# MATH3401

Lecture 2

## Complex Numbers

• 1545: Cardano, roots of $$x^3+ax+b$$
$$5+\sqrt{-15} \ra$$ "mental torture"
• 1575: $$i$$, rules for operations in $$\C$$
• 1629: $$a+\sqrt{-b}$$ Girard - "Solutions impossibles"
• 1637: Descartes - imaginary numbers
• 1831: Complex numbers

## Introduction

• $$\N$$: $$\{1, 2, 3, \ldots \}$$ natural numbers
• $$\mathbb N_0$$: $$\{0, 1, 2, 3, \ldots \}$$
• $$\Z$$: $$\{0, \pm 1, \pm 2, \ldots \}$$ integers
• $$\Q$$: $$\{p/q \; | \; p, q \in \mathbb Z, q\neq 0 \}$$ rational numbers
• $$\R$$: real numbers
• $$\C$$: complex numbers

#### Equivalent representations of $$\C$$

$$z = (x,y)$$ belongs to the complex plane with $$x,y\in \R$$.

So $$z=x\left(1,0\right)+y\left(0,1\right)$$, i. e. $z = x+i\, y$

where $$x$$ and $$y$$ are the real and imaginary parts, respectively.

#### Equivalent representations of $$\C$$

$$i$$ is the complex number represented by $$\left(0,1\right)$$.

We say $$\R \subset \C$$ by identifying the complex number $$x+i\cdot 0$$ with the real number $$x$$.

### Addition in $$\mathbb C$$

Set $\left(x_1,y_1\right)+\left(x_2,y_2\right) = \left(x_1+x_2,y_1+y_2\right)$

That is

$$\left(x_1 + i\, y_1\right)+\left(x_2 + i \,y_2\right) = \left(x_1+x_2\right)+i\left(y_1+y_2\right)$$

### Multiplication in $$\C$$

We can use $$\times$$ or $$\cdot$$ or juxtaposition.

${\small \left(x_1, y_1\right) \cdot \left(x_2, y_2\right) = \left(x_1 x_2 - y_1 y_2, y_1 x_2 + x_1 y_2\right)}$

That is

$$\left(x_1 + i\, y_1\right) \left(x_2 + i \,y_2\right) = \left(x_1 x_2 - y_1 y_2\right)+i\left(y_1 x_2 + x_1 y_2\right)$$

### Note 2.1

The definition of $$\times$$ formally applies if we use the usual rules for algebra in $$\R$$ and set $$i=\sqrt{-1}$$.

With the operations $$+$$ and $$\times$$, $$\C$$ is a field.

Exercise: Check that $$\C$$ is closed under $$+$$ and $$\times$$.

### Note 2.1

(F2)(i) Existence of additive identity: $$0 = 0+i\,0$$

(F2)(ii) Existence of additive inverse: $-z = -(x+i\,y) = (-x)+i\,(-y)$

### Note 2.1

(F5)(i) Existence of multiplicative identity: $$1 = 1+i\,0$$

(F5)(ii) Existence of multiplicative inverse: $$z = x+i\,y \neq 0$$, \begin{align*} z^{-1} &= \dfrac{1}{x+i\,y}\cdot \dfrac{x-i\,y}{x-i\,y} \\ &= \dfrac{x}{x^2+y^2} - i \dfrac{y}{x^2 + y^2} \end{align*}

### Note 2.1

Since $$\C$$ is a field, there holds: $z_1 z_2 = 0 \implies \text{ either } z_1=0 \text{ or } z_2=0$

(called null-factor or cancelation law).

### Note 2.2

\begin{align*} \left(z_1 z_2\right)^{-1} &= z_1^{-1}z_2^{-1} \\ &= \dfrac{1}{z_1}\cdot \dfrac{1}{z_2}=\dfrac{1}{z_1 z_2}, \text{ with } z_1,z_2\neq 0 \end{align*}

### Note 2.3

$$i^2=-1$$, $$\left(-i\right)^2=-1$$.

These are the only two solutions of $$z^2=-1$$ in $$\C$$.

### Other functions

• Modulus: $$|z|=\sqrt{x^2+y^2}$$
$\text{e. g. } |7+i\,3| = \sqrt{49+9} = \sqrt{58}$ The modulus is a one-to-one function from $$\C$$ to $$\R$$. Indeed $$|z|:\mathbb C \ra \left[0,\infty\right)$$.

### Other functions

• $$\Re (z) =$$ real part of $$z$$.
• $$\Im(z) =$$ imaginary part of $$z$$.
• $$\Re : \C \ra \R$$ and $$\Im : \C \ra \R$$
• Examples: $\Re(7+3i) = 7, \quad \Im(7+3i)= 3.$