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Unformatted text preview: MAT 1332: Calculus for Life Sciences A course based on the book Modeling the dynamics of life by F.R. Adler Supplementary material University of Ottawa Frithjof Lutscher, with Jing Li and Robert Smith? February 21, 2010 MAT 1332: Additional Course Notes 1 Complex numbers Introductory consideration We can easily solve the equation x 2 4 = 0 . The answer is x = ± 2; in particular, x is a rational number, even an integer. The equation x 2 2 = 0 is a bit more tricky. The solution x = ± √ 2 is not a rational number. Instead, we have defined the square root of a positive number as the real number that gives the original number back when multiplied by itself. But what should we do with the equation x 2 + 1 = 0? The answer cannot be a real number. (Why?) Can we do the same as above and define a number whose square equals 1? Indeed, this is what mathematicians did in the eighteenth century (it was a daring act and caused a lot of controversy), and they call that number ‘i’ or imaginary . (We will see that complex numbers are hardly more imaginary than √ 2 . ) Definition A complex number z is a number of the form z = a + b i with real numbers a,b and the symbol i that satisfies i 2 = 1 . We call a = Re( z ) the real part of z and b = Im( z ) the imaginary part of z . The real number a can be considered the complex number a + 0i . A complex number of the form z = b i is called purely imaginary . Addition, subtraction, and multiplication of complex numbers Complex numbers are easily added, subtracted and multiplied, if we keep the rule i 2 = 1 in mind and use the distributive laws. ( a + b i) ± ( c + d i) = ( a ± c ) + ( b ± d )i ( a + b i) × ( c + d i) = ac + bc i + ad i + bd i 2 = ( ac bd ) + ( ad + bc )i , since i 2 = 1. Examples 1. (3 + 5i) + (2 7i) = 5 2i 2. (0 . 5 + 1 . 7i) (0 . 8 2 . 6i) = . 3 + 4 . 3i 3. ( 3 + 2i) × (4 5i) = ( 12 ( 10)) + (15 + 8)i = 2 + 23i 4. (2 . 5i) × (3 + 4i) = (6 ( 2)) + ( 1 . 5 + 8)i = 8 + 6 . 5i 5. (9 + 2i) + 5 = (9 + 2i) + (5 + 0i) = 14 + 2i 6. 3i + (2 + 3i) = (0 3i) + (2 + 3i) = 2 + 0i = 2 1 MAT 1332: Additional Course Notes 2 7. 2 × (3 5i) = 6 10i 8. 3i × ( 1 + 4i) = 12 3i Before we look at inverses and division of complex numbers, we introduce the complex conju gate of a complex number. Definition and observation The complex conjugate of z = a + b i is ¯ z = a b i , i.e., we simply change the sign of the imaginary part. Since the multiplication z ¯ z = ( a + b i)( a b i) = a 2 + b 2 always produces a nonnegative real number, we can take the square root. We define the modulus or absolute value of z = a + b i as  z  = √ z ¯ z = p a 2 + b 2 . From the identity z ¯ z =  z  2 , we find the inverse of z to be 1 z = z 1 = ¯ z/  z  2 ....
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This note was uploaded on 03/19/2011 for the course MAT 1332 taught by Professor Munteanu during the Spring '07 term at University of Ottawa.
 Spring '07
 MUNTEANU
 Calculus, Algebra, Addition

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