Complex Conjugate Calculator
Result
How to Use
- Enter the real part of your complex number
- Enter the imaginary part of your complex number
- Click the "Calculate" button
- View the complex conjugate and related calculations
About Complex Conjugates
A complex conjugate is formed by changing the sign of the imaginary part of a complex number:
- For a complex number z = a + bi, its conjugate is z* = a - bi
- The product of a complex number and its conjugate is always a real number
- Complex conjugates are useful in division of complex numbers
- They help in finding the magnitude of complex numbers
Properties:
- (z*)* = z (conjugate of conjugate is the original number)
- (z₁ + z₂)* = z₁* + z₂* (conjugate of sum is sum of conjugates)
- (z₁ × z₂)* = z₁* × z₂* (conjugate of product is product of conjugates)
- z × z* = |z|² (product of number and its conjugate is square of magnitude)
Common Examples
| Complex Number | Conjugate | Product (z × z*) |
|---|---|---|
| 3 + 4i | 3 - 4i | 25 |
| 1 - 2i | 1 + 2i | 5 |
| -2 + 3i | -2 - 3i | 13 |
| 5i | -5i | 25 |
Applications
- Division of complex numbers
- Finding magnitude of complex numbers
- Solving complex equations
- Signal processing and electrical engineering
- Quantum mechanics and wave functions
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