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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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Seventy-twelve

Impossible, Imaginary, UsefulComplex 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

  • Why are they so useful in REAL world problems


Remember Cardano’scubic x3 + cx + d = 0


Finding imaginary answers


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

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

  • 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


Again lets look at

We got

So Is There A Real Solution to this equation


But WaitThis Can’t Be True

I say let us try x = 4


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 AmazingThe WonderfulEuler Relation


Useful complex


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


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Imaginary to an Imaginary is


Why are complex numbers so useful

  • Differential Equations

  • To find solutions to polynomials

  • Electromagnetism

  • Electronics(inductance and capacitance)


So who uses them

  • Engineers

  • Physicists

  • Mathematicians

  • Any career that uses differential equations


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

  • (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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