About this calculator
The complex conjugate calculator is used to calculate the complex conjugate z̄ = a - bi of any complex number z = a + bi. As long as you enter the real and imaginary parts of a complex number, the tool will automatically give the conjugate result and preserve the standard algebraic form of the complex number, making it easy to proceed with addition, multiplication, division, modulus, or equation solving.
Complex conjugate numbers are very common in complex number arithmetic. It can convert complex numbers in the denominator to real numbers, and can also be used to calculate the modulus square: z·z̄ = a² + b². Complex conjugates are fundamental operations in signal processing, circuit analysis, Fourier transforms, and complex plane geometry.
From a geometric point of view, a conjugate complex number represents a symmetry point on the complex plane about the real axis. For example, the complex conjugate of 3 + 4i is 3 - 4i. Both points are the same distance from the origin and have opposite signs. This calculator is suitable for students to check their homework, and is also suitable for quickly sorting out complex expressions in engineering calculations.
What it calculates
The complex conjugate calculator changes the sign of the imaginary part of a complex number. For z = a + bi, the conjugate is a - bi.
Formula
If z = a + bi, then conj(z) = a - bi.
- The real part a stays the same.
- The imaginary part b changes sign.
- z × conj(z) = a^2 + b^2.
Inputs
- a: the real part of the complex number.
- b: the imaginary part, or the coefficient of i.
Example
| Original number | Conjugate | Note |
|---|---|---|
| 3 + 4i | 3 - 4i | Imaginary part 4 becomes -4 |
| -2 + 5i | -2 - 5i | Real part stays -2 |
| 7 - 3i | 7 + 3i | Imaginary part -3 becomes 3 |
| 6 | 6 | A real number has imaginary part 0 |
How to interpret the result
The conjugate is the mirror image of the complex number across the real axis. Its modulus stays the same, while the argument changes sign.
Common mistakes
- Do not change the sign of the real part.
- A negative imaginary part becomes positive after conjugation.
- The coefficient of i is the imaginary part.
- The conjugate of 0 is still 0.
How to use
When using the conjugate complex number calculator, first fill in a in the real part input box and b in the imaginary part input box. Complex numbers are usually written a + bi, for example 3 + 4i has a real part of 3 and an imaginary part of 4. After clicking "Calculate", the result will be displayed as a - bi.
If the input is 5 - 2i, you can fill in the real part with 5 and the imaginary part with -2, and the calculated result is 5 + 2i. When the imaginary part is 0, the complex number itself is a real number, and the conjugate complex number is still equal to itself.
When performing complex division, you can first find the conjugate of the denominator, and then multiply both the numerator and denominator by the conjugate complex number. For example, 1/(2+3i) can be multiplied by 2-3i to change the denominator to 13 to obtain the standard form.
Main features
Supports input of real and imaginary parts in the form of positive numbers, negative numbers, and decimals.
Automatically output the standard form a + bi or a - bi, suitable for complex number simplification, complex number division, modular length calculation and complex plane analysis.
The page provides the formulas, geometric meanings and FAQs of conjugate complex numbers, which can be used directly in the browser without installing software or registering an account.
Use cases
Conjugate complex numbers are commonly used in high school and college mathematics to simplify complex numbers, solve complex equations, and analyze complex plane geometry. Students can use it to check whether conjugate calculations are correct and understand the geometric significance of real axis symmetry.
In engineering and physics, conjugate complex numbers appear in alternating current phasors, impedance calculations, signal spectra, quantum mechanical inner products, and Fourier analysis. Get accurate conjugate values quickly and reduce hand-calculated sign errors.
In programming and data analysis, complex arrays, complex matrices, and frequency domain data often need to be conjugated. This tool is suitable for quickly verifying the results of a single complex number operation in the browser.