Seventy-twelve Impossible, Imaginary, Useful Complex Numbers By:Daniel

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 cubic x3 cx d 0 x 3 d 2 d 4 2 c 27 3 3 d 2 d 4 2 c 27 3

Finding imaginary answers 3 x 15x 4 0 x 3 4 2 x 4 4 3 x 2 2 3 2 15 27 3 3 3 4 125 121 3 2 4 2 4 4 2 4 125 2 15 27 121 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 3 x 15x 4 0 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 3 x 15x 4 0 3 4 15(4 ) 4 0 64 60 4 0 w orks

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 e i c o s ( ) i s in ( ) e c o s ( ) s in ( ) e i i i e 2 i e 2i

Useful complex s in ( ) c o s ( ) e ( ) e e i( ) i e 2i e 4i i i( ) 2 i s in ( ) 2 i s in ( ) 4i 1 1 s in ( ) s in ( ) 2 2 e i e e 2 i i( )

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 i i 2 i i i e 2 e 2 e 2 0.2078.

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 1673 1637 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 Algebra of complex numbers as number pairs (x iy) 1835 1

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