Conjugate Calculator - Complex Conjugate, Modulus, and Hermitian

A conjugate calculator is a small tool that turns a complex number z = a + bi into its complex conjugate z̄ = a - bi by negating the imaginary part, then reports |z| and z·z̄ so you can read the modulus and the squared modulus without doing the algebra by hand.

• • Find a - bi for any a + bi: Type the real and imaginary parts and read the conjugate as a - bi in a single line, so you can drop the result into a worksheet or a textbook problem set.
• • Read |z| and z·z̄ without retyping: Use the same input row to read the modulus and the real product z·z̄, which removes the chance of transcribing the wrong number into a follow-up step.
• • Sanity-check homework or exam answers: When a problem says "find the conjugate of 3+4i" or asks for |z| and z·z̄, use the calculator as a quick second opinion before turning in the work.

The word conjugate shows up in two settings that students compute by hand: the complex conjugate z̄ of a single complex number, and the conjugate transpose A* of a small complex matrix. The group-theoretic conjugate g·h·g⁻¹ lives in abstract algebra and uses a different workflow, so it sits outside the form on this page. This calculator focuses on the single-number case, where you enter a and b, and read z̄ = a - bi along with the modulus |z| and the product z·z̄. The conjugate transpose A* is described in the key concepts below, and a dedicated 2x2 matrix Hermitian tool is a good follow-up if you need the matrix form.

Because z·z̄ is always a nonnegative real number, the calculator uses that product as a check: when the displayed product matches a² + b², you have confirmed that the conjugate is correct. The same idea carries over to matrices, where A·A* is a Hermitian positive semidefinite matrix.

When you also need to add, subtract, multiply, or divide the same complex number, pair this tool with our Complex Number Calculator so the conjugate result stays in one place.

Behind the form, the conjugate calculator applies a single rule: keep the real part fixed and flip the sign of the imaginary part. Every other result on the page is a small arithmetic follow-up to that one step, so the output never depends on a hidden or hard-coded constant.

• a: Real part of the input complex number, the term that does not get flipped when you take the conjugate.
• b: Imaginary part of the input complex number; the conjugate negates this term so b becomes -b.
• z̄: Complex conjugate of z, written a - bi, equal to z reflected across the real axis on the complex plane.
• |z|: Modulus of z, the distance from the origin to the point (a, b) in the complex plane.
• z·z̄: Product of z and its conjugate; the imaginary parts cancel and the result equals a² + b², a nonnegative real number.

Once you enter a and b, the script sets a_conjugate = a and b_conjugate = -b, so the conjugate is the tuple (a, -b). It then computes |z| = sqrt(a² + b²) and z·z̄ = a² + b² from the same a and b, with no hidden constants and no normalization step.

According to Wikipedia, the complex conjugate of a+bi is a-bi, obtained by negating the imaginary part while keeping the real part fixed.

If you also want the angle Arg(z) for the same input, send the result through our Cartesian to Polar Calculator once you have a and b.

Four short ideas carry the whole calculator. Once you have each of them in mind, the output panel stops looking like a wall of numbers and starts looking like a small set of named objects.

The reflection property is also the reason nth roots of a complex number come in conjugate pairs, which you can see in the Complex Root Calculator when the number of roots is even.

Pick the mode that matches your problem, enter the real and imaginary parts, and read the result. The default values are the classic 3 + 4i example, so the page is correct out of the box.

1. 1 Enter the real part a: Type the real component of z = a + bi into the Real part box. Positive and negative values are both fine, and a can be 0 for a purely imaginary input.
2. 2 Enter the imaginary part b: Type the imaginary component into the Imaginary part box. Use a negative value for inputs like 3 - 4i, since the calculator negates b to form the conjugate.
3. 3 Read the conjugate from the result panel: The hero line shows the full conjugate as a - bi, so 3 + 4i becomes 3 - 4i. The Modulus and Product rows update at the same time.
4. 4 Check the z·z̄ row for a real number: If z·z̄ ever comes back with a nonzero imaginary part, the input is not in the expected form. Re-enter a and b as the real and imaginary parts of z and the product will return to a real value.
5. 5 Copy the result into your work: Use the displayed z̄, |z|, and z·z̄ directly in the next step. The hero line uses the same a - bi form you would write by hand, so you do not need to reformat anything before pasting it into a solution.

Try the default example first: a = 3 and b = 4. The result panel should read z̄ = 3 - 4i, |z| = 5, and z·z̄ = 25. If those match the 3-4-5 right triangle, the calculator is set up correctly and you can swap in your own a and b.

If you only need |z| and want a refresher on the absolute value of a real number, our Absolute Value Calculator covers the real-number case with the same sqrt(a²) approach.

The calculator is small, but it removes three classes of small mistakes that show up over and over again when students work through complex number problems by hand.

• • Removes sign errors on the imaginary part: Hand-calculations often slip on whether b becomes -b or +b. The script enforces the rule once and shows the result as a - bi, so the imaginary part is always the negation of the input.
• • Gives the modulus and the product in one place: You get |z| and z·z̄ alongside the conjugate, so you do not have to set up the same expression a second time just to read the squared modulus.
• • Works the same for integers, decimals, and zero: The form accepts a = 0, b = 0; a = 7, b = 0 (a real number); a = 0, b = 6 (a purely imaginary number); and any decimal combination, so the page covers the common edge cases without a special toggle.
• • Catches the case where z·z̄ should be real: Because the calculator always returns z·z̄ = a² + b², a nonzero imaginary component in the product row is a clear signal that the input was misread, which turns the product row into a built-in error check.

If you also need the angle of a unit complex number, the Arccos Calculator handles the inverse cosine for the real and imaginary parts of the same input.

Three small factors decide what the output looks like, plus two approximation caveats worth knowing about before you paste the result into a follow-up calculation.

• • This page covers single complex numbers in the form z = a + bi; for the 2x2 conjugate transpose A* (Hermitian adjoint) you would need a separate linear-algebra tool that stores a full complex matrix.
• • The modulus and product are rounded to 4 decimal places in the result panel, even though the underlying arithmetic uses full double precision. If you need more than 4 decimal places, repeat the calculation with a higher-precision environment.

According to Wolfram MathWorld, z * conj(z) equals a^2 + b^2, the squared modulus of z, and is always a nonnegative real number.

According to Wikipedia, the conjugate transpose of a complex matrix A is written A* or A^H and is the transpose with each entry's imaginary part negated.