the modulus

The modulus of a complex number z, written |z|, is simply its distance from the origin in the complex plane. If z = x + i y sits at the point (x, y), then by the Pythagorean theorem that distance is |z| = the square root of (x^2 + y^2). It is the length of the arrow from 0 to z. The modulus is always a real number that is zero or positive, and it is zero only for z = 0. For a real number it reduces to the ordinary absolute value, which is why the same notation is used.

A slicker way to compute it uses the conjugate: |z|^2 = z times z-bar = x^2 + y^2, with no square root in sight until the end. The modulus behaves beautifully under multiplication and division: |z w| = |z| |w| and |z/w| = |z|/|w|. (Addition is the exception — |z + w| is generally NOT |z| + |w|; that gap is the triangle inequality.) For z = 3 + 4i, for instance, |z| = the square root of (9 + 16) = 5.

The modulus is how 'size' enters complex analysis, and it is everywhere. Convergence of sequences and series is measured by it; the modulus of a function controls deep results like the maximum modulus principle; and Cauchy's estimates bound derivatives by the modulus of the function. Whenever you need to say one complex quantity is small or large, you mean its modulus — because, unlike z itself, |z| is an honest real number you can compare.