Calculus 2 - Course Notes 2

# Calculus 2 - Course Notes 2 - MAT 1332 Calculus for Life...

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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 February 24, 2009

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MAT 1332: Frithjof Lutscher 14 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 defned 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 defne 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 . ) Defnition A complex number z is a number of the form z = a + b i with real numbers a, b and the symbol i that satis±es 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 bd i 2 =( ac - bd ad + bc )i Examples 1. (3 + 5i) + (2 - 7i) = 5 - 2i 2. (0 . 5+1 . 7i) - (0 . 8 - 2 . 6i) = - 0 . 3+4 . 3i 3. ( - 3 + 2i) * (4 - 5i) = ( - 12 - ( - 10)) + (15 + 8)i = - 2 + 23i 4. (2 - 0 . 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 7. 2 * (3 - 5i) = 6 - 10i 14
MAT 1332: Frithjof Lutscher 15 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. Defnition 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 non-negative real number, we can take the square root. We de±ne the modulus or absolute value of z = a + b i as | z | = z ¯ z = ± a 2 + b 2 . From the identity z ¯ z = | z | 2 , we ±nd the inverse of z to be 1 z = z - 1 z/ | z | 2 . Example 1 Start with z = 3 + 4i and w =2 - i . The complex conjugates are ¯ z =3 - 4i and ¯ w = 2 + i . The absolute values are | z | = 5 and | w | = 5 . The inverses are z - 1 = 1 25 (3 - 4i) ,w - 1 = 1 5 (2 + i) . Finally, we can divide z w = z ¯ w | w | 2 = 1 5 (2 + 11i) , w z = w ¯ z | z | 2 = 1 25 (2 - 11i) . Example 2 Start with z =1 - 4i and w =0 . 5+3i . The complex conjugates are ¯ z = 1+4i and ¯ w . 5 - 3i . The absolute values are | z | = 17 and | w | = ± 37 / 4 . The inverses are z - 1 = 1 17 (1 + 4i) - 1 = 4 37 (0 . 5 - 3i) .

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Calculus 2 - Course Notes 2 - MAT 1332 Calculus for Life...

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