MATH 415 Lecture 1

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Complex Numbers

• $z\in \mathbb{ℂ}$
• $z=a+bi$
  • $\mathrm{\Re }z=a$
  • $\mathrm{\Im }z=b$

Plot complex numbers on a Cartesian coordinate plane: $x$-axis is real, $y$-axis is imaginary

Magnitude of complex number:

$|z|=d=\sqrt{a^2+b^2}$

Polar Coordinate Form

$a=|z|+\cos \theta$ $b=|z|+\sin \theta$ $0\le \theta <2\pi$

Thus

$z=|z|(\cos \theta +i\sin \theta )$

It's worth noting Euler's formula: ${\mathrm{e}}^{i\theta }=\cos \theta +i\sin \theta$, so

$z=|z|{\mathrm{e}}^{i\theta }$

Multiplication of Complex Numbers

$|z_1{z}_2|=|z_1||z_2|$

$z_1{z}_2=|z_1|{\mathrm{e}}^{i{\theta }_1}|z_2|{\mathrm{e}}^{i{\theta }_2}=|z_1{z}_2|{\mathrm{e}}^{i({\theta }_1+{\theta }_2)}$

Complex Conjugate

$\frac{z_1}{z_2}=\frac{a+bi}{c+di}=\frac{a+bi}{c+di}\frac{a-bi}{c-di}=\frac{(a+bi)(c-di)}{c^2+d^2}$

Algebra on Circles

Unit circle defined as

$U=\{z\in \mathbb{ℂ}\mid |z|=1\}$

So all points on the circle satisfy $z={\mathrm{e}}^{i\theta }$

Notice $U$ is closed on multiplication: $z_1{z}_2\in U$ for all $z_1,z_2\in U$

Isomorphism

Also, every complex number $z$ on the circle corresponds exactly to a unique angle $\theta$ (one-to-one bijection). Furthermore, if $z_1$ corresponds to ${\theta }_1$ and $z_2$ corresponds to ${\theta }_2$, then the product $z_1{z}_2$ corresponds to the sum ${\theta }_1{+}_{2\pi }{\theta }_2$ (see section below)

This isomorphism applies to the sets $U$ and $[0,2\pi )$

Modular addition

$\theta \in [0,2\pi )={\mathbb{ℝ}}_{2\pi }$

We define ${+}_c$ as the modular operation:

$x{+}_{c}y=\{\begin{array}{ll}x+y& x+y<c\\ x+y-c& x+y\ge c\end{array}\mathrm{\forall }x,y\in [0,c)$

For example,

$\frac{3\pi }{2}{+}_{2\pi }\frac{5\pi }{4}=\frac{11\pi }{4}-2\pi =\frac{3\pi }{4}$

Roots of Unity

$U_n=\{z\in \mathbb{ℂ}\mid z^n=1\}n\in \mathbb{\mathbb{N}}$

We define the root of unity $\zeta$ to be an $n$^th of a circle:

$\zeta =\cos \left(\frac{2\pi }{n}\right)+i\sin \left(\frac{2\pi }{n}\right)$

The set $U_n\subset U$ consists of the points $1={\zeta }^0$, $\zeta ={\zeta }^1$, ${\zeta }^2$, etc. (or $\cos \left(\frac{2\pi \cdot m}{n}\right)+i\sin \left(\frac{2\pi \cdot m}{n}\right)$ for integers $0\le m<n-1$

These points define an $n$-gon subset of the unit circle, so the cardinality of $U_n$ (denoted $\left|U_n\right|$) is $m$.

For example, $U_4$ for $n=4$ consists of the points $\{1,i,-1,-i\}$

Notice that the set $U_n$ has a one-to-one correspondence to ${\mathbb{\mathbb{Z}}}_n$, so there is this isomorphism between $U_n$ and ${\mathbb{\mathbb{Z}}}_n$

Binary Operations

A Binary operation $*$ on a set $S$ is a function mapping $S\times S$ onto $S$: $*((a,b))=a*b\in S$ for all $(a,b)\in S^2$

Addition, subtraction, mutiplication, division, etc. are all binary operations over the real numbers.

Example from linear algebra:

• Let $M_n(\mathbb{ℝ})$ be square matrices of size $n\times n$ over the real numbers. These matrices can be added and multiplied.
• However, let $M_{m\times n}(\mathbb{ℝ})$ be matrices of size $m\times n$ over real numbers, where $m\ne n$. These matrices may be added, but multiplication is not defined (only over matrices of size $n\times k$.

Given $(S,*)$ and $H\subset S$

We say $H$ is closed under $*$ if for all $a,b\in H$ we have $a*b\in H$.

Composition

Given function $F:\mathbb{ℝ}\to \mathbb{ℝ}$ we can perform all standard arithmetic operations (including division if $F$ is non-zero. We add to this a special composition operation:

$F_1\circ F_2=F_1(F_2(x))$

Properties

• $*$ is commutative if $a*b=b*a$ for all $a,b\in S$.
  • addition and multiplication are both commutative in $\mathbb{\mathbb{Z}}$, $\mathbb{\mathbb{Q}}$, $\mathbb{ℝ}$, and $\mathbb{ℂ}$
  • but multiplication of (square) matrices is not commutative
• $*$ is associative if $(a*b)*c=a*(b*c)$ for all $a,b\in S$
  • All arithmetic operations including composition of functions are associative

Tables

Only applicable to finite sets (e.g. $S$)

Rows and columns of table labeled with elements of $S$. cell in $i$^th row and $j$^th column has value $i*j$.