Chap 1 The Number System

22 2 4 3 i sin 2 4 3 5 2 2 cos 12 i sin 5 12 on

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Unformatted text preview: least the length of the third side. We have shown earlier that the cartesian coordinates (x, y ) are very useful for adding two complex numbers, whereas multiplication of complex numbers has a rather messy formula in cartesian coordinates. Let us use polar coordinates instead. Suppose that z = r(cos θ + i sin θ) where r, s, θ, φ ∈ R and r, s > 0. Then (9) zw = rs(cos θ + i sin θ)(cos φ + i sin φ) = rs((cos θ cos φ − sin θ sin φ) + i(cos θ sin φ + sin θ cos φ)) = rs(cos(θ + φ) + i sin(θ + φ)). and w = s(cos φ + i sin φ), It follows that if we represent complex numbers in polar coordinates, then multiplication of complex numbers simply means essentially multiplying the moduli and adding the arguments. On the other hand, it is not difficult to show that (10) z r = (cos(θ − φ) + i sin(θ − φ)). w s √ Example 1.6.3. Suppose that z = 1 + i and w = −1 − i 3. Since z= √ 2 cos π π + i sin 4 4 and w = 2 cos − 2π 3 + i sin − 2π 3 , it follows from (9) th...
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