Complex Conjugate Calculator
Find the complex conjugate of a + bi, compute the product z·z̄, and find the modulus.
This free online complex conjugate calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Results
Conjugate real part
0
Conjugate imaginary part
0
z · z̄
0
|z| (modulus)
0
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Complex Conjugate Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Complex Conjugate Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Complex Conjugate Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Complex Conjugate Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
About This Calculator
The Complex Conjugate Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Find the complex conjugate of a + bi, compute the product z·z̄, and find the modulus. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
About Complex Conjugate Calculator
The Complex Conjugate Calculator finds the conjugate of a complex number z = a + bi, which is z̄ = a - bi. The conjugate is obtained by negating the imaginary part while keeping the real part unchanged. Complex conjugates are fundamental in complex analysis, quantum mechanics, signal processing, and electrical engineering. The product z · z̄ = a² + b² = |z|² is always a non-negative real number, which is why conjugates are used to rationalize complex denominators (divide both numerator and denominator by the conjugate of the denominator). The Complex Conjugate Root Theorem states that for polynomials with real coefficients, complex roots always come in conjugate pairs: if a + bi is a root, then a - bi is also a root.
The Math Behind It
Formula Reference
Complex Conjugate
z̄ = a - bi when z = a + bi
Variables: The imaginary part is negated
Product with Conjugate
z · z̄ = a² + b² = |z|²
Variables: Always a non-negative real number
Worked Examples
Example 1: Find conjugate and modulus
Find the conjugate, product z·z̄, and modulus of 3 + 4i
z̄ = 3 - 4i, z·z̄ = 25, |z| = 5
Example 2: Rationalizing a denominator
Simplify (2 + i)/(3 - i)
1/2 + (1/2)i
Common Mistakes & Tips
- !Negating the real part instead of the imaginary part: the conjugate of a+bi is a-bi, not -a+bi
- !Forgetting that z·z̄ is always real and non-negative, never complex
- !Not applying the conjugate to the entire denominator when rationalizing
- !Assuming conjugation changes the modulus: |z| = |z̄| always
Related Concepts
Used in These Calculators
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Frequently Asked Questions
Why do complex roots of real polynomials come in conjugate pairs?
Because conjugation distributes over addition and multiplication and fixes real numbers. If P(z) = 0 and P has real coefficients, then P(z̄) = P(z)̄ = 0̄ = 0, so z̄ is also a root.
What is the geometric meaning of conjugation?
In the complex plane, the conjugate of z is the reflection of z across the real axis. It has the same distance from the origin (same modulus) but the opposite angle (negated argument).