Complex Numbers

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Complex Numbers
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Presentation Outline

• What are complex numbers?
• Where did they come from?
• What properties do they have?
• Are they actually useful?

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What are complex numbers?

• a bi where a,b and i v-1

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What are complex numbers?

• a bi where a,b and i v-1
• ordered pairs of real numbers (a, b)

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What are complex numbers?

• a bi where a,b and i v-1
• ordered pairs of real numbers (a, b)
• C is a field

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What are complex numbers?

• a bi where a,b and i v-1
• ordered pairs of real numbers (a, b)
• C is a field
• R is a subfield of C

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Where did they come from?

• Heron

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Heron of Alexandria (between 150 BC and 300 AD)
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Where did they come from?

• Heron
• Negative numbers?

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Where did they come from?

• Heron
• Negative numbers?
• Cardona fictitious numbers

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Girolamo Cardano (1501 - 76)
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Where did they come from?

• Heron
• Negative numbers?
• Girolamo Cardona fictitious numbers
• René Descartes imaginary numbers

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René Descartes (1596 1650)
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Where did they come from?

• Heron
• Negative numbers?
• Girolamo Cardona fictitious numbers
• René Descartes imaginary numbers
• Leonhard Euler i

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Leonhard Euler (1707 1783)
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Where did they come from?

• Heron
• Negative numbers?
• Girolamo Cardona fictitious numbers
• René Descartes imaginary numbers
• Leonhard Euler
• Caspar Wessel Carl Friedrich Gauss

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Caspar Wessel, (1745 - 1818) Johann Carl
Friedrich Gauss (1777 - 1855)

• ?

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Where did they come from?

• Heron
• Negative numbers?
• Girolamo Cardona fictitious numbers
• René Descartes imaginary numbers
• Leonhard Euler
• Caspar Wessel Carl Friedrich Gauss
• Jean Robert Argand

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Jean Robert Argand (1768 - 1822)
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What properties do they have?

• Argand diagram

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What properties do they have?

• Argand diagram
• Addition / subtraction

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Addition / subtraction
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What properties do they have?

• Argand diagram
• Addition / subtraction
• Negation

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Negation
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What properties do they have?

• Argand diagram
• Addition / subtraction
• Negation
• Absolute value

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Absolute value
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What properties do they have?

• Argand diagram
• Addition / subtraction
• Negation
• Absolute value
• Multiplication

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(No Transcript)
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(No Transcript)
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What properties do they have?

• Argand diagram
• Addition / subtraction
• Negation
• Absolute value
• Multiplication
• Reciprocal and conjugate

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(No Transcript)
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What properties do they have?

• Argand diagram
• Addition / subtraction
• Negation
• Absolute value
• Multiplication
• Reciprocal and conjugate
• Nice properties

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Nice properties
(cosx isinx)n cos(nx) isin(nx).
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Are they actually useful?
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Are they actually useful?

• Improper integrals

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(No Transcript)
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Improper integrals
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Are they actually useful?

• Improper integrals
• Quantum mechanics
• Differential equations
• Fluid dynamics
• Phasors
• Fractals

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Pretty fractal of Mandelbrot set to finish
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Fractal nutters

• Fri. 4th March The Molong Music and Arts Festival
  Molong
• The Australian band GangGajang (http//www.ganggaj
  ang.com/) has a song Time (and the Mandelbrot
  set) .