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## PowerPoint Slideshow about 'Impossible, Imaginary, Useful Complex Numbers' - emmy

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Presentation Transcript

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

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”

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

- 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

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 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

- (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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