How this instrument works
A complex number a+bi can be pictured as a point on a flat plane, with the real part a as its horizontal coordinate and the imaginary part b as its vertical coordinate. Its modulus is simply that point's distance from the origin, found via the Pythagorean theorem exactly as this site's Euclidean Distance calculator finds the distance between any two points. Its argument is the angle that position makes with the positive real axis, found with the same atan2 function used for reading angles off ordinary coordinates.
Together, the modulus and argument describe a complex number's POLAR form — an alternative to the ordinary a+bi (rectangular) description, useful because multiplying complex numbers in polar form is remarkably simple: moduli multiply together, and arguments simply add.
This site's companion Polar Form calculator runs the identical relationship in reverse, starting from a modulus and an argument and converting back to a+bi — the two pages together cover both directions of the same conversion.
- Enter the complex number's real part into the a field.
- Enter its imaginary part into the b field.
- Read Modulus: the number's distance from the origin.
- Read Argument: the angle its position makes with the positive real axis.
Worked example — the complex number 3+4i
3+4i has a modulus of √(3²+4²)=5 (a 3-4-5 right triangle) and an argument of atan2(4,3)≈0.927 radians, about 53.13° — the angle its position, plotted at (3,4), makes with the positive real axis.
The real number 1 (written as 1+0i) has a modulus of 1 and an argument of exactly 0 — sitting right on the positive real axis with no imaginary component at all. The pure imaginary number i (0+1i) has a modulus of 1 and an argument of exactly π⁄2 (90°) — sitting straight up the imaginary axis.
Questions
What is the modulus of a complex number?
Its distance from the origin when plotted on the complex plane, found via the Pythagorean theorem from its real and imaginary parts — modulus = √(a²+b²).
What is the argument of a complex number?
The angle its position makes with the positive real axis, found using atan2 on its imaginary and real parts — the same angle-finding function used for ordinary coordinate points.
Why convert a complex number into modulus and argument at all?
That form, called polar form, makes multiplying complex numbers together far simpler: moduli multiply and arguments simply add, rather than needing the full distributive expansion that multiplying a+bi forms directly would require.
How is this different from the Polar Form calculator on this site?
This page starts from a+bi and finds the modulus and argument; that page runs the identical relationship in reverse, starting from a modulus and argument and converting back into a+bi.
What is the modulus of a purely real or purely imaginary number?
It's just the absolute value of whichever part is nonzero — a purely real number a+0i has modulus |a|, and a purely imaginary number 0+bi has modulus |b|.