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 $$\pi$$.↩︎