Impossible, Imaginary, Useful Complex Numbers
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Transcript 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
1 2.45i
Overview
Introduction
Cardano
Bombelli
De Moivre & Euler
Berkeley, Argand, and Gauss
Hamilton
Timeline
GIROLAMO CARDANO
1545
Published The Great Art
Formula
2
3
2
3
c
c
b
c
c
b
x3
3
2
4 27
2
4 27
Works for many cubics….but WAIT!
Example:
x 15 x 4
3
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
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
1 u 3u v 1 3u v 1 v
1 u 3uv 3u v v 1
1 u u 3v v3u v 1
1
1 u 3u v 1 3u v 1 v 1
3
3
3
3
3
3
2
2
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 Moivre’s 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 multidimensional 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: 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.