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Introduction to Complex Numbers

YouTube Workbook

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Language:  English
This ebook makes learning "complex" numbers easy through an interactive, fun and personalized approach. Features include: live YouTube video streams and closed captions that translate to 90 languages!
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This ebook makes learning "complex" numbers easy through an interactive, fun and personalized approach. Features include: live YouTube video streams and closed captions that translate to 90 languages!

Complex numbers "break all the rules" of traditional mathematics by allowing us to take a square root of a negative number. This "radical" approach has fundamentally changed the capabilities of science and engineering to enhance our world through such applications as: signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. A particularly beautiful connection between art and complex numbers lies in fractals, such as the Mandelbrot set.

  1. What is a complex number?
    1. Video 1: Complex numbers are AWESOME
  2. Basic operations involving complex numbers
    1. Video 2: How to add/subtract two complex numbers
    2. Video 3: How to multiply a real number with a complex number
    3. Video 4: How to multiply complex numbers together
    4. Video 5: How to divide complex numbers
    5. Video 6: Complex numbers: Quadratic formula
  3. What is the complex conjugate?
    1. Video 7: What is the complex conjugate?
    2. Video 8: Calculations with the complex conjugate
    3. Video 9: How to show a number is purely imaginary
    4. Video 10: How to prove the real part of a complex number is zero
    5. Video 11: Complex conjuage and linear systems
    6. Video 12: When are the squares of z and its conjugate equal?
    7. Video 13: Conjugate of products is product of conjugates
    8. Video 14: Why complex solutions appear in conjugate pairs
  4. How big are complex numbers?
    1. Video 15: How big are complex numbers?
    2. Video 16: Modulus of a product is the product of moduli
    3. Video 17: Square roots of complex numbers
    4. Video 18: Quadratic equations with complex coefcients
    5. Video 19: Show real part of complex number is zero
  5. Polar trig form
    1. Video 20: Polar trig form of complex number
  6. Polar exponential form
    1. Video 21: Polar exponential form of a complex number
    2. Revision Video 22: Intro to complex numbers + basic operations
    3. Revision Video 23: Complex numbers and calculations
    4. Video 24: Powers of complex numbers via polar forms
  7. Powers of complex numbers
    1. Video 25: Powers of complex numbers
    2. Video 26: What is the power of a complex number?
    3. Video 27: Roots of comples numbers
    4. Video 28: Complex numbers solutions to polynomial equations
    5. Video 29: Complex numbers and tan (π/12)
    6. Video 30: Euler’s formula: A cool proof
  8. De Moivre’s formula
    1. Video 31: De Moivre’s formula: A cool proof
    2. Video 32: Trig identities from De Moivre’s theorem
    3. Video 33: Trig identities: De Moivre’s formula
  9. Connecting sin, cos with e
    1. Video 34: Trig identities and Euler’s formula
    2. Video 35: Trig identities from Euler’s formula
    3. Video 36: How to prove trig identities WITHOUT trig!
    4. Revision Video 37: Complex numbers + trig identities
  10. Regions in the complex plane
    1. Video 38: How to determine regions in the complex plane
    2. Video 39: Circular sector in the complex plane
    3. Video 40: Circle in the complex plane
    4. Video 41: How to sketch regions in the complex plane
  11. Complex polynomials
    1. Video 42: How to factor complex polynomials
    2. Video 43: Factorizing complex polynomials
    3. Video 44: Factor polynomials into linear parts
    4. Video 45: Complex linear factors
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About the Author

Christopher C. Tisdell

“With more than a million YouTube hits, Dr Chris Tisdell is the equivalent of a best-selling author or chart-topping musician. And the unlikely subject of this mass popularity? University mathematics.” [Sydney Morning Herald, 14/6/2012 http://ow.ly/o7gti].

Chris Tisdell has been inspiring, motivating and engaging large mathematics classes at UNSW, Sydney for over a decade. His lectures are performance-like, with emphasis on contextualisation, clarity in presentation and a strong connection between student and teacher.

He blends the live experience with out-of-class learning, underpinned by flexibility, sharing and openness. Enabling this has been his creation, freely sharing and management of future-oriented online learning resources, known as Open Educational Resources (OER). They are designed to empower learners by granting them unlimited access to knowledge at a time, location and pace that suits their needs. This includes: hundreds of YouTube educational videos of his lectures and tutorials; an etextbook with each section strategically linked with his online videos; and live interactive classes streamed over the internet.

His approach has changed the way students learn mathematics, moving from a traditional closed classroom environment to an open, flexible and forward-looking learning model.

Indicators of esteem include: a prestigious educational partnership with Google; an etextbook with over 500,000 unique downloads; mathematics videos enjoying millions of hits from over 200 countries; a UNSW Vice-Chancellor’s Award for Teaching Excellence; and 100% student satisfaction rating in teaching surveys across 15 different courses at UNSW over eight years.

Chris has been an educational consultant to The Australian Broadcasting Corporation and has advised the Chief Scientist of Australia on educational policy.