## Complex Numbers

Table of Contents

## Imaginary number

Imaginary number (when squared give a negative result)

i is the unit of imaginary number which is square root of -1

i = √-1, because i^{2}=-1

So, (7i)^{2 }= -7

## Definition of Complex Number

A Complex Number is a combination of a Real Number and an Imaginary Number

4 + 7i, where 4 is Real Number, and 7i is Imaginary Number

## Either Part Can Be Zero

- a Complex Number has a real part and an imaginary part.
- Either part can be 0

So,

- all Real Numbers are Complex numbers
- all Imaginary Numbers are Complex Numbers.

| Real Part | Imaginary Part | Complex Number |

Combination | 4 | 7i | 4 + 7i |

Pure Real | 4 | 0i | 4 |

Pure Imaginary | 0 | 7i | 7i |

We often use z for a complex number. And Re() for the real part and Im() for the imaginary part, like this:

*z = a + bi*, where *Re*_{(z)}* = a, Im*_{(z)}* = b*

## Adding 2 complex numbers

To add 2 complex numbers, add each part separately

(a+bi) + (c+di) = (a+c) + (b+d)i

## Multiplying 2 complex numbers

To multiply 2 complex numbers, each part of the first complex number gets multiplied by

each part of the second complex number

(a+bi)(c+di) = ac + adi + bci + bdi^{2}

Because i^{2} = -1, so

(a+bi)(c+di) = ac – bd + adi + bci

## Dividing 2 complex numbers

To divide 2 complex numbers, multiply both top and bottom by the conjugate of the bottom.

Note: A conjugate is where we change the sign in the middle

Complex number | conjugate | Multiplying By the Conjugate |

a + bi | a – bi | a^{2} + b^{2} |

a – bi | a + bi | a^{2} + b^{2} |

So, to divide *a + bi by c – di*:

Or, to divide *a + bi by c + di:*

_{ } | Application | Result |

z_{1} | Given Re(z_{1}) = 4, Im(z_{2})=7 | 4+7i |

z_{2} | Given Re(z_{1}) = 4, Im(z_{2})=-7 | 4-7i |

z_{1}+z_{2} | (4+4) + (7-7)i | 8 + 0i = 8 |

z_{1 }x z_{2} | 4×4 – 7x(-7) + 4x(-7)i + 4x7i | 44 + 0i = 44 |

z_{1 }/ z_{2} |