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’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

i æ ö p ( ) i - p - p 1 2 i - = = = = = i ç ÷ 1 i e e e 0.2078 . 2 2 2 è ø 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 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

Impossible, Imaginary, Useful Complex Numbers