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Complex Numbers 2 www.mathxtc.com

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Complex Numbers

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Complex Numbers What is truth?

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Complex Numbers Who uses them in real life?

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Complex Numbers Who uses them in real life? Heres a hint….

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

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Can you see a problem here? -2

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Who goes first? -2

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Complex numbers do not have order -2

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

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

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

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

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Complex i is an imaginary number i is an imaginary number

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Complex Or a complex number Or a complex number

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

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

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Some observations In the beginning there were counting numbers In the beginning there were counting numbers 1 2

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

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

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

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

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Some observations In the beginning there were counting numbers In the beginning there were counting numbers And then we needed integers And then we needed integers And rationals And rationals And irrationals And irrationals And reals And reals 1 2 -3 0.41 0

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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. 1 2 -3 0.41 3 + 4i 2i2i 0

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

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A number such as 6 is a purely real number A number such as 6 is a purely real number

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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 6 is a purely real number 6 + 3i is a complex number 6 + 3i is a complex number

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

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

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

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Worked Examples 1. Simplify

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Worked Examples 1. Simplify

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Worked Examples 1. Simplify 2. Evaluate

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Worked Examples 1. Simplify 2. Evaluate

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Worked Examples 3.Simplify

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Worked Examples 3.Simplify

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Worked Examples 3.Simplify 4.Simplify

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Worked Examples 3.Simplify 4.Simplify

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Worked Examples 3.Simplify 4.Simplify 5.Simplify

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

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Division Division 6.Simplify

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

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

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Solving Quadratic Functions Solving Quadratic Functions

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Powers of i Powers of i

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Developing useful rules Developing useful rules

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

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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 Argand Diagram

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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 Argand Diagram His work on the bell curve His work on the bell curve

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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 Argand Diagram His work on the bell curve His work on the bell curve Very little is known about Argand. No likeness has survived. Very little is known about Argand. No likeness has survived.

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Argand Diagrams x y 1 23 1 2 3 2 + 3i

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Argand Diagrams x y 1 23 1 2 3 2 + 3i We can represent complex numbers as a point.

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Argand Diagrams x y 1 23 1 2 3

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x y 1 23 1 2 3 We can represent complex numbers as a vector. O A

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Argand Diagrams x y 1 23 1 2 3 O A B

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x y 1 23 1 2 3 O A B C

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x y 1 23 1 2 3 O A B C

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x y 1 23 1 2 3 O A B C

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x y 1 23 1 2 3 O A B C

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x y 1 23 1 2 3 O A B C

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x y 1 23 1 2 3 O A B C

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x y 1 23 1 2 3 O A B C

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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.

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De Moivres Theorem This remarkable formula works for all values of n.

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Enter Leonhard Euler…..

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

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One last amazing result Have you ever thought about i i ?

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

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i i = 0.20787957635076190855

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i i = 111.31777848985622603

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So i i is an infinite number of real numbers

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The End

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