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Complex Numbers

Complex Numbers What is truth?

Complex Numbers Who uses them in real life?

Complex Numbers Who uses them in real life? Here’s a hint….

Complex Numbers Who uses them in real life? Here’s a hint….

Complex Numbers Who uses them in real life? The navigation system in the space shuttle depends on complex numbers!

Can you see a problem here? -2 Can you see a problem here?

-2 Who goes first?

Complex numbers do not have order -2 Complex numbers do not have order

What is a complex number? It is a tool to solve an equation.

What is a complex number? It is a tool to solve an equation. It has been used to solve equations for the last 200 years or so.

What is a complex number? It is a tool to solve an equation. It has been used to solve equations for the last 200 years or so. It is defined to be i such that ;

What is a complex number? It is a tool to solve an equation. It has been used to solve equations for the last 200 years or so. It is defined to be i such that ; Or in other words;

Complex i is an imaginary number

Complex i is an imaginary number Or a complex number

Complex i is an imaginary number Or a complex number Or an unreal number

Complex? i is an imaginary number Or a complex number Or an unreal number The terms are inter-changeable unreal complex imaginary

Some observations In the beginning there were counting numbers 1 2

Some observations In the beginning there were counting numbers And then we needed integers 1 2

Some observations In the beginning there were counting numbers And then we needed integers 1 2 -1 -3

Some observations In the beginning there were counting numbers And then we needed integers And rationals 1 0.41 2 -1 -3

Some observations In the beginning there were counting numbers And then we needed integers And rationals And irrationals 1 0.41 2 -1 -3

Some observations In the beginning there were counting numbers And then we needed integers And rationals And irrationals And reals 1 0.41 2 -1 -3

So where do unreals fit in ? We have always used them. 6 is not just 6 it is 6 + 0i. Complex numbers incorporate all numbers. 3 + 4i 2i 1 0.41 2 -1 -3

A number such as 3i is a purely imaginary number

A number such as 3i is a purely imaginary number A number such as 6 is a purely real number

A number such as 3i is a purely imaginary number A number such as 6 is a purely real number 6 + 3i is a complex number

A number such as 3i is a purely imaginary number A number such as 6 is a purely real number 6 + 3i is a complex number x + iy is the general form of a complex number

A number such as 3i is a purely imaginary number A number such as 6 is a purely real number 6 + 3i is a complex number x + iy is the general form of a complex number If x + iy = 6 – 4i then x = 6 and y = -4

A number such as 3i is a purely imaginary number A number such as 6 is a purely real number 6 + 3i is a complex number x + iy is the general form of a complex number If x + iy = 6 – 4i then x = 6 and y = – 4 The ‘real part’ of 6 – 4i is 6

Worked Examples Simplify

Worked Examples Simplify

Worked Examples Simplify Evaluate

Worked Examples Simplify Evaluate

Worked Examples 3. Simplify

Worked Examples 3. Simplify

Worked Examples 3. Simplify 4. Simplify

Worked Examples 3. Simplify 4. Simplify

Worked Examples 3. Simplify 4. Simplify 5. Simplify

Addition Subtraction Multiplication 3. Simplify 4. Simplify 5. Simplify

Division 6. Simplify

Division 6. Simplify The trick is to make the denominator real:

Division 6. Simplify The trick is to make the denominator real:

Solving Quadratic Functions

Powers of i

Powers of i

Powers of i

Powers of i

Powers of i

Developing useful rules

Developing useful rules

Developing useful rules

Developing useful rules

Argand Diagrams Jean Robert Argand was a Swiss amateur mathematician. He was an accountant book-keeper.

Argand Diagrams Jean Robert Argand was a Swiss amateur mathematician. He was an accountant book-keeper. He is remembered for 2 things His ‘Argand Diagram’

Argand Diagrams Jean Robert Argand was a Swiss amateur mathematician. He was an accountant book-keeper. He is remembered for 2 things His ‘Argand Diagram’ His work on the ‘bell curve’

Argand Diagrams Jean Robert Argand was a Swiss amateur mathematician. He was an accountant book-keeper. He is remembered for 2 things His ‘Argand Diagram’ His work on the ‘bell curve’ Very little is known about Argand. No likeness has survived.

Argand Diagrams x y 1 2 3 2 + 3i

Argand Diagrams x y 1 2 3 2 + 3i We can represent complex numbers as a point.

Argand Diagrams x y 1 2 3

Argand Diagrams y x We can represent complex numbers as a vector. 1 2 3 A O We can represent complex numbers as a vector.

Argand Diagrams x y 1 2 3 B A O

Argand Diagrams C x y 1 2 3 B A O

Argand Diagrams C x y 1 2 3 B A O

Argand Diagrams C x y 1 2 3 B A O

Argand Diagrams C x y 1 2 3 B A O

Argand Diagrams C x y 1 2 3 B A O

Argand Diagrams C x y 1 2 3 B A O

Argand Diagrams C x y 1 2 3 B A O

De Moivre Abraham De Moivre was a French Protestant who moved to England in search of religious freedom. He was most famous for his work on probability and was an acquaintance of Isaac Newton. His theorem was possibly suggested to him by Newton.

This remarkable formula works for all values of n. De Moivre’s Theorem This remarkable formula works for all values of n.

Enter Leonhard Euler…..

Euler who was the first to use i for complex numbers had several great ideas. One of them was that eiq = cos q + i sin q Here is an amazing proof….

One last amazing result Have you ever thought about ii ?

One last amazing result What if I told you that ii is a real number?

ii = 0.20787957635076190855

ii = 111.31777848985622603

So ii is an infinite number of real numbers

The End