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Seventy-twelve Impossible, Imaginary, Useful Complex Numbers By:Daniel Fulton Eleventeen • • • • • 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 3 cubic x + cx + d = 0 d x 2 3 2 3 d c d 3 4 27 2 2 d c 4 27 3 Finding imaginary answers x 15x 4 0 3 x 4 4 15 4 4 15 3 2 4 27 2 4 27 2 3 3 2 x 3 2 4 125 3 2 4 125 x 3 2 121 3 2 121 3 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 “ 9 is either +3 or –3, for a plus [times a plus] or a minus times a minus yields a plus. Therefore 9 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 x 15x 4 0 3 We got x 3 2 121 3 2 121 So Is There A Real Solution to this equation But Wait This Can’t Be True I say let us try x = 4 x 15x 4 0 3 4 15( 4) 4 0 3 64 60 4 0 works 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 x 3 2 121 3 2 121 ( 2 1) ( 2 1) 4 The Amazing The Wonderful Euler Relation i e cos( ) i sin( ) cos( ) sin( ) e e i i i e 2 i e 2i Useful complex e i e i e i e i sin( ) cos( ) 2i 2 e( ) e i ( ) e i ( ) e i ( ) 4i 2i sin( ) 2i sin( ) 4i 1 1 sin( ) sin( ) 2 2 Learning to add and multiply again 1. 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 ( 1) 1 ip 2p p i i i e 2 e 2 e 2 0.2078. 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“ John Wallis 1637 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 1 The memoirs of Augustin-Louis Cauchy 1777 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 Algebra of complex numbers as number pairs (x + iy) 1835 References • • • • • • • (Photograph of Thinker by Auguste Rodin http://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