1.1. Definition and Simple Properties 7 Part (h) follows directly from Part (g) and the other parts of the theorem. In fact, |z + w|2 = (z + w)(z + w) = |z|2 + 2 Re(zw) + |w|2 ≤ |z|2 + 2|z||w| + |w|2 = (|z| + |w|)2. On taking square roots, we conclude |z + w| ≤ |z| + |w|, which is Part (h). The inequalities in the following theorem are used extensively – particularly in the next section. The proofs are very simple and are left as an exercise (Exercise 1.1.12). Theorem 1.1.8. If z = x + iy, then max{|x|, |y|} ≤ |z| ≤ |x| + |y|. Inversion and Division. Recall that the inverse of a non-zero complex number z = x + iy is z−1 = x − iy x2 + y2 = z |z|2 = z zz . Stating this in the last form makes the identity zz−1 = 1 obvious. This also suggests the right way to do complex division problems in general: to express w/z as a complex number in standard form (as a real number plus i times a real number), simply multiply both numerator and denominator by z. That is, w z = wz zz = wz |z|2 . The number wz is then easily put in standard form and the problem is finished by dividing by the real number |z|2. Example 1.1.9. Express 1 2 + 3i in the standard form x + yi. Solution: 1 2 + 3i = 2 − 3i (2 + 3i)(2 − 3i) = 2 − 3i |2 + 3i|2 = 2 13 − 3 13 i. Example 1.1.10. Express 3 + 4i 3 − 4i and 3 − 4i 3 + 4i in standard form. Solution: 3 + 4i 3 − 4i = (3 + 4i)2 |3 + 4i|2 = − 7 25 + 24 25 i, 3 − 4i 3 + 4i = (3 − 4i)2 |3 − 4i|2 = − 7 25 − 24 25 i.

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