Modulus and Amplitude (Argument) of a Complex Number

So far complex numbers have been purely algebraic objects. This section attaches a size and a direction to each one, which is what eventually lets us picture them as points in a plane.

The modulus (or absolute value) of $z = x+iy$ is defined as $|z| = \sqrt{x^2+y^2}$, always a non-negative real number, with $|z|=0$ exactly when $z=0$. If you think of $(x,y)$ as coordinates of a point, $|z|$ is simply that point's distance from the origin — the same Pythagorean quantity as for real coordinates.

The modulus satisfies several useful identities (each provable directly from the definition and the conjugate rules of Section 1.3):

and the triangle inequality $|\alpha+\beta| \le |\alpha| + |\beta|$, along with the parallelogram-style identity $|\alpha+\beta|^2 + |\alpha-\beta|^2 = 2\big(|\alpha|^2+|\beta|^2\big)$. The triangle inequality is the same intuitive fact as in ordinary geometry — going 'directly' from $0$ to $\alpha+\beta$ can never be longer than going via two separate legs of lengths $|\alpha|$ and $|\beta|$.

Now think of $z=x+iy$ (with $z\ne 0$) as a point $P(x,y)$ in a plane with perpendicular axes, and let $r = |z|$ be its distance from the origin $O$. Let $\theta$ be the angle that $OP$ makes with the positive $x$-axis (measured the usual trigonometric way). Basic right-triangle trigonometry — dropping a perpendicular from $P$ to the $x$-axis and checking all four quadrants — gives:

Any such $\theta$ is called an amplitude (or argument) of $z$; since $\theta$ and $\theta+2\pi k$ describe the same point for any integer $k$, a nonzero complex number has infinitely many arguments, differing by multiples of $2\pi$. To pin down a single value, we single out the one lying in $(-\pi, \pi]$ and call it the principal amplitude, written $\operatorname{Arg}(z)$.

Substituting $x=r\cos\theta$, $y=r\sin\theta$ back into $z=x+iy$ gives the polar form (also called the modulus–amplitude form):

often abbreviated $z = r\,\mathrm{cis}\,\theta$. This form is especially convenient for multiplication and division: if $z_1 = r_1\,\mathrm{cis}\,\theta_1$ and $z_2 = r_2\,\mathrm{cis}\,\theta_2$, expanding the product using the compound-angle formulas for sine and cosine shows

$z_1 z_2 = r_1 r_2\,\mathrm{cis}(\theta_1+\theta_2), \qquad \frac{z_1}{z_2} = \frac{r_1}{r_2}\,\mathrm{cis}(\theta_1-\theta_2)\ \ (z_2\ne 0)$ …