Complex Conjugate Calculator
Find complex conjugates and explore their properties
What is a Complex Conjugate?
The complex conjugate of a complex number z = a + bi is z̄ = a - bi, where we keep the real part (a) the same but change the sign of the imaginary part (b). Complex conjugates are fundamental in complex analysis and have important applications in mathematics, physics, and engineering.
Properties of Complex Conjugates
- The product of a complex number and its conjugate is always real and equals the square of its modulus: z × z̄ = |z|²
- The sum of a complex number and its conjugate is twice the real part: z + z̄ = 2a
- The conjugate of a sum is the sum of conjugates: (z₁ + z₂)̄ = z̄₁ + z̄₂
- The conjugate of a product is the product of conjugates: (z₁ × z₂)̄ = z̄₁ × z̄₂
- The conjugate of the conjugate is the original number: (z̄)̄ = z
Applications
Complex conjugates are used in many areas:
- Rationalizing denominators with complex numbers
- Finding modulus and argument of complex numbers
- Solving complex equations
- Signal processing and electrical engineering
- Quantum mechanics calculations
- Wave function analysis
Examples
Example 1: Basic Conjugate
- Given: z = 3 + 2i
- Conjugate: z̄ = 3 - 2i
- Product: z × z̄ = (3 + 2i)(3 - 2i) = 13
- Sum: z + z̄ = (3 + 2i) + (3 - 2i) = 6
Example 2: Pure Imaginary Number
- Given: z = 5i
- Conjugate: z̄ = -5i
- Product: z × z̄ = 25
- Sum: z + z̄ = 0
How to Find Complex Conjugates
- Identify the real and imaginary parts of the complex number
- Keep the real part exactly the same
- Change the sign of the imaginary part (+ to - or - to +)
- Write the new complex number with the changed sign
Common Mistakes to Avoid
- Changing the sign of the real part instead of the imaginary part
- Forgetting that real numbers are their own conjugates
- Incorrectly applying conjugate properties to other operations
- Not considering the distributive property when finding conjugates of expressions
- Forgetting to change all imaginary parts in a complex expression