Modulus and Argument

Complex Analysis · Axiom Academy

Every complex number has a length and a direction — learn to read both straight off the plane. 1. The Modulus Is the Hypotenuse The modulus of z = a + bi is its distance from the origin in the complex plane. Drop the point onto the real and imaginary axes and you get a right triangle: a horizontal leg of length a , a vertical leg of length b , and a hypotenuse running from the origin out to z . That hypotenuse is |z| , so Pythagoras hands you the formula directly. The modulus is the hypotenuse of the a, b right triangle The argument of z is the angle swept counterclockwise from the positive real axis to the ray from the origin to z . In the right triangle, the opposite side is b and the adjacent side is a , so — and the angle is recovered with the arctangent. is measured counterclockwise from the positive real axis The bare formula only lands in the right place for Quadrant I . Because always returns an angle between -90° and 90° , you must adjust for the signs of a and b in the other quadrants — that is what the next step is about. Four points can share the same legs |a| = |b| = 2 yet point in four completely different directions. Watch one point of fixed length rotate through all four quadrants: the angle keeps climbing counterclockwise, but alone would collapse 135° and -45° onto the same value. The signs of a and b are what tell the quadrants apart.

This is the written version of the interactive lesson above. See the full Complex Analysis course.