Complex Numbers

$\mathbb{C}=\{a+bi:a,b\in \mathbb{R}\}$
Basically union of Real Numbers and Imaginary Numbers

• Basically a Vector in ${\mathbb{R}}^2$, which allows us to plot complex numbers in a graph, where the real part is in the x axis, and the imaginary part is in the y axis
  • The only difference is that $\mathbb{C}$ is defined in multiplication
• Do not use the normal quadratic formula, but instead, replace the determinant with $w^2=b^2-4ac$ since square root is undefined for complex numbers so far

Polar Coordinates

Operations

• complex numbers are equal if their components are equal
• Real part is $\mathrm{Re}(z)=\mathrm{\Re }(z)$, imaginary part is $\mathrm{Im}(z)=\mathrm{\Im }(z)$
• You can do most operations as though $i$ was a variable
• Multiplication is $(xa-yb)+(xb+ya)i$
  • Geometrically
    • multiplying by a real number scales
    • Multiplying by $i$ is a rotation $90\text{°}$ counterclockwise
• If $z\ne 0$, then it has a unique multiplicative inverse, defined as $\frac{a}{a^2+b^2}-\frac{b}{a^2+b^2}i=\frac{a-bi}{a^2+b^2}$
  • Derive this by using the conjugate
• $|z|=\sqrt{\bar{z}z}=\sqrt{x^2+y^2}$ (modulus or Absolute Value)
  • $|zw|=|z||w|$
  • $|\frac{z}{w}|=\frac{|z|}{|w|}$
  • $|z+w|\le |z|+|w|$ (triangle inequality)
• Division can then be written as $\frac{z}{w}=\frac{z\bar{w}}{|w{|}^2}$

Conjugate

The conjugate of $z=x+yi$ is $\bar{z}=x-yi$, then:

• $\overline{z\pm w}=\bar{z}\pm \bar{w}$
• $\overline{zw}=\bar{z}\bar{w}$