# Impossible, Imaginary, Useful Complex Numbers - PowerPoint PPT Presentation

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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

### 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

• 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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### 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 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