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
x 
3

d
2

d
4
2

c
27
3

3

d
2

d
4
2

c
27
3
Finding imaginary answers
x  15 x  4  0
3
x 
3
4
4

2
x 
2
4
2
3
x 
3
2

 15
3

4
3
27
2
4  125 
 121 

3
3
2
2
4
2

4
 15
27
4  125
 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  15 x  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  15 x  4  0
3
4  15( 4 )  4  0
3
64  60  4  0
w orks
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
e
i
 cos( )  i sin( )
c o s(  ) 
e
i
 e
 i
2
sin (  ) 
e
i
 e
2i
 i
Useful complex
sin(  ) cos(  ) 
i
e
e
 i

e
i
2i

e
(   )
e
 i(   )
e
i(   )
4i

2 i sin(    )  2 i sin(    )
4i

1
2
sin(    ) 
1
2
sin(    )
e
 i
2
e
 i(   )
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