📐 Complex Conjugate Calculator
Find the conjugate of a complex number
z = a + bi
How to Use This Calculator
Enter Real Part (a)
Type the real part of your complex number. For 3 + 4i, enter 3.
Enter Imaginary Part (b)
Type the imaginary coefficient. For 3 + 4i, enter 4.
Click Find Conjugate
Press the button to calculate the conjugate and related properties.
Review Results
See the conjugate, magnitude, and product z × z̄.
Formula
If z = a + bi, then z̄ = a - bi
where a is the real part and b is the imaginary part
Example 1: Find conjugate of 3 + 4i
z = 3 + 4i
z̄ = 3 - 4i
Conjugate: 3 - 4i
Example 2: Find conjugate of -2 + 5i
z = -2 + 5i
z̄ = -2 - 5i
Conjugate: -2 - 5i
Example 3: Find conjugate of 7 - 3i
z = 7 - 3i (already the conjugate of 7 + 3i)
z̄ = 7 + 3i
Conjugate: 7 + 3i
About Complex Conjugate Calculator
The Complex Conjugate Calculator finds the conjugate of any complex number. The conjugate of a complex number z = a + bi is z̄ = a - bi, obtained by changing the sign of the imaginary part. Conjugates have important properties in complex arithmetic and are used extensively in mathematics and engineering.
When to Use This Calculator
- Complex Arithmetic: Find conjugates for division and other operations
- Rationalization: Eliminate imaginary parts from denominators
- Signal Processing: Work with complex exponentials and Fourier transforms
- Quantum Mechanics: Calculate probabilities and expectation values
- Circuit Analysis: Analyze AC circuits with complex impedances
Why Use Our Calculator?
- ✅ Instant Calculation: Get conjugate immediately
- ✅ Shows Magnitude: Displays |z| and |z̄|
- ✅ Product z × z̄: Shows important relationship
- ✅ Clear Display: Easy-to-read format
- ✅ Educational: Perfect for learning complex numbers
- ✅ Free Tool: No registration required
Key Properties of Complex Conjugates
- (z̄̄) = z: Conjugate of conjugate is original number
- z + z̄ = 2a: Sum is twice the real part
- z - z̄ = 2bi: Difference is twice the imaginary part
- z × z̄ = a² + b² = |z|²: Product equals magnitude squared
- (z₁ + z₂)̄ = z̄₁ + z̄₂: Conjugate of sum equals sum of conjugates
- (z₁ × z₂)̄ = z̄₁ × z̄₂: Conjugate of product equals product of conjugates
Real-World Applications
Division: Divide complex numbers by multiplying numerator and denominator by the conjugate of the denominator to get a real result.
Root Finding: Polynomials with real coefficients have complex roots in conjugate pairs.
Signal Processing: FFT operations use conjugate symmetry properties for efficient computation.
Tips for Using This Calculator
- Remember: conjugate changes only the imaginary part's sign
- The real part stays the same
- Conjugate is its own inverse: taking conjugate twice returns original
- Useful for rationalizing complex fractions
- Product z × z̄ is always a real, non-negative number
- Geometrically, conjugate is reflection across real axis in complex plane
Frequently Asked Questions
What is a complex conjugate?
The conjugate of z = a + bi is z̄ = a - bi, obtained by changing the sign of the imaginary part. Both numbers have the same magnitude but opposite imaginary components.
Why is the product z × z̄ always real?
Multiplying (a + bi)(a - bi) = a² - (bi)² = a² - b²i² = a² + b², which is always real and equals |z|².
What's the geometric meaning of conjugate?
In the complex plane, the conjugate is the reflection of the number across the real (x) axis. If z is at (a, b), then z̄ is at (a, -b).
Can I use conjugates to divide complex numbers?
Yes! To divide (a+bi)/(c+di), multiply numerator and denominator by (c-di). This eliminates the imaginary part in the denominator.
What happens when I take the conjugate twice?
You get the original number back: (z̄̄) = z. Taking the conjugate is its own inverse operation.
Are conjugates used in engineering?
Yes! In AC circuit analysis, impedance is complex. Conjugates help calculate power and analyze phase relationships in electrical systems.