arg}z=\varphi$$

Complex Conjugate

$$\overline{a+bi}=(a+bi{)}^{\ast }=a-bi$$ $$\overline{(z_1+z_2)}=\overline{z_1}+\overline{z_2}$$ $$\overline{(z_1{z}_2)}=\overline{z_1}\overline{z_2}$$ $$\overline{A\vec{z}}=A\overline{\vec{z}}$$

$\text{Im}(z\overline{z})=0$ $z\overline{z}=|z{|}^2$ Reflects across the Real axis.

Magnitude

Also called “Modulus”

$$|a+bi|=\sqrt{a^2+b^2}=\sqrt{(a+bi)(a-bi)}$$

The magnitude is the $r$ in the figure.