Download Impossible, Imaginary, Useful Complex Numbers

Seventy-twelve
Impossible, Imaginary, Useful
Complex Numbers
By:Daniel Fulton
Eleventeen
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Where did the idea of imaginary
numbers come from
Descartes, who contributed the term
"imaginary"
Euler called sqrt(-1) = i
Who uses them
Why are they so useful in REAL world
problems
Remember Cardano’s
3
cubic x + cx + d = 0
d
x  
2
3
2
3
d
c
d
3

  
4
27
2
2
d
c

4
27
3
Finding imaginary answers
x  15x  4  0
3
x
4
4
 15
4
4
 15
3





2
4
27
2
4
27
2
3
3
2
x  3 2  4  125  3 2  4  125
x  3 2   121  3 2   121
3
Inseparable Pairs
• Complex numbers always appear as pairs in
solution
• Polynomials can’t have solutions with only
one complex solution
Imaginary answers to a problem
originally meant there was no
solution
As Cardano had stated “ 9 is either +3 or –3,
for a plus [times a plus] or a minus times a
minus yields a plus. Therefore  9 is neither
+3 or –3 but in some recondite third sort of
thing.
Leibniz said that complex numbers were a
sort of amphibian, halfway between
existence and nonexistence.
Descartes pointed out
• To find the
intersection of a circle
and a line
• Use quadratic equation
• Which leads to
imaginary numbers
• Creates the term
“imaginary”
Wallis draws a clear picture
Again lets look at
x  15x  4  0
3
We got
x  3 2   121  3 2   121
So Is There A Real Solution to this equation
But Wait
This Can’t Be True
I say let us try x = 4
x  15x  4  0
3
4  15( 4)  4  0
3
64  60  4  0
works
Thank Heavens For Bombelli
He used plus of minus for adding a square
root of a negative number, which finally
gave us a way to work with these
imaginary numbers.
He showed
x  3 2   121  3 2   121  ( 2   1)  ( 2   1)  4
The Amazing
The Wonderful
Euler Relation
i
e  cos( )  i sin( )
cos( ) 
sin( ) 
e
e
i
i
 i
e
2
 i
e
2i
Useful complex
e i  e  i e i  e  i
sin( ) cos(  ) 

2i
2
e(   )  e  i (   )  e i (   )  e  i (   )

4i

2i sin(   )  2i sin(   )
4i
1
1
 sin(   )  sin(   )
2
2
Learning to add and multiply
again
1. Adding or subtracting complex numbers involves
adding/subtracting like terms.
(3 - 2i) + (1 + 3i) = 4 + 1i = 4 + i
(4 + 5i) - (2 - 4i) = 2 + 9i
(Don't forget subtracting a negative is adding!)
2. Multiply: Treat complex numbers like binomials, use the FOIL
method, but simplify i2.
(3 + 2i)(2 - i) = (3 • 2) + (3 • -i) + (2i • 2) + (2i • -i)
= 6 - 3i + 4i - 2i2
= 6 + i - 2(-1)
=8+i
Imaginary to an Imaginary is
(  1)
1
ip 
2p

p
i
i
 i   e 2   e 2  e 2  0.2078.
 
i
Why are complex numbers so
useful
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Differential Equations
To find solutions to polynomials
Electromagnetism
Electronics(inductance and capacitance)
So who uses them
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Engineers
Physicists
Mathematicians
Any career that uses differential equations
Timeline
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Brahmagupta writes Khandakhadyaka
665
Solves quadratic equations and allows for the possibility of negative solutions.
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Girolamo Cardano’s the Great Art
1545
General solution to cubic equations
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Rafael Bombelli publishes Algebra
1572
Uses these square roots of negative numbers
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Descartes coins the term "imaginary“
John Wallis
1637
1673
Shows a way to represent complex numbers geometrically.
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Euler publishes Introductio in analysin infinitorum
1748
Infinite series formulations of ex, sin(x) and cos(x), and deducing
the formula, eix = cos(x) + i sin(x)
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Euler makes up the symbol i for  1
The memoirs of Augustin-Louis Cauchy
1777
1814
Gives the first clear theory of functions of a complex variable.
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De Morgan writes Trigonometry and Double Algebra
1830
Relates the rules of real numbers and complex numbers
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Hamilton
1833
Introduces a formal algebra of real number couples using rules
which mirror the algebra of complex numbers
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Hamilton's Theory of Algebraic Couples
Algebra of complex numbers as number pairs (x + iy)
1835
References
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(Photograph of Thinker by Auguste Rodin
http://www.clemusart.com/explore/work.asp?searchText=thinker&recNo=1&tab=2&display=
http://history.hyperjeff.net/hypercomplex.html
http://mathworld.wolfram.com/ComplexNumber.html (Wallis picture)
Nahin, Paul. An Imaginary Tale Princeton, NJ: Princeton University Press,1998
Maxur, Barry. Imagining Numbers. New York:Farrar Straus Giroux, 2003
Berlinghoff, William and Gouvea, Fernando. Math through the Ages. Maine: Oxton House
Publishers, 2002
Katz, Victor. A History of Mathematics. New York: Pearson, 2004