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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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
Inseparable Pairs
• Complex numbers always appear as pairs in solution
• Polynomials can’t have solutions with only one complex 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
• Which leads to imaginary numbers
• Creates the term “imaginary”
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

Learning to add and multiply again
• (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

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
• (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