## Summary

IIntroduction to complex numbersAThe complex numbers systemBOperations with complex numbersCGraphic représentation of complex numbersIIComplex numbers and equationsIIITrigonometric form, exponential form and polar form of complex numbersATrigonometric form (and exponential form) of a complex numberBPolar coordinates and polar form## Introduction to complex numbers

### The complex numbers system

#### The number i

*i* is a number such that :

i^2=-1

Or equivalently:

i=\sqrt { -1 }

*i* is one of only a few numbers that has a name. Other named numbers are \pi and *e*. It creates the basis for the complex system and gives rise to solutions for polynomial equations.

*i* is a solution to the quadratic equation x^2=-1.

#### Complex Numbers

Let *a* and *b* be two real numbers. A complex number is written:

** z=a+bi **

The following numbers are complex numbers :

- z_1=1+2i
- z_2=3i-4
- z_3=5i

z=a+bi is called the **algebric form** of a complex number.

#### Real part and imaginary part

Let *a* and *b* be two real numbers and *z *a complex number such that:

z=a+ib

*a*is called the real part of*z.**b*is called the imaginary part of*z.*

Let *z* be the complex number such that z=2+4i :

- The real part of
*z*is 2 - The imaginary part of
*z*is 4

Let *z* be the complex number such that z=-2i+5 :

- The real part of
*z*is 5 - The imaginary part of
*z*is -2

The number *i* is never a component of the imaginary part.

- The imaginary part of 5i is 5.
- The imaginary part of 2-3i is -3.

#### Equality of Complex Numbers

Two complex numbers are equal if and only if the real and imaginary parts are equal.

- 2+8i \text{ is equal to }8i+2 \text{ because the real and imaginary parts are equal. }
- 7+3i \text{ is not equal to } 7-3i \text{ because the imaginary parts are not equal. }

### Operations with complex numbers

Let a_1, a_2, b_1 and b_2 be real numbers and let z_1 and z_2 be complex numbers with :

- z_1=a_1+b_1i
- z_2=a_2+b_2i

Then:

** z_1+z_2=\left(a_1+b_1i\right)+\left(a_2+b_2i\right)=\left(a_1+a_2\right)+\left(b_1+b_2\right)i **

If :

- z_1=5+2i
- z_2=1-3i

z_1+z_2=\left(5+2i\right) +\left(1-3i\right)=\left(5+1\right)+\left(2-3\right)i=6-i

Let a_1, a_2, b_1 and b_2 be real numbers and let z_1 and z_2 be complex numbers with :

- z_1=a_1+b_1i
- z_2=a_2+b_2i

Then:

** z_1-z_2=\left(a_1+b_1i\right)-\left(a_2+b_2i\right)=\left(a_1-a_2\right)+\left(b_1-b_2\right)i **

If :

- z_1=8+3i
- z_2=1+2i

z_1-z_2=\left(8+3i\right)-\left(1+2i\right)=\left(8-1\right)+\left(3-2\right)i=7+i

#### Complex Conjugate

Let *a* and *b* be two real numbers and *z* a complex number such that:

z=a+bi

The conjugate of *z* is the following complex number:

** \overline{z}=a-bi **

Consider the complex number z=2+5i. The conjugate is :

\overline{z}=2-5i

If *z* is written as z=ib+a, then the conjugate of *z* is:

** \overline{z}=-ib+a=a-bi **

Considering the complex number z=i-3. The conjugate of *z* is :

\overline{z}=-i-3=-3-i

Let a_1, a_2, b_1 and b_2 be real numbers and let z_1 and z_2 be complex numbers with :

- z_1=a_1+b_1i
- z_2=a_2+b_2i

Then:

** z_1.z_2=\left(a + bi\right) \times \left(c + di\right) = \left(ac-bd\right) + i\left(bc + ad\right) **

If z_1=2+3i and z_2=3-3i, then:

\begin{aligned}z_1.z_2&=\left(2+3i\right) \times \left(3-3i\right) \\ &= \left(6-\left(-9\right)\right)+\left(-6+9\right)i \\&=15+3i \end{aligned}

Let z_1 and z_2 be two complex numbers such that:

- z_1=a+bi
- z_2=c+di

*a*, *b*, *c,* and *d* being four real numbers with z_2\neq 0.

The division of *z _{1}* by

*z*is given by :

_{2}** \begin{aligned}\dfrac{z_1}{z_2} &= \dfrac {\left(a + bi\right)} {\left(c + di\right)} = \dfrac {\left(a + bi\right)} {\left(c + di\right)} \times \dfrac {\left(c - di\right)} {\left(c - di\right)}= \dfrac {\left(ac + bd\right) + \left(bc-ad\right) i} {\left(c ^ 2 + d ^ 2\right)} \end{aligned} **

If :

- z_1=1+2i
- z_2=2+3i

\begin{aligned}\dfrac{z_1}{z_2}&= \dfrac{\left(1 + 2i\right)\left(2-3i\right)}{\left(2 + 3i\right)\left(2-3i\right)}\\&= \dfrac {\left(2 + 6\right) + \left(4-3\right) i} {2 ^ 2 + 3 ^ 2} \\&= \dfrac {8 + i} {13} \\&= \dfrac{8}{13}+\dfrac{i}{13}\end{aligned}

### Graphic représentation of complex numbers

Complex numbers can be represented on a two-dimensional graph:

- The horizontal axis represents the real part.
- The vertical axis represents the imaginary part.

Consider the following complex numbers:

- z_1=-1+4i
- z_2=3+2i

z_1 and z_2 can be represented by A_1\left(-1{,}4\right) and A_2\left(3{,}2\right) :

Let a,b be real numbers and consider the complex number z=a+bi. The reflection of z across the real axis is where the complex conjugate \overline{z}=a-bi is located on the complex plane.

The following graphic contains the location of 2+3i and its complex conjugate 2-3i.

Observe that 2-3i is the reflection of the point 2+3i across the real axis.

## Complex numbers and equations

### Discriminant

The discriminant of the quadratic equation ax^2+bx+c=0 is :

** b^2-4ac **

Consider the equation:

x^2-5x+2=0

The discriminant is:

\left(-5\right)^2-4\left(1\right)\left(2\right)=17

### Quadratic Formula

Let *a*, *b,* and *c* be three real numbers. Given a quadratic equation of the form ax^2+bx+c=0 , the solutions are given by:

** x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a} **

Consider the equation 2x^2-5x+4=0. The solutions are given by:

x=\dfrac {- \left(- 5\right) \pm \sqrt {\left(-5\right) ^ 2-4 \left(4\right)}} {2 \left(4\right)} \\\\=\dfrac {- \left(- 5\right) \pm \sqrt {25-16}} {2 \left(4\right)} \\ = \dfrac {5 \pm \sqrt {9}} { 8} \\ = \dfrac {5 \pm 3} 8

Therefore, the equation has two solutions :

- x_1=\dfrac{5-3}{8}=\dfrac{2}{8}=\dfrac{1}{4}
- x_2=\dfrac{5+3}{8}=\dfrac{8}{8}=1

In the complex system, square roots of negative numbers can be found. Hence, quadratic equations with negative discriminant will also have solutions.

Consider the equation :

2x^2+x+1=0

The square root of the discriminant is :

\sqrt{1^2-4\left(1\right)\left(1\right)}=\sqrt{-3}=\sqrt{3}i

The quadratic formula provides the solution to a quadratic equation of the form ax^2+bx+c=0 :

- A positive discriminant will provide two real solutions to the given equation.
- A discriminant will be zero when there is only one real solution.
- A negative discriminant will provide two complex solutions that are conjugates of each other (the imaginary part will be non-zero).

Consider the equation x^2+4x+5=0. The discriminant is equal to:

4^2-4\left(1\right)\left(5\right)=16-20=-4.

The discriminant is negative and therefore the quadratic has two complex roots :

- z_1=\dfrac{-4+\sqrt{-4}}{2}=\dfrac{-4+2i}{2}=-2+i
- z_2=\dfrac{-4-\sqrt{-4}}{2}=\dfrac{-4-2i}{2}=-2-i

### Conjugate Theorem

If a polynomial with real coefficients has a root of the form z=a+bi, then its complex conjugate \overline{z}=a-bi is also a root of the polynomial.

Consider the following equation:

2x^2-5x+7=0

Assume that \dfrac {5 + i \sqrt {31}} {4} is a solution to the equation. Then the other solution is:

\overline{\dfrac {5 + i \sqrt {31}} {4}}=\dfrac {5 - i \sqrt {31}} {4}

### Fundamental Theorem of Algebra

Every polynomial having real coefficients and a degree greater than or equal to one has at least one complex root.

2x+1=0 has a root when x=\dfrac {-1} {2}.

x^2+2x+1=0 has a root when x=-1.

2x^2-5x+7=0 has roots when x=\dfrac {5 \pm i \sqrt {31}} {4}.

## Trigonometric form, exponential form and polar form of complex numbers

A complex number is determined by its magnitude and the angle that the complex number makes with the positive real axis. Therefore, in addition to the algebraic form, a given complex number can be written in forms that use those two elements.

Consider the complex number 2+2i.

The magnitude of 2+2i is \sqrt{2^2+2^2}=2\sqrt{2}, and the angle that 2+2i makes with the positive real axis is \theta=\dfrac{\pi}{4}.

### Trigonometric form (and exponential form) of a complex number

#### Absolute value of *z*

Let z=a+bi. \left| z \right| is called the absolute value of *z* and equals:

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

|4-3i|=\sqrt{4^2+3^2}=\sqrt{25}=5

|2+3i|=\sqrt{2^2+3^2}=\sqrt{13}

Let *z* be a complex number. Then:

** |z|=\sqrt{z\cdot \bar{z}} **

Consider the complex number 2+i. Then:

|2+i|=\sqrt{\left(2+i\right)\left(2-i\right)}=\sqrt{4-2i+2i+1}=\sqrt{5}

#### Trigonometric form of a complex number

Suppose a,b are real numbers and z=a+bi. Let \theta be the angle which z makes with the real axis as in the following graphic:

The trigonometric form of *z* is:

** z=|z|\left(\cos\left(\theta\right)+i\sin\left(\theta\right)\right) **

Consider the complex number z=2+2i. Then:

- |z|=\sqrt{2^2+2^2}=2\sqrt{2}
- The angle z makes with the real axis is \dfrac{\pi}{4} radians.

Therefore:

2+2i=|2+2i|\left(\cos\left(\dfrac{\pi}{4}\right)+\sin\left(\dfrac{\pi}{4}\right)\right)

2+2i=\sqrt{2}\left(\cos\left(\dfrac{\pi}{4}\right)+\sin\left(\dfrac{\pi}{4}\right)\right)

#### Exponential form of a complex number

Let a,b be real numbers and z=a+bi. Let \theta be the angle that z makes with the real axis as in the following graphic:

The exponential form of *z* is:

** z=|z|e^{i\theta} **

Consider the complex number z=-1. Then:

- |z|=|-1|=1
- The angle z=-1 with the positive real axis is \pi.

Therefore:

z=e^{i\pi}

### Polar coordinates and polar form

#### Polar form

Let z be a complex number such that \theta is the angle that z makes with the positive real axis as in the following graphic:

The polar form of z is:

** \left(|z|,\theta\right) **

The numbers |z| and \theta are the **polar coordinates** of z.

Consider the complex number z=3-3i. Then:

- |z|=\sqrt{3^2+\left(-3\right)^2}=3\sqrt{2}.
- z makes an angle of \dfrac{7\pi}{4} with the positive real axis.

Therefore, the polar form of *z* is \left(3\sqrt{2},\dfrac{7\pi}{4}\right).