Argand’s diagram. Let \(P\) (Fig. 24) be the point \((x, y)\), \(r\) the length \(OP\), and \(\theta\) the angle \(XOP\), so that \[x = r\cos\theta,\quad y = r\sin\theta,\quad r = \sqrt{x^{2} + y^{2}},\quad \cos\theta : \sin\theta : 1 :: x : y : r.\]

We denote the complex number \(x + yi\) by \(z\), as in § 43, and we call \(z\) the complex variable. We call \(P\) the point \(z\), or the point corresponding to \(z\); \(z\) the argument of \(P\), \(x\) the real part, \(y\) the imaginary part, \(r\) the modulus, and \(\theta\) the amplitude of \(z\); and we write \[x = \mathbb{R}(z),\quad y = \mathbb{I}(z),\quad r = |z|,\quad \theta = \operatorname{am} z.\]

When \(y = 0\) we say that \(z\) is real, when \(x = 0\) that \(z\) is purely imaginary. Two numbers \(x + yi\), \(x – yi\) which differ only in the signs of their imaginary parts, we call conjugate. It will be observed that the sum \(2x\) of two conjugate numbers and their product \(x^{2} + y^{2}\) are both real, that they have the same modulus \(\sqrt{x^{2} + y^{2}}\) and that their product is equal to the square of the modulus of either. The roots of a quadratic with real coefficients, for example, are conjugate, when not real.

It must be observed that \(\theta\) or \(\operatorname{am} z\) is a many-valued function of \(x\) and \(y\), having an infinity of values, which are angles differing by multiples of \(2\pi\).1 A line originally lying along \(OX\) will, if turned through any of these angles, come to lie along \(OP\). We shall describe that one of these angles which lies between \(-\pi\) and \(\pi\) as the principal value of the amplitude of \(z\). This definition is unambiguous except when one of the values is \(\pi\), in which case \(-\pi\) is also a value. In this case we must make some special provision as to which value is to be regarded as the principal value. In general, when we speak of the amplitude of \(z\) we shall, unless the contrary is stated, mean the principal value of the amplitude.

Fig 24 is usually known as Argand’s diagram.

  1. It is evident that \(|z|\) is identical with the polar coordinate \(r\) of \(P\), and that the other polar coordinate \(\theta\) is one value of \(\operatorname{am} z\). This value is not necessarily the principal value, as defined below, for the polar coordinate of § 22 lies between \(0\) and \(2\pi\), and the principal value between \(-\pi\) and \(\pi\).↩︎

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