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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

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 s cubic x 3 cx d 0
Remember Cardano’scubic x3 + cx + d = 0



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 solution

  • 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 solution

We got

So Is There A Real Solution to this equation


But wait this can t be true
But Wait solutionThis Can’t Be True

I say let us try x = 4


Thank heavens for bombelli
Thank Heavens For Bombelli solution

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 Amazing solutionThe WonderfulEuler Relation


Useful complex
Useful complex solution


Learning to add and multiply again
Learning to add and multiply again solution

  • 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

i solution

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Imaginary to an Imaginary is


Why are complex numbers so useful
Why are complex numbers so useful solution

  • Differential Equations

  • To find solutions to polynomials

  • Electromagnetism

  • Electronics(inductance and capacitance)


So who uses them
So who uses them solution

  • Engineers

  • Physicists

  • Mathematicians

  • Any career that uses differential equations


Timeline
Timeline solution

  • 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 solution

  • (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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