Complex Numbers

 

Sets reminder

 

 

 

 

 

 

 

 

 

 

Example

 

 

 

 

 

·   Arithmetic operations

 

 

 

 

 

Examples

 

Solve the equation x²-2x +5 =0

 

 

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Alternatively,

 

 

  

 

 

 

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Inverse

 

 

 

 

 

 

 

 

·     Argand diagrams

 

z = x +iy  can be represented on the complex plane

 by the point P(x,y)

 

 

Points on the x axis are real.

Points on the y axis are imaginary.

 

z = x +iy  can also  be represented

by the vector 

 

 

The length of    , r, is called the modulus of z.

 

 

By Pythagoras’ Theorem

 

 

 

The size of the rotation is called the amplitude or

argument of z.

 

Arg z = θ + 2nπ

 

 

The principal argument is denoted arg z and lies

in the range  –π< θ ≤ π

 

 

Example

 

Find the modulus and argument of the complex

number z = 3 +4i

 

 

 

 

·   Loci

 

Given that z =x +iy, find the equation of the locus of

 

 

 

 

 

 

 

 

 

 

 

 

 

·         Polar form 

 

 

 

Example

 

Express z = 3 +4i  in polar form

 

     

 

 

Note

 

 

 

 

·        De moivre's theorem

 

 

 

Example

 

 Given z = 3 + 4i  , calculate z⁵

 

 

     

 

 

 

 

·        Roots of a complex number

 

 

 

 

 

Example

 

Solve the equation  

 

     

 

 

 

·   Complex roots

 

If the root of a polynomial is unreal,

it has complex roots

 

r(cosθ +isinθ)  and r(cosθ -isinθ) 

 

 

A polynomial of degree n will have n complex roots.

 

Example

 

Find the roots of the equation

 z³-6z²+13z-20=0 , given  z=1+2i is a root

 

If z=1+2i is a root , then so is its conjugate z=1-2i

 

Factors are z -1 -2i  and z -1+2i