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Chap 1 The Number System

# 4 show that if x y a r satisfy x y and a 0 then ax

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Unformatted text preview: iting z = x + iy , where x, y ∈ R, equation (20) becomes |(x − 1) + iy | = |(x + 1) + iy |, so that squaring both sides, we obtain (x − 1)2 + y 2 = (x + 1)2 + y 2 . On simplifying, we obtain x = 0. Interpreted geometrically, note that |z − 1| represents the distance between the points z and 1 on the Argand plane, while |z + 1| represents the distance between the points z and −1 on the Argand plane. Equation (20) thus asserts that z is equidistant from 1 and from −1. To achieve this, z must lie on the y -axis; in other words, we must have x = 0. Example 1.8.3. Consider a parallelogram OABC , where OB is a diagonal and AC is the other diagonal. We now place the parallelogram on the Argand plane so that the vertex O is precisely at the point 0. Suppose that the points A and C are represented by the complex numbers z and w respectively. Then the vertex B is represented by the complex number z + w. It is not diﬃcult to see that the midpoint of the diagonal OB is represented by the co...
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