Impossible imaginary useful complex numbers
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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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Impossible, Imaginary, Useful Complex Numbers

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Impossible imaginary useful complex numbers

Seventy-twelve

Impossible, Imaginary, UsefulComplex Numbers

By:Daniel Fulton

Eleventeen


Impossible imaginary useful complex numbers

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


Remember cardano s cubic x 3 cx d 0

Remember Cardano’scubic x3 + cx + d = 0


Finding imaginary answers

Finding imaginary answers


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”


Wallis draws a clear picture

Wallis draws a clear picture


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


The amazing the wonderful euler relation

The AmazingThe WonderfulEuler Relation


Useful complex

Useful complex


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 Khandakhadyaka665

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

  • Girolamo Cardano’s the Great Art1545

    General solution to cubic equations

  • Rafael Bombelli publishes Algebra1572

    Uses these square roots of negative numbers

  • Descartes coins the term "imaginary“1637

  • John Wallis1673

    Shows a way to represent complex numbers geometrically.

  • Euler publishes Introductio in analysin infinitorum1748

    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 Cauchy1814

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

  • De Morgan writes Trigonometry and Double Algebra1830

    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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