`cmath`

The Python cmath module provides mathematical functions for complex numbers. Unlike the math module, which raises an exception for operations like the square root of a negative number, cmath always returns a complex result.

Here’s a quick example:

>>> import cmath

>>> cmath.sqrt(-1)
1j
>>> cmath.sqrt(-4)
2j

Key Features

Computes square roots, logarithms, and exponentials for complex numbers
Converts between rectangular and polar coordinate representations
Provides trigonometric and hyperbolic functions for complex arguments
Includes classification functions to test for infinity, NaN (not a number), and closeness
Accepts integers, floats, and complex numbers as arguments
Always returns a complex number, even when the imaginary part is zero
Exposes mathematical constants including pi, e, tau, inf, and nan

Frequently Used Classes and Functions

Object	Type	Description
`cmath.sqrt()`	Function	Returns the square root of a complex number
`cmath.exp()`	Function	Returns e raised to the power of a complex number
`cmath.log()`	Function	Returns the natural (or given-base) logarithm of a complex number
`cmath.phase()`	Function	Returns the phase (argument) of a complex number as a float
`cmath.polar()`	Function	Converts a complex number to polar coordinates as an (r, phi) tuple
`cmath.rect()`	Function	Converts polar coordinates to a complex number
`cmath.isclose()`	Function	Tests whether two complex values are close within a given tolerance
`cmath.isfinite()`	Function	Returns True if both real and imaginary parts are finite

Examples

Converts between rectangular and polar coordinate forms:

>>> import cmath

>>> z = 3 + 4j
>>> r, phi = cmath.polar(z)
>>> r
5.0
>>> cmath.rect(r, phi)
(3.0000000000000004+3.9999999999999996j)

Verifies Euler’s formula using the isclose() function:

>>> cmath.isclose(cmath.exp(cmath.pi * 1j) + 1, 0, abs_tol=1e-9)
True

Checks whether a complex value is finite or NaN:

>>> cmath.isfinite(2 + 3j)
True
>>> cmath.isnan(complex(float("nan"), 0))
True

Common Use Cases

The most common tasks for cmath include:

Computing square roots and logarithms of negative or complex numbers
Converting between rectangular and polar representations of complex numbers
Evaluating trigonometric functions with complex arguments
Testing the validity of complex values in numerical computing

Real-World Example

Converting a set of phasors from rectangular to polar form to inspect their magnitudes and phases:

>>> import cmath

>>> signals = [1 + 1j, 3 + 4j, -1 + 0j]

>>> for z in signals:
...     r, phi = cmath.polar(z)
...     print(f"z={z}  magnitude={r:.4f}  phase={phi:.4f}")
...
z=(1+1j)  magnitude=1.4142  phase=0.7854
z=(3+4j)  magnitude=5.0000  phase=0.9273
z=(-1+0j)  magnitude=1.0000  phase=3.1416

The polar form gives the amplitude and phase angle of each signal directly, which is a common requirement in signal processing and electrical engineering.

Tutorial

Simplify Complex Numbers With Python

In this tutorial, you'll learn about the unique treatment of complex numbers in Python. Complex numbers are a convenient tool for solving scientific and engineering problems. You'll experience the elegance of using complex numbers in Python with several hands-on examples.

intermediate python

For additional information on related topics, take a look at the following resources: