Math

Instant, private, and free

Complex Conjugate Calculator.

Find the complex conjugate of a+bi.

On-device calculationNo signup
01

Set your values

Results update as you type.

Real part: 3

Real part

3.000000
Imaginary part: -4

Imaginary part

-4.000000

Results update automatically as you type.

Use Cases

Simplifying division of complex numbers

When dividing by a complex number, multiply numerator and denominator by the conjugate of the denominator to get a real denominator.

Example: For (2+3i)/(1-i), multiply by (1+i) to get a real denominator.

Finding the magnitude of a complex number

The magnitude (or modulus) of a+bi is sqrt(a^2+b^2), which can be found by multiplying the number by its conjugate and taking the square root.

Example: For 3+4i, conjugate is 3-4i, product is 25, magnitude is 5.

Frequently Asked Questions

What is a complex conjugate?
The complex conjugate of a complex number a+bi is a-bi. It has the same real part but the opposite sign on the imaginary part. For example, the conjugate of 3+4i is 3-4i.
How do I use this calculator?
Enter the real part in the 'Real part' field and the imaginary part in the 'Imaginary part' field. The calculator will output the conjugate in the form a-bi.
Why is the complex conjugate useful?
Complex conjugates are used to simplify expressions with complex numbers, especially when dividing or finding magnitudes. Multiplying a complex number by its conjugate yields a real number.

Tips & Common Mistakes

Tips

  • Remember to change the sign of the imaginary part only; the real part stays the same.
  • If the imaginary part is negative, the conjugate will have a positive imaginary part. For example, 5-2i becomes 5+2i.
  • Use the conjugate to rationalize denominators in complex fractions.
  • Check your input: the imaginary part should be the coefficient of i, not including the i itself.

Common Mistakes to Avoid

  • Forgetting to change the sign of the imaginary part.
  • Entering the imaginary part with the 'i' included (e.g., typing '4i' instead of '4').
  • Confusing the real and imaginary parts when entering values.

Last updated: August 13, 2026