Complex Conjugate Calculator
Use our complex conjugate calculator to calculate complex conjugate values instantly. Includes what is a complex conjugate, how to find complex conjugate, the complex conjugate of a complex number (a+bi → a−bi), a complex conjugate calculator with steps, and tips for how to calculate complex numbers (Excel and TI-84).
What is a Complex Conjugate?
A complex number is written as a + bi, where a is the real part and b is the imaginary part (and i is the imaginary unit, i² = −1).
The complex conjugate of a complex number is found by changing the sign of the imaginary part. So the complex conjugate of a complex number a + bi is a − bi.
If you’re asking what is a complex conjugate, it’s a paired number that mirrors the complex number across the real axis on the complex plane. Complex conjugates are used in simplifying division of complex numbers, finding magnitudes, and solving many algebra and calculus problems involving complex numbers.
Complex Conjugate Formula
To calculate complex conjugate, keep the real part the same and flip the sign of the imaginary part.
Only the sign of the imaginary part changes.
This is why conjugates help when dividing complex numbers.
Same real part, opposite imaginary sign.
That’s the full procedure for finding the conjugate.
How to Use the Complex Conjugate Calculator
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Enter the real part (a).
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Enter the imaginary part (b).
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The calculator returns the complex conjugate of a complex number: a − bi.
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It can also show the magnitude-related identity a² + b² when needed.
Frequently Asked Questions
The complex conjugate of a + bi is a − bi. It keeps the real part and flips the sign of the imaginary part.
Write the complex number in the form a + bi, then change the sign of the imaginary part: a + bi → a − bi.
It is 7 + 3i (flip the sign on the imaginary part).
First compute the conjugate, then use identities like (a+bi)(a−bi)=a²+b² to simplify expressions or rationalize denominators.
Add/subtract by combining real parts and imaginary parts separately. Multiply using FOIL and i² = −1. Divide by multiplying numerator and denominator by the complex conjugate of the denominator.
Excel supports complex numbers with functions like COMPLEX(), IMCONJUGATE(), IMSUM(), IMPRODUCT(), and IMDIV(). IMCONJUGATE() returns the complex conjugate.
Put the TI-84 in a complex number mode (a+bi). Then you can enter numbers like 3+4i and use built-in operations. Many TI-84 menus also include conjugate-related functions or you can manually flip the sign.
Yes—the page includes the rule a+bi → a−bi and examples so you can follow the steps.