Complex Conjugate Calculator

Enter the real and imaginary parts of a complex number z = a + bi to find its complex conjugate a − bi, along with the modulus, the product z·z̄, and the argument.

Quick Facts

Formula
z̄ = a − bi for z = a + bi
z and z̄ share the same modulus; z · z̄ = a² + b² is always real.

Your Results

Calculated
Complex conjugate (z̄)
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a − bi, reflected across the real axis
Modulus |z|
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√(a² + b²), same for z and z̄
Product z · z̄
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a² + b² = |z|² (always real)
Argument of z̄
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Angle from positive real axis

Ready

Enter the real and imaginary parts, then press Calculate.

How the Complex Conjugate works

Every complex number z can be written as z = a + bi, where a is the real part and b is the imaginary part (the coefficient of i, with i² = −1). The complex conjugate, written z̄ or z*, flips the sign of only the imaginary part: z̄ = a − bi. On the complex plane this is a mirror reflection of the point (a, b) across the horizontal (real) axis to (a, −b) — the real part never changes.

Formula and method

Given a and b, this calculator computes four related quantities using the standard identities:

  • Conjugate: z̄ = a − bi
  • Modulus: |z| = √(a² + b²), which is identical for z and z̄ since squaring removes the sign of b
  • Product: z · z̄ = (a + bi)(a − bi) = a² + b², a real, non-negative number equal to |z|²
  • Argument of z̄: atan2(−b, a), the angle from the positive real axis to z̄ — the negative of the argument of z, since reflecting across the real axis reverses the angle's sign

Common sources of error

  • Sign slip: the conjugate only negates the imaginary part — the real part a is always left unchanged.
  • Confusing conjugate with negation: −z = −a − bi flips both signs and points the opposite direction; z̄ = a − bi flips only b and reflects across the real axis. They are different operations.
  • Degrees vs. radians: the argument is only meaningful with its unit attached — check which one the calculator (or your textbook) is using.

Checking your result

A few identities make it easy to verify an answer by hand: taking the conjugate of z̄ should return the original z (the operation is its own inverse); the modulus of z̄ should exactly match the modulus of z; and the product z · z̄ should always come out non-negative and free of any i term. If any of those checks fail, recheck the sign entered for b.

Applications

Conjugates are the standard tool for dividing complex numbers — multiplying numerator and denominator by the conjugate of the denominator clears the imaginary part from the bottom. They also appear when computing the modulus of a complex number, in AC circuit analysis (impedance and power calculations), in signal processing, and in the fact that polynomials with real coefficients always have complex roots in conjugate pairs.

Frequently Asked Questions

What is the complex conjugate of a + bi?
The complex conjugate of a + bi is a − bi: the real part stays the same and the sign of the imaginary part flips. Geometrically this reflects the point across the real axis on the complex plane, so z and its conjugate sit the same distance from the origin but on opposite sides of the horizontal axis.
Why does z times its conjugate always give a real number?
(a+bi)(a−bi) expands to a² − (bi)², and since i² = −1 that becomes a² + b², a real number with no imaginary part. This product equals the square of the modulus, |z|², which is why multiplying by the conjugate is the standard way to clear an imaginary term from a denominator.
How is the modulus of z related to its conjugate?
A complex number and its conjugate always have the same modulus: |z| = |z̄| = √(a² + b²). Only the argument (angle) changes — the argument of the conjugate is the negative of the argument of z, since reflecting across the real axis flips the angle's sign.
What is the complex conjugate used for?
Conjugates are used to divide complex numbers by multiplying the numerator and denominator by the conjugate of the denominator, which clears the imaginary part from the bottom. They are also used to compute the modulus of a complex number and to show that a polynomial with real coefficients has complex roots in conjugate pairs.

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