Impossible, Imaginary, Useful Complex Numbers

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Impossible, Imaginary, Useful Complex Numbers. Ch. 17 Chris Conover &amp; Holly Baust. SOLVE. Solve the equation x 2 +2x+7 Use the quadratic formula. Solve on the calculator using a+bi mode. Overview. Introduction Cardano Bombelli De Moivre &amp; Euler Berkeley, Argand, and Gauss Hamilton

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### Impossible, Imaginary, Useful Complex Numbers

Ch. 17

Chris Conover & Holly Baust

SOLVE
• Solve the equation
• x2+2x+7

Solve on the calculator using a+bi mode

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

Works for many cubics….but WAIT!

Example:

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 operation
BOMBELLI
• WARNING!!!
• Not numbers
• Used to simplify complicated expressions
• From previous example combined with the NEW language:

WILD IDEA→

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 Moivre’s formula:

(cos(x)+isin(x))n = cos(nx)+isin(nx)

• Euler
BERKELEY, ARGAND, and GAUSS
• Bishop George Berkeley
• Would say that all numbers were useful functions
• J.R. Argand
• 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

TIMELINE
• 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
• 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.