Complex Numbers

• What are complex numbers

  • recall in Eulers thesis (or what ever the fuck it was called)
    • we denote $\sqrt{-1}=i$
      • same rules as regualar algebra apply to it

• A complex number is a number of the form $a+bi,(a,b\in \mathbb{R})$

  • a is the Real part of them number
  • b is the imaginary part
    • complex is $\mathbb{C}$ where as reals are $\mathbb{R}$

• ​ dont really work as $\sqrt{ab}=/=\sqrt{a}\sqrt{b}$

Solving polynomials with complex numbers

• For $n>0$ we say $\sqrt{-n}=(\sqrt{n})i$

Other properties of complex numbers

• if $z=a+bi$ is a complex number, then we define the absolute value or the modulus of z as $|z|=\sqrt{a^2+b^2}$ and the conjugate of z as $\bar{z}=a-bi$

  • properties:

    • $\bar{z}=z$ if and only if $z\in \mathbb{R}$
    • $\overline{z_1\pm z_2}=\overline{z_1}\pm \overline{z_2}$
    • $\overline{z_1{z}_2}=\overline{z_1}\ast \overline{z_2}$
    • for any interger, n, $\overline{z^n}={\bar{z}}^n$
    • $\bar{z}z=|z{|}^2$

sketching complex numbers in the complex plane

• A complex number $a+bi$ can be drawn like the vector $(a,b)$ in the $xy-plane$

Examples

• Solving polynomial

  • $z^2-2z+2=0$

    • recall: $a{z}^2+bz+c=0$

$$\begin{array}{r}{z}^2-2z+2=0\\ z=\frac{2\pm \sqrt{4-8}}{2}\\ ⟹z_1=1+i\\ ⟹z_2=1-i\end{array}$$

• Note: every poly nomial over $\mathbb{R}$ and $\mathbb{C}$ can be factored into linear factors in $\mathbb{C}$ (Fundamental Theorem of Algebra)