Complex Numbers 2

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Complex Numbers 2
www.mathxtc.com
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Complex Numbers
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Complex NumbersWhat is truth?
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Complex NumbersWho uses themin real life?
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Complex NumbersWho uses themin real
life?Heres a hint.
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Complex NumbersWho uses themin real
life?Heres a hint.
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Complex NumbersWho uses themin real life?

• The navigation system in the space shuttle
  depends on complex numbers!

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Can you see a problem here?
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Who goes first?
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Complex numbers do not have order
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What is a complex number?

• 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 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 has been used to solve equations for the last
  200 years or so.
• 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 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

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Complex

• i is an imaginary number

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Complex

• i is an imaginary number
• Or a complex number

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Complex

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

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Complex?

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

unreal
complex
imaginary
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Some observations

• In the beginning there were counting numbers

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2
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Some observations

• In the beginning there were counting numbers
• And then we needed integers

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2
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Some observations

• In the beginning there were counting numbers
• And then we needed integers

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2
-1
-3
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Some observations

• In the beginning there were counting numbers
• And then we needed integers
• And rationals

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0.41
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-1
-3
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Some observations

• In the beginning there were counting numbers
• And then we needed integers
• And rationals
• And irrationals

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0.41
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-1
-3
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Some observations

• In the beginning there were counting numbers
• And then we needed integers
• And rationals
• And irrationals
• And reals

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0.41
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-1
0
-3
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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.

3 4i
2i
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0.41
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-1
0
-3
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• A number such as 3i is a purely imaginary number

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

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• 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

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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

• 6. Simplify

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Division

• 6. Simplify
• The trick is to make the denominator real

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Division

• 6. Simplify
• The trick is to make the denominator real

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Solving Quadratic Functions
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Powers of i
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Powers of i
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Powers of i
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Powers of i
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Powers of i
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Developing useful rules
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Developing useful rules
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Developing useful rules
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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

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

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Argand Diagrams
2 3i
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Argand Diagrams
2 3i
We can represent complex numbers as a point.
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Argand Diagrams
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Argand Diagrams
A
O
We can represent complex numbers as a vector.
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Argand Diagrams
B
A
O
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Argand Diagrams
C
B
A
O
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Argand Diagrams
C
B
A
O
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Argand Diagrams
C
B
A
O
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Argand Diagrams
C
B
A
O
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Argand Diagrams
C
B
A
O
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Argand Diagrams
C
B
A
O
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Argand Diagrams
C
B
A
O
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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 eiq cos q i sin q Here is an amazing
proof.
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One last amazing result

• Have you ever thought about ii ?

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One last amazing result

• What if I told you that ii is a real number?

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• ii 0.20787957635076190855

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• ii 111.31777848985622603

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• So ii is an infinite number of real numbers

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The End