Complex Conjugate Calculator
Find the complex conjugate, absolute value, and explore complex number properties
Calculate Complex Conjugate
Enter the real and imaginary parts of your complex number in the form a + bi:
Any real number
Coefficient of i
Your complex number:
Results
Complex Conjugate
The imaginary part changes sign
Absolute Value (Modulus)
Distance from origin on complex plane
Complex Plane Visualization
Step-by-Step Solution
Step 1: Identify Complex Number
Start with the complex number in standard form:
Where a is the real part and b is the imaginary part.
Step 2: Apply Conjugation Rule
The complex conjugate changes the sign of the imaginary part:
The real part stays the same, only the imaginary part changes sign.
Step 3: Calculate Absolute Value
The modulus (absolute value) is calculated as:
This represents the distance from the origin to the point on the complex plane.
Step 4: Verify Using Multiplication
Multiplying a complex number by its conjugate gives the square of the modulus:
This equals |z|² = 3.605551² = 13.000000
Complex Conjugate Properties
Double Conjugation
The conjugate of a conjugate returns the original number
(2 − 3i)̄ = 2 + 3i
Geometric Interpretation
Conjugation reflects the point across the real axis
Original: (2, 3), Conjugate: (2, -3)
Same Modulus
A complex number and its conjugate have the same absolute value
|2 + 3i| = |2 − 3i| = 3.605551
Example Problems
Example 1
Example 2
Example 3
Example 4
Quick Reference
Key Properties
Complex Plane
Horizontal axis = Real part
Vertical axis = Imaginary part
Conjugate = reflection across x-axis
Distance from origin = |z|
Applications
Understanding Complex Conjugates
What is a Complex Conjugate?
The complex conjugate of a complex number z = a + bi is the number z̄ = a - bi. It's obtained by changing the sign of the imaginary part while keeping the real part unchanged. This simple operation has profound mathematical significance and geometric interpretation.
Geometric Meaning
On the complex plane, conjugation represents a reflection across the real axis (x-axis). If you imagine placing a mirror along the horizontal axis, the conjugate would be the reflection of the original complex number.
Why Are Conjugates Important?
- •They help convert complex division to real arithmetic
- •Essential in finding polynomial roots
- •Used in quantum mechanics and signal processing
- •Help calculate distances and magnitudes
Key Mathematical Properties
Multiplicative Property
When you multiply a complex number by its conjugate, you always get a real number:
Conjugate Pairs
Complex roots of polynomials with real coefficients come in conjugate pairs:
Division Simplification
Conjugates help rationalize complex fractions:
Real-World Applications
Electrical Engineering
AC circuit analysis, impedance calculations, and power factor corrections all rely heavily on complex conjugates.
Quantum Physics
Wave function normalization and probability calculations require complex conjugates to ensure real-valued results.
Signal Processing
Fourier transforms, filter design, and signal analysis use conjugates for frequency domain calculations.