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Chapter 1 : The Number System page 14 of 19 First Year Calculus c W W L Chen, 1982, 2005 equations, since each of the real part and the imaginary part of the complex equation gives rise to a real equation. It follows that to obtain a genuine locus, these two equations should be essentially the same. Here, we shall restrict our discussion to three examples. The reader is advised to draw some pictures. Example 1.8.1. The equation of a circle can be given by (18) |z − c| = r. To see this, suppose that z = x + iy and c = a + ib, where x, y, a, b ∈ R. Then |z − c|2 = |(x + iy ) − (a + ib)|2 = |(x − a) + i(y − b)|2 = (x − a)2 + (y − b)2 , so that we have the equation (x − a)2 + (y − b)2 = r2 . Note that the equation (18) can also be written in the form (19) (z − c)(z − c) = r2 . Note also that equation (19) is in invariant under conjugation; in other words, the conjugate of (19) is exactly the same as (19). Example 1.8.2. The equation (20) |z − 1| = |z + 1| represents a straight line. For wr...
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