Math
Complex Conjugate Calculator
Type a complex number such as 3 - 4i and the complex conjugate calculator flips the sign of the imaginary part to give 3 + 4i.
It then shows the product z · z̄ = a^2 + b^2 = |z|^2, which is the reason conjugates are used to divide complex numbers and to rationalize denominators, plus the modulus, the reciprocal, and the sum and difference with the conjugate.
Write it as a + bi. Fractions and decimals are kept exact.
Try an example
Result
Conjugate z̄
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- z · z̄ = a² + b² = |z|²
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- Modulus |z|
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- Reciprocal 1/z
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- z + z̄
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- z − z̄
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- Arguments of z and z̄
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More details (1 more)
- Statement
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Study path
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What the complex conjugate calculator solves
The conjugate of z = a + bi is z̄ = a - bi: the same real part, the opposite imaginary part. On the complex plane the conjugate is the mirror image of z across the real axis, so it has the same modulus and the opposite argument.
Why the product has no i
FOIL gives a^2 - abi + abi - b^2 i^2. The middle terms cancel and i^2 = -1 turns -b^2 i^2 into +b^2. That is why multiplying a The bottom number of a fraction; it says how many equal parts make one whole. by its conjugate clears the imaginary part when dividing complex numbers.
| z | z̄ | z · z̄ |
|---|---|---|
| 3 - 4i | 3 + 4i | 25 |
| -2i | 2i | 4 |
| 5 | 5 | 25 |
| 1/2 + i | 1/2 - i | 5/4 |
How to use it
- Type the complex number in a + bi form.
- Read the conjugate at the top, then the product z · z̄, which equals |z|^2.
- Use the The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5. line when you need 1/z, and the steps to see the FOIL work.
How to read the answer
The conjugate keeps the real part and negates the imaginary part. The product z · z̄ is always a non-negative real number equal to the square of the modulus; it is 0 only when z = 0. The sum z + z̄ is twice the real part and the difference is twice the imaginary part times i.
Common mistakes and edge cases
- Negating both parts: that gives -z, not the conjugate. Only the sign of b changes.
- Writing z · z̄ = a^2 - b^2. The minus from i^2 flips the sign, so it is a^2 + b^2.
- Forgetting that a real number is its own conjugate and a pure imaginary number bi has conjugate -bi.
- Trying to take 1/z when z = 0, which is undefined.
Worked examples
Typical
conj(3 - 4i) = 3 + 4i, product 25
Conjugate z̄
3 + 4i
Pure imaginary
conj(-2i) = 2i
Conjugate z̄
2i
Real number
conj(5) = 5
Conjugate z̄
5
Fractions
conj(1/2 + i) = 1/2 - i, product 5/4
Conjugate z̄
1/2 - i
Decimal input
conj(2.5 + 1.5i) = 5/2 - (3/2)i
Conjugate z̄
5/2 - (3/2)i
Not a complex number
Letters other than i are rejected
Conjugate z̄
Error
Frequently asked questions
What is the conjugate of a complex number?+
Flip the sign of the imaginary part: the conjugate of a + bi is a - bi. Geometrically it is the reflection of the point across the real axis.
Why is z times its conjugate a real number?+
(a + bi)(a - bi) = a^2 + b^2. The cross terms -abi and +abi cancel, and i^2 = -1 makes the last term positive. The result is |z|^2, the square of the distance from the origin.
How is the conjugate used to divide complex numbers?+
Multiply the numerator and denominator by the conjugate of the denominator. The denominator becomes the real number c^2 + d^2, so the quotient can be split into real and imaginary parts.
What is the conjugate of a real number?+
Itself. A real number has imaginary part 0, and -0 is still 0. Likewise the conjugate of a pure imaginary bi is -bi.
Does the conjugate change the modulus?+
No. |z̄| = |z| because a^2 + (-b)^2 = a^2 + b^2. Only the argument changes sign.
How do I find 1/z with the conjugate?+
1/z = z̄/(z · z̄) = z̄/|z|^2. For z = 3 - 4i that is (3 + 4i)/25 = 3/25 + (4/25)i.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
- OpenStax, College Algebra 2e, Section 2.4 “Complex Numbers” — Complex conjugates and dividing complex numbers.
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Last updated: September 4, 2026