MATH 415 Lecture 1

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Complex Numbers

• $z\in \mathbb {C}$
• $z=a+b\,i$
  • $\Re {z}=a$
  • $\Im {z}=b$

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

$a=|z|+\cos {\theta }$ $b=|z|+\sin {\theta }$ $0\leq \theta <2\pi$

Thus

$z=|z|\,\left(\cos {\theta }+i\,\sin {\theta }\right)$

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

${\frac {z_{1}}{z_{2}}}={\frac {a+b\,i}{c+d\,i}}={\frac {a+b\,i}{c+d\,i}}\,{\frac {a-b\,i}{c-d\,i}}={\frac {\left(a+b\,i\right)\,\left(c-d\,i\right)}{c^{2}+d^{2}}}$

Algebra on Circles

Unit circle defined as

$U=\left\{z\in \mathbb {C} ~\mid ~|z|=1\right\}$

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 $\left[0,2\pi \right)$

Modular addition

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

We define $+_{c}$ as the modular operation:

$x+_{c}y={\begin{cases}x+y&x+y<c\\x+y-c&x+y\geq c\end{cases}}\quad \forall x,y\in \left[0,c\right)$

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

$U_{n}=\left\{z\in \mathbb {C} ~\mid ~z^{n}=1\right\}\quad n\in \mathbb {N}$

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 $0\leq m<n-1$

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 $\left\{1,i,-1,-i\right\}$

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

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 $m\neq n$. 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 $\mathbb {Z}$, $\mathbb {Q}$, $\mathbb {R}$, 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 $i*j$.