Impossible, Imaginary, Useful Complex Numbers Ch. 17 Chris Conover & Holly Baust
SOLVE n Solve the equation qx 2+2 x+7 q. Use the quadratic formula Solve on the calculator using a+bi mode
Overview Introduction n Cardano n Bombelli n De Moivre & Euler n Berkeley, Argand, and Gauss n Hamilton n Timeline n
GIROLAMO CARDANO n 1545 q n Published The Great Art Formula Works for many cubics…. but WAIT! Example: The process of dealing with the square root of negative o “as refined as it is useless. ”
RAFAEL BOMBELLI n 1560 s q Operating with the “new kind of radical” q Invented NEW LANGUAGE n Old language q“two plus square root of minus 121” n New Language q“two plus of minus square root of 121” q“plus of minus” became code n Explained the rules of operation
BOMBELLI n n WARNING!!! q Not numbers q Used to simplify complicated expressions From previous example combined with the NEW language: WILD IDEA→
BOMBELLI n n 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 n De Moivre n n At this time mathematicians knew that: q (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) n n 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 Moivre’s formula: (cos(x)+isin(x))n = cos(nx)+isin(nx) n Euler
BERKELEY, ARGAND, and GAUSS n Bishop George Berkeley q n J. R. Argand q q n 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 q q 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- n n n dimensional geometry. Worked for 8 years to apply to the 3 rd dimension, only to realize that it only existed in the 4 th. 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
TIMELINE n n n n n 1545: Cardano’s The Great Art 1560: Bombelli’s new language 1629: Girard assumption of roots and coefficients 1637: René Decartes coined the term “imaginary” 1730: De Moivre’s formula (cos(x)+isin(x))n = cos(nx)+isin(nx) 1748: Euler’s formula eix = cos(x)+isin(x) 1806: Argand’s 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 n n n Baez, John. Octonions. May 16, 2001. University of California. http: //math. ucr. edu/home/baez/octonions. Berlinghoff, William P. , and Fernando Q. Gouvêa. 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.
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