Complex Conjugate Calculator – Guide & Formulas
Find the complex conjugate of any complex number instantly. Supports addition, subtraction, multiplication, and division of complex conjugate pairs.
Find the complex conjugate of any complex number with our free calculator. Enter real and imaginary parts to get the conjugate, modulus, and perform conjugate arithmetic operations.
Key Takeaway
Use the free Complex Conjugate Calculator to find the complex conjugate of any complex number instantly. supports addition, subtraction, multiplication, and division of complex conjugate pairs. Get instant results with step-by-step explanations.
How to Use the Complex Conjugate Calculator
- Enter the real part (a) of the complex number a + bi.
- Enter the imaginary part (b) of the complex number a + bi.
- Review the conjugate a - bi, modulus, and product (a² + b²).
- Use the optional second complex number to perform conjugate operations.
The Formula
Variable Definitions
- z: A complex number a + bi
- z̄: The complex conjugate of z
- a: The real part of the complex number
- b: The imaginary part of the complex number
- |z|: The modulus (magnitude) of the complex number: √(a² + b²)
Finding the Conjugate of 3 + 4i
Compute the conjugate, modulus, and product of z = 3 + 4i.
- Step 1: Identify a = 3, b = 4.
- Step 2: Conjugate: z̄ = 3 - 4i.
- Step 3: Modulus: |z| = √(3² + 4²) = √(9 + 16) = √25 = 5.
- Step 4: Product: z × z̄ = 3² + 4² = 9 + 16 = 25.
Frequently Asked Questions
What is a complex conjugate?
The complex conjugate of a complex number a + bi is a - bi. It is obtained by changing the sign of the imaginary part while keeping the real part the same.
Why are complex conjugates useful?
Complex conjugates are used to simplify fractions with complex denominators (rationalizing), compute the modulus squared of a complex number, and find roots of polynomials with real coefficients.
What happens when you multiply a number by its conjugate?
Multiplying z by z̄ gives a real number: z × z̄ = a² + b² = |z|². This eliminates the imaginary part entirely.
Is the conjugate of a real number the same?
Yes. If b = 0, then z = a + 0i = a, and its conjugate is a - 0i = a. Real numbers are their own conjugates.
How do you divide complex numbers using conjugates?
To divide (a+bi)/(c+di), multiply numerator and denominator by the conjugate of the denominator (c-di). This rationalizes the denominator to c² + d².
What is the conjugate of zero?
The conjugate of 0 + 0i is 0 - 0i = 0. Zero is its own conjugate.
Does conjugation change the modulus?
No. The modulus of z and z̄ are equal: |z| = |z̄| = √(a² + b²). Conjugation reflects the number across the real axis without changing its distance from the origin.
What are conjugate pairs in polynomial roots?
If a polynomial with real coefficients has a complex root a + bi, then its conjugate a - bi is also a root. This ensures all non-real roots come in conjugate pairs.