AMS10sn1 - ams 10/10A supplementary notes ucsc A Primer on...

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ams 10/10A supplementary notes ucsc A Primer on Complex Numbers c 2009, Yonatan Katznelson 1. Imaginary and complex numbers. One of the fundamental properties of the real numbers is that the square of a real number is always nonnegative . I.e., if x is a real number, then x 2 0. This implies, among other things, that certain quadratic equations don’t have real solutions. In particular, the equation x 2 = - 1, has no solution in real numbers. This doesn’t mean however, that the equation cannot have a solution. By the 16th century, it became apparent to various mathematicians that - 1 would be very useful and this new number, and its multiples, entered slowly into common use. As with all important constants, a special symbol was eventually designated to represent - 1. Definition 1. An imaginary number is a number of the form b i , where b is real and (1.1) i = - 1 . With the set of imaginary numbers in hand, we can find a square root for every real number, positive or negative. Indeed, following the usual algebraic rules, we have ( b i ) 2 = b 2 · i 2 = b 2 · ( - 1) = - b 2 , for any real number b . Now, if α < 0 and b = p | α | (remember, | α | > 0 so it has a real square root), then - b 2 = α , so b i = α. Thus, for example, - 9 = 3 i , - 100 = 10 i and - 2 = 2 · i . More is true. By combining real and imaginary numbers, we can solve any qua- dratic equation. Example 1.1. Solve the equation x 2 + 2 x + 2 = 0. Using the quadratic formula, we find the two solutions z 1 = - 2 + 4 - 8 2 = - 1 + 2 i and z 2 = - 2 - 4 - 8 2 = - 1 - 2 i . Note that the two solutions are neither real numbers, nor are they purely imaginary. 1
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2 Definition 2. A complex number is a number of the form z = a + b i , , where a and b are real numbers, and i = - 1 . The (real) numbers a and b are called the real and imaginary parts of z , respectively, and we often use the notation a = Re ( z ) and b = Im ( z ) . If Re ( z ) = 0, then z is an imaginary number and if Im ( z ) = 0, then z is a real number. In other words, the real numbers and the imaginary numbers are subsets of the complex numbers. The boldfaced letter R is used to denote the set of real numbers and the boldfaced letter C is used to denote the set of all complex numbers. The two solutions of the quadratic equation in Example 1.1 have the same real part and their imaginary parts are opposite. I.e., using the notation above, we have Re ( z 1 ) = Re ( z 2 ) and Im ( z 1 ) = - Im ( z 2 ). This is not a coincidence, and there is also a name for this. Definition 3. The complex conjugate of a + b i is the number a - b i . We use a bar over the number to denote the conjugate, i.e., a + b i = a - b i . If a, b and c are real numbers, and b 2 - 4 ac < 0, then the quadratic equation (1.2) ax 2 + bx + c = 0 has no real solutions. But, as in Example 1.1, there are always complex solutions and these solutions come in conjugate pairs. Namely, the solutions to equation (1.2) are given by α + β i and α - β i , where (from the quadratic formula) α = - b 2 a and β = p | b 2 - 4 ac | 2 a .
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