Complex Conjugate Calculator
Find the conjugate of a complex number, graph it on the complex plane, and see modulus, argument, product, and reciprocal details.
Complex conjugate in rectangular and polar form
A complex conjugate calculator changes the sign of the imaginary part of a complex number. If z = a + bi, then the complex conjugate is z-bar = a - bi.
Calculate a complex conjugate by changing the sign of the imaginary part. For a complex number a + bi, the complex conjugate is a - bi. For example, the conjugate of 3 + 4i is 3 - 4i, and the conjugate of 7 - 2i is 7 + 2i.
Use rectangular input for real and imaginary parts, expression input for values like 3+4i or -2-5i, or polar input for magnitude and angle. The calculator also shows the modulus, argument, reciprocal, and the real product z x z-bar.
On the complex plane, a conjugate is the reflection of the original point across the real axis. The real part stays the same, while the imaginary coordinate switches sign.
Complex conjugate
--
Original number
--
Rectangular form a + bi.
Conjugate
--
Imaginary part changes sign.
Modulus and argument
--
--
Reciprocal
--
1 / z, when z is not zero.
Step-by-step work
--
--
--
Graph note: The original complex number and its conjugate are mirror images across the real axis. If the number is purely real, both points overlap.
How to use the complex conjugate calculator
- Choose an input method: Enter real and imaginary parts, type a complex expression, or use polar form.
- Enter the complex number: For z = a + bi, enter a as the real part and b as the imaginary coefficient.
- Calculate: The result shows z-bar, the modulus, argument, product z x z-bar, reciprocal, and graph.
- Check the sign: Only the imaginary part changes sign. The real part remains unchanged.
- Use the graph: The graph confirms that the conjugate is reflected across the real axis.
Complex conjugate formula
The conjugate keeps the real part and reverses the sign of the imaginary part. This operation is often used to simplify division by complex numbers and to turn a complex product into a real number.
If z = a + bi, then z-bar = a - bi
z x z-bar = a^2 + b^2 = |z|^2
1 / z = z-bar / |z|^2, when z is not 0
In polar form, the magnitude stays the same and the angle changes sign. If z = r at angle t, then z-bar has the same r at angle -t.
Reference: NIST Digital Library of Mathematical Functions - complex variables and conjugates.
Complex conjugate examples
| Original z | Conjugate z-bar | Product z x z-bar | Comment |
|---|---|---|---|
| 3 + 4i | 3 - 4i | 25 | Classic 3-4-5 modulus example. |
| -2 - 5i | -2 + 5i | 29 | Only the imaginary sign flips. |
| 7i | -7i | 49 | Purely imaginary numbers reflect across the origin vertically. |
| 6 | 6 | 36 | Real numbers are their own conjugates. |
Batch conjugate checker
Paste several complex numbers to build a quick answer table for homework, worksheets, or checking repeated calculations. Separate values with commas, semicolons, or new lines.
| Input | Conjugate | |z| | z x z-bar |
|---|---|---|---|
| Click Build Batch Table to calculate the examples. | |||
Where complex conjugates are used
Complex conjugates appear anywhere complex arithmetic needs to be simplified or related back to real quantities.
Division
Multiply numerator and denominator by the denominator's conjugate to remove i from the denominator.
Modulus
The product z x z-bar equals |z|^2, a real number that measures squared distance from the origin.
Roots and polynomials
For polynomials with real coefficients, non-real complex roots occur in conjugate pairs.
Conjugate vs similar operations
| Operation | For z = a + bi | Example z = 3 + 4i | What changes |
|---|---|---|---|
| Conjugate | a - bi | 3 - 4i | Only the imaginary part changes sign. |
| Negative | -a - bi | -3 - 4i | Both real and imaginary parts change sign. |
| Reciprocal | z-bar / |z|^2 | 3/25 - 4i/25 | Uses the conjugate to divide by a complex number. |
| Modulus | sqrt(a^2 + b^2) | 5 | Returns the distance from the origin, not a complex number. |
Worked example: divide by a complex denominator
One of the most common reasons to use a complex conjugate is to simplify a fraction that has an imaginary term in the denominator. The method is the same every time: multiply the numerator and denominator by the denominator's conjugate, then simplify.
Example
(5 + 2i) / (3 - 4i)
The denominator is 3 - 4i, so its conjugate is 3 + 4i.
Multiply top and bottom by 3 + 4i:
((5 + 2i)(3 + 4i)) / ((3 - 4i)(3 + 4i)) = (7 + 26i) / 25
Final result: 7/25 + (26/25)i.
This is useful because the denominator becomes a real number, a^2 + b^2. Once the denominator is real, the fraction is easier to compare, graph, or use in another algebra equation.
Division reference: Imperial College London - division and the complex conjugate.
Input forms and how to read the output
| Input type | Example input | Calculator interpretation | Conjugate output |
|---|---|---|---|
| Real and imaginary parts | a = 3, b = 4 | 3 + 4i | 3 - 4i |
| Typed expression | -2-5i | Real part -2, imaginary coefficient -5 | -2 + 5i |
| Pure imaginary number | i or -i | 0 + 1i or 0 - 1i | -i or i |
| Polar form | r = 5, angle = 53.13 deg | Approximately 3 + 4i | Approximately 3 - 4i |
Tip: the calculator accepts i and j in typed expressions, but the displayed result uses i because that is the standard notation in most algebra and mathematics courses.
Conjugate identities and quick checks
These identities help you verify a result, simplify an expression by hand, or spot whether a calculator output makes sense before using it in a larger solution.
Conjugate of a sum
conj(z + w) = conj(z) + conj(w). You can conjugate each term separately when simplifying a longer expression.
Conjugate of a product
conj(zw) = conj(z)conj(w). This is useful when a binomial expression contains several complex factors.
Real and imaginary checks
z + z-bar = 2a and z - z-bar = 2bi. These formulas recover the real part and imaginary part from a number and its conjugate.
Modulus check
z x z-bar = |z|^2. If your product still contains i, a sign or multiplication step probably needs another look.
Practice reference: Ohio State Ximera - complex arithmetic.
Interesting fact
Complex conjugates help reduce duplicate information in signal processing. For a real-valued signal, the negative-frequency Fourier terms mirror the positive-frequency terms as complex conjugates, so a real FFT only needs `n//2 + 1` output values. That means a 1024-sample real signal has 513 unique frequency outputs instead of 1024. Source: NumPy documentation for real FFT.
Frequently Asked Questions
What is a complex conjugate in mathematics?
In mathematics, the complex conjugate of a complex number a + bi is a - bi. The real part a stays the same, while the imaginary part changes sign. For example, the conjugate of 3 + 4i is 3 - 4i, so the formula keeps the number's horizontal position and flips its vertical direction.
How do I use the calculator input?
Enter the complex number as separate real and imaginary parts, as a typed expression such as 5-2i, or in polar form. The calculator reads the input, identifies the imaginary coefficient b, and returns the output in standard a + bi form. If the expression is already a - bi, the result changes it to a + bi.
Why does the conjugate formula make z times z-bar real?
When the binomial expression (a + bi)(a - bi) is multiplied out, the middle imaginary terms cancel. The equation simplifies to a^2 + b^2, a real and nonnegative result. This is why z times z-bar equals the squared modulus of the complex number.
What is the conjugate of a polar expression?
In polar form, the conjugate keeps the same magnitude and reverses the angle. If z has magnitude r and angle t, then the conjugate has magnitude r and angle -t. The calculator converts that expression back to rectangular form when it displays the final answer.
Is a real number its own conjugate?
Yes. A real number has imaginary part 0, so changing the sign of the imaginary part does not change the value. For example, the conjugate of 8 is still 8. In algebra, this is the special case a + 0i, where the complex number lies directly on the real axis.
How does a conjugate simplify a fraction with a complex denominator?
To divide by a complex number, multiply the numerator and denominator of the fraction by the denominator's conjugate. This simplification makes the denominator real because z times z-bar equals a^2 + b^2. After that, the answer can be written as a standard complex number in a + bi form.
What does the output look like on a complex plane graph?
On the complex plane, the original number and its conjugate have the same horizontal coordinate and opposite vertical coordinates. That means the output is a reflection across the real axis. If z is above the real axis, z-bar is the matching point below it, with the same real part and the opposite imaginary part.
Why do complex roots of an equation come in conjugate pairs?
For a polynomial equation with real coefficients, non-real complex roots occur in conjugate pairs. If a + bi is a solution for the variable, then a - bi is also a solution. This pairing keeps the polynomial's coefficients real after the matching algebra factors are multiplied together.
Other useful calculators
Disclaimer: This complex conjugate calculator is for general educational and informational use only. It provides algebraic results based on user-entered values and rounded output. It is not a substitute for classroom instructions, textbook notation, exam requirements, engineering analysis, signal-processing review, or professional mathematical verification. Always confirm exact form, rounding rules, angle units, and notation required by your course, software, or project before relying on the result.
Last updated: May 22, 2026