44. Argand’s diagram

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

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 $$x=\mathbb{R}(z),y=\mathbb{I}(z),r=|z|,\theta =\mathrm{am}z.$$

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 $\mathrm{am}z$ 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.