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Tutorial Reference

How to Calculate Square Root of Negative Numbers in Python

Learn how to compute square roots of negative numbers in Python using cmath for single values and numpy.emath for arrays, avoiding ValueError and nan results.

By default, calculating the square root of a negative value using the standard math library raises a ValueError: math domain error. This is because real number mathematics doesn't define square roots for negatives. To handle this, you need complex numbers using Python's specialized modules.

Single Values with cmath

The cmath (complex math) module is part of the standard library and returns complex results when appropriate:

import cmath

val = -16

result = cmath.sqrt(val)

print(f"Result: {result}")               # (4j)
print(f"Real part: {result.real}")       # 0.0
print(f"Imaginary part: {result.imag}")  # 4.0

Comparison with math Module

import math
import cmath

try:
    math.sqrt(-16)
except ValueError as e:
    print(f"math.sqrt error: {e}")

result = cmath.sqrt(-16)
print(f"cmath.sqrt result: {result}")  # 4j

Output:

math.sqrt error: math domain error
cmath.sqrt result: 4j

Arrays with numpy.emath

For NumPy arrays, regular np.sqrt() returns nan for negative values. Use np.emath.sqrt() for complex support:

import numpy as np

arr = np.array([4, -4, 25, -25])

regular = np.sqrt(arr)
print(f"np.sqrt: {regular}")

complex_result = np.emath.sqrt(arr)
print(f"np.emath.sqrt: {complex_result}")

Output:

<main.py>:6: RuntimeWarning: invalid value encountered in sqrt
np.sqrt: [ 2. nan  5. nan]
np.emath.sqrt: [2.+0.j 0.+2.j 5.+0.j 0.+5.j]

Working with Complex Results

import cmath

result = cmath.sqrt(-9)  # 3j

print(f"Real: {result.real}")       # 0.0
print(f"Imaginary: {result.imag}")  # 3.0
print(f"Magnitude: {abs(result)}")  # 3.0
print(f"Phase: {cmath.phase(result)}")  # 1.5707... (π/2 radians)

print(f"Squared: {result ** 2}")    # (-9+0j)

Output:

Real: 0.0
Imaginary: 3.0
Magnitude: 3.0
Phase: 1.5707963267948966
Squared: (-9+0j)

Creating Complex Numbers

z1 = 3 + 4j

z2 = complex(3, 4)

import cmath
z3 = cmath.sqrt(-25)  # 5j

print(z1, z2, z3)

Output:

(3+4j) (3+4j) 5j

Library Comparison

Module Result for √-1 Use Case
math ❌ ValueError Real numbers only
cmath ✅ 1j Single complex values
numpy 🟡 nan + warning Clean real-number arrays
numpy.emath ✅ 1j Mixed arrays with negatives

Quick Reference

Goal Method
Single negative sqrt cmath.sqrt(x)
Array with negatives np.emath.sqrt(arr)
Get real part result.real
Get imaginary part result.imag
Get magnitude abs(result)

Summary

Use cmath.sqrt() for single negative values, it returns proper complex numbers instead of raising errors. For NumPy arrays with potential negative values, use np.emath.sqrt() to avoid silent nan corruption. Both modules seamlessly handle the transition between real and complex results.

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