# Impossible, Imaginary, Useful Complex Numbers Impossible, Imaginary, Useful Complex Numbers Ch. 17 Chris Conover & Holly Baust SOLVE Solve the equation x2+2x+7 Use the quadratic formula 24 1 2 Solve on the calculator using a+bi mode 12.45i Overview

Introduction Cardano Bombelli De Moivre & Euler Berkeley, Argand, and Gauss Hamilton Timeline GIROLAMO CARDANO 1545 Published The Great Art Formula 2 3 c

c b c 3 3 x 2 4 27 2 Works for many cubics.but WAIT! 3 Examp x 15 x 4 le: c 2 b3 4 27 The process of dealing with the square root of negative one is as refined as it is useless.

RAFAEL BOMBELLI 1560s Operating with the new kind of radical Invented NEW LANGUAGE Old language two plus square root of minus 121 New Language two plus of minus square root of 121 plus of minus became code Explained the rules of 2 121 BOMBELLI

3 WARNING!!! Not numbers Used to simplify complicated expressions From previous example combined with the NEW language: 2 121 3 2 11 1 WILD IDEA u v u v u v u v u v 1 3 2 11 1 3 3 3 3

2 1 u 3 u v 1 3u v 1 v 1 u 3uv 3u v v 1 1 u u 3v v3u v 1 2 1 1 u 3u v 1 3 u v 1 v 1 3 3 3 2 2 2 2 2 2

3 2 2 3 3 BOMBELLI Negative numbers can lead to real solutions so appearance can be tricky! USEFUL And although to many this will appear an extravagant thing, because even I held this opinion some time ago, since it appeared to me more sophistic than true, nevertheless I searched hard and found the demonstration, which will be noted below. ... But let the reader apply all his strength of mind, for [otherwise] even he will find himself deceived.

DE MOIVRE & EULER De Moivre At this time mathematicians knew that: (a+bi)(c+di) = (ac-bd) + i(bc+da) If you think of this in the right frame of mind you can see the similarities in the REAL parts in the formula: cos(x+y) = cos(x)cos(y)-sin(x)sin(y) Similarly, you can notice the relationship between imaginary parts of formula: sin(x+y) = sin(x)cos(y) + sin(y)cos(x) From here it is not hard to see De Moivres formula: (cos(x)+isin(x))n = cos(nx)+isin(nx) Euler BERKELEY, ARGAND, and

GAUSS Bishop George Berkeley J.R. Argand Would say that all numbers were useful functions First to suggest the mystery of these fictitious or monstrous imaginary numbers could be eliminated by geometrically representing them on a plane Published booklet in 1806 Points Results ignored until Gauss suggested a similar idea Gauss

Proposed similar idea and showed it could be useful mathematically in 1831 Coined the term Complex number SIR WILLIAM ROWAN HAMILTON Interested in applying complex numbers to multi-dimensional geometry. Worked for 8 years to apply to the 3rd dimension, only to realize that it only existed in the 4th. Quaternions q = w+xi+yj+zk, where i, j, and k are all different square roots of -1 and w, x, y, and z are real numbers i 2 j 2 k 2 ijk 1 TIMELINE

1545: Cardanos The Great Art 1560: Bombellis new language 1629: Girard assumption of roots and coefficients 1637: Ren Decartes coined the term imaginary 1730: De Moivres formula (cos(x)+isin(x))n = cos(nx) +isin(nx) 1748: Eulers formula eix = cos(x)+isin(x) 1806: Argands booklet on graphing imaginary numbers 1831: Gauss coined the term complex number 1831: Gauss found complex numbers useful in mathematics 1843: Hamilton discovered quaternions Works Cited

Baez, John. Octonions. May 16, 2001. University of California. http://math.ucr.edu/home/baez/octonions . Berlinghoff, William P., and Fernando Q. Gouva. Math Through the Ages: a Gentle History for Teachers and Others. Farmington: Oxton House, 2002. 141-146. Hahn, Liang-Shin. Complex Numbers & Geometry. Washington, DC: The Mathematical Association of America, 1994. Hawkins, F M., and J Q. Hawkins. Complex Numbers & Elementary Complex Functions. New York: Gordon and Breach Science, 1968. Lewis, Albert C. "Complex Numbers and Vector Algebra." Campanion Encyclopedia of the History and Philosophy of the Mathematical Sciences. 2 vols. New York: Routledge, 1994.