Chap 1 The Number System

# It follows that a a2 b2 a a2 b2 b a bi i

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Unformatted text preview: the complex number z = cos θ + i sin θ, noting that |z | = cos2 θ + sin2 θ = 1. Remarks. (1) Formally, Proposition 1E is proved by induction; see Example 1.2.4. (2) In the notation eiθ = cos θ + i sin θ, de Moivre’s theorem is the observation that einθ = (eiθ )n . Example 1.6.5. We have cos 3θ + i sin 3θ = (cos θ + i sin θ)3 = cos3 θ + 3i cos2 θ sin θ + 3i2 cos θ sin2 θ + i3 sin3 θ = (cos3 θ − 3 cos θ sin2 θ) + i(3 cos2 θ sin θ − sin3 θ). It follows that cos 3θ = cos3 θ − 3 cos θ sin2 θ and sin 3θ = 3 cos2 θ sin θ − sin3 θ. Remark. It can be shown that the conclusion of de Moivre’s theorem remains true for every n ∈ Q. 1.7. Finding Roots Let us try to ﬁnd the square roots of the complex number a + bi, where a, b ∈ R. We are therefore looking for complex numbers x + y i, where x, y ∈ R and (x + y i)2 = a + bi. We may assume that b = 0, otherwise the solution is trivial. Since (x + y i)2 = (x2 − y 2 ) + 2xy i, we must have (11) (12) I...
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