Download 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
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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
x3


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  3u v  1   3u v  1  v
 1   u  3uv   3u v  v   1 
 1   u u  3v   v3u  v   1 

 1
 1  u  3u v  1  3u  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
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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

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

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

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
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
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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
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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.