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Seventy-twelve. Impossible, Imaginary, Useful Complex Numbers. By:Daniel Fulton. Eleventeen. Why imagine the imaginary. Where did the idea of imaginary numbers come from Descartes, who contributed the term "imaginary" Euler called sqrt(-1) = i Who uses them

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Presentation Transcript
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Why imagine the imaginary
  • Where did the idea of imaginary numbers come from
  • Descartes, who contributed the term "imaginary"
  • Euler called sqrt(-1) = i
  • Who uses them
  • Why are they so useful in REAL world problems
inseparable pairs
Inseparable Pairs
  • Complex numbers always appear as pairs in solution
  • Polynomials can’t have solutions with only one complex solution
imaginary answers to a problem originally meant there was no solution
Imaginary answers to a problem originally meant there was no solution

As Cardano had stated “ is either +3 or –3, for a plus [times a plus] or a minus times a minus yields a plus. Therefore is neither +3 or –3 but in some recondite third sort of thing.

Leibniz said that complex numbers were a sort of amphibian, halfway between existence and nonexistence.

descartes pointed out
Descartes pointed out
  • To find the intersection of a circle and a line
  • Use quadratic equation
  • Which leads to imaginary numbers
  • Creates the term “imaginary”
again lets look at
Again lets look at

We got

So Is There A Real Solution to this equation

but wait this can t be true
But WaitThis Can’t Be True

I say let us try x = 4

thank heavens for bombelli
Thank Heavens For Bombelli

He used plus of minus for adding a square root of a negative number, which finally gave us a way to work with these imaginary numbers.

He showed

learning to add and multiply again
Learning to add and multiply again
  • Adding or subtracting complex numbers involves adding/subtracting like terms.
  • (3 - 2i) + (1 + 3i) = 4 + 1i = 4 + i (4 + 5i) - (2 - 4i) = 2 + 9i(Don't forget subtracting a negative is adding!)
  • 2. Multiply: Treat complex numbers like binomials, use the FOIL method, but simplify i2.
  • (3 + 2i)(2 - i) = (3 • 2) + (3 • -i) + (2i • 2) + (2i • -i)
  • = 6 - 3i + 4i - 2i2
  • = 6 + i - 2(-1)
  • = 8 + i
imaginary to an imaginary is
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Imaginary to an Imaginary is
why are complex numbers so useful
Why are complex numbers so useful
  • Differential Equations
  • To find solutions to polynomials
  • Electromagnetism
  • Electronics(inductance and capacitance)
so who uses them
So who uses them
  • Engineers
  • Physicists
  • Mathematicians
  • Any career that uses differential equations
timeline
Timeline
  • Brahmagupta writes Khandakhadyaka 665

Solves quadratic equations and allows for the possibility of negative solutions.

  • Girolamo Cardano’s the Great Art 1545

General solution to cubic equations

  • Rafael Bombelli publishes Algebra 1572

Uses these square roots of negative numbers

  • Descartes coins the term "imaginary“ 1637
  • John Wallis 1673

Shows a way to represent complex numbers geometrically.

  • Euler publishes Introductio in analysin infinitorum 1748

Infinite series formulations of ex, sin(x) and cos(x), and deducing

the formula, eix = cos(x) + i sin(x)

  • Euler makes up the symbol i for 1777
  • The memoirs of Augustin-Louis Cauchy 1814

Gives the first clear theory of functions of a complex variable.

  • De Morgan writes Trigonometry and Double Algebra 1830

Relates the rules of real numbers and complex numbers

  • Hamilton 1833

Introduces a formal algebra of real number couples using rules

which mirror the algebra of complex numbers

  • Hamilton's Theory of Algebraic Couples 1835

Algebra of complex numbers as number pairs (x + iy)

references
References
  • (Photograph of Thinker by Auguste Rodinhttp://www.clemusart.com/explore/work.asp?searchText=thinker&recNo=1&tab=2&display=
  • http://history.hyperjeff.net/hypercomplex.html
  • http://mathworld.wolfram.com/ComplexNumber.html (Wallis picture)
  • Nahin, Paul. An Imaginary Tale Princeton, NJ: Princeton University Press,1998
  • Maxur, Barry. Imagining Numbers. New York:Farrar Straus Giroux, 2003
  • Berlinghoff, William and Gouvea, Fernando. Math through the Ages. Maine: Oxton House Publishers, 2002
  • Katz, Victor. A History of Mathematics. New York: Pearson, 2004
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