Complex Conjugate Calculator
Welcome to the complex conjugate calculator, where we'll learn all about the conjugate of a complex number. We'll start slow with an introduction to complex numbers and the conjugate definition. Then, we'll see some conjugation examples and see how conjugate pairs behave, i.e., what happens when we multiply by the conjugate. If you've been asking yourself, "What is conjugation?" or "What is a complex conjugate?", you've come to the right place!
Complex numbers
At the very beginning of your mathematical education, you were told how to add and subtract numbers. But there's a catch: you can't subtract, say, 5 from 2. A couple of years later, they tell you that, actually, you can, and the result is a negative number. After all, what is a debt if not a negative number?
When you get to multiplication and division, they teach you that 7 is not divisible by 4 — until you meet fractions and find that you can divide them. And once you reach exponents and roots, one thing seems certain: the square root of a negative number doesn't exist… Or does it?
Complex numbers are of the form z = a + bi, where a and b are real numbers, and i denotes the square root of −1 (i = √−1). The a is called the real part of z, while b is called the imaginary part. Whenever b = 0, z is simply a real number — so complex numbers are just an extension of the numbers we already know. They are extremely important in modern physics and mathematics, appearing in:
- Quadratic and higher-power polynomials;
- Wavelength; and
- Diffraction.
What is conjugation? The conjugate definition
Let's start with the conjugate definition because it's so simple. The conjugate of a complex number z = a + bi is the number a − bi. We denote this operation by putting a horizontal line over the value, like z. In other words, to conjugate a complex number you simply flip the sign of the imaginary part and leave the real part untouched.
Conjugation examples, conjugate pairs
| Number z | Conjugate z |
|---|---|
| 3 + 4i | 3 − 4i |
| −2 + 5i | −2 − 5i |
| 7i | −7i |
| 6 (real) | 6 (unchanged) |
A number and its conjugate form a conjugate pair. On the complex (Argand) plane, the conjugate is simply the mirror reflection of z across the real axis. Conjugate pairs have a lovely property — when you multiply them together, the imaginary parts cancel out:
z · z = (a + bi)(a − bi) = a² + b² = |z|²
The result is always a non-negative real number — exactly the square of the modulus of z. That's why multiplying by the conjugate is the standard trick for dividing complex numbers and for rationalizing denominators. A few more identities: z + z = 2a (twice the real part) and z − z = 2bi (twice the imaginary part).
Using the complex conjugate calculator
- Enter the real part a of your complex number.
- Enter the imaginary part b (the coefficient in front of i). Both parts may be negative or zero.
- Hit Calculate and the tool instantly returns the conjugate z = a − bi.
- You'll also get the modulus, the argument, the products and sums with the conjugate, and an Argand diagram showing the reflection across the real axis — plus a clear step-by-step solution.