complex_no_notes_4_6_06 - EM Research Group Complex Numbers...

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EM Research Group Complex Numbers Edward J. Rothwell John R. Deller Prepared by the Faculty of the Department of Electrical and Computer Engineering 6 April 2006
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EM Research Group A little history Negative numbers (such as displacement) were known to the ancients As early as 50 AD the Greeks ( Heron of Alexandria ) were contemplating the square roots of negative numbers Indian mathematicians ( Mahavira , AD 850) decided that a negative number does not have a square root In 1545 Girolamo Cardano used completing the square to solve the two simultaneous equations Which is equivalent to the quadratic equation He got the solution 10 40 x y xy + = = 5 15 5 15 x y = + - = - - 2 10 40 0 x x - + = And when he plugged the solution back in, it worked!
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EM Research Group A little more history Cardano also considered the solution to the cubic equation He found the solutions Cardano thought that the third solution was “fictitious” due to and didn’t pursue this topic 30 years later, Rafael Bombelli tried to understand Cardano’s solution. He speculated that these numbers could be written as and that these numbers OBEY THE USUAL RULES OF ALGEBRA He set and solved for 3 15 4 x x = + 3 3 4, 2 3, 2 121 2 121 x x x = = - ± = + - + - - 121 - a b + - 3 3 2 121 2 121 a b a b + - = + - - - = - - 2, 1 a b = = Thus, the third solution to the cubic equation is which made sense to Bombelli (2 1) (2 1) 4 x = + - + - - =
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EM Research Group A little more history After Bombelli , work in complex number theory flourished! 1 - In 1620, Albert Girard theorized that a polynomial has as many roots as its degree Rene Descarte coined the term “imaginary” for the square root of a negative number, and said that a polynomial may have roots that are complex numbers von Leibniz and Bernoulli used imaginary numbers to help perform integrations Euler chose the symbol i to represent In 1797 Gauss published the first correct proof to the fundamental theorem of algebra which states that every polynomial of degree n with complex coefficients has n complex roots Caspar Wessel plotted complex numbers in a plane In 1833 Hamilton expressed complex numbers as pairs of real numbers In the 1800’s, Cauchy and others developed the theory of functions of a complex variable ( , ) a b
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EM Research Group 1. Equality : 2. Addition: 3. Negation: 4. Multiplication: Let be complex numbers. Let be real numbers. Definition : are ordered pairs that obey the following algebraic rules : Complex number as an ordered pair , z w , , , x y u v ( , ), ( , ) z x y w u v = = Notation: is called the real part of z is called the imaginary part of z , z w iff x u y v = = = ( , ) z w x u y v + = + + ( , ) w u v - = - - ( , ) z w x u y v - = - - ( , ) z w xu yv xv yu = - + Re{ } x z = Im{ } y z =
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EM Research Group Note: We can develop all sorts of rules from the previous laws. For example, multiplication of a complex number by a real number is The law for division can be obtained as: We also find that complex numbers obey the associative , commutative , and distributive laws of multiplication and addition: Complex number as an ordered pair 2 2 2 2 ( , ) ( , ) ( , ) ( , ) , ( , ) ( , ) ( , ) ( , ) z x y x y u v xu yv xv yu xu yv xv yu w u v u v u v uu vv uv uv u v u v - + - + + - + ° ± = = = = ² ³ - + - + + + ´ µ ( , ) ( ,0) ( , ) ( 0 , 0 ) ( , ) r z r x y
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