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Unformatted text preview: and diagram: 0 p p p p pθ r pp p x p p p z
y We shall study more carefully the triangle shown. By Pythagoras’s theorem, we have (6) Also (7) x = r cos θ and y = r sin θ. r2 = x2 + y 2 . Definition. Suppose that z = x + y i, where x, y ∈ R. We write |z | = x2 + y 2 and call this the modulus of z . On the other hand, any number θ ∈ R satisfying the equations (7) is called an argument of z , and denoted by arg z . Remarks. (1) Note that for a given z ∈ C, arg z is not unique. Clearly we can add any integer multiple of 2π to θ without aﬀecting (7). We sometimes call a real number θ ∈ R the principal argument of z if θ satisﬁes the equations (7) and −π < θ ≤ π . Note that it follows from (7) that y/x = tan θ. However, even with this restriction on θ, it is not meaningful to write (8) θ = tan−1 y . x To see this, draw ﬁrst of all the complex number z = 1 + i on the Argand diagram. Clearly the equations √ (7) are satisﬁed with r = 2 and θ = π/4. Further...
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