Complex Conjugate Calculator
This complex conjugate calculator takes one number and gives back everything the conjugate brings with it: z̄ itself, the product z·z̄, the sum, the difference and the reciprocal — all exact.
The conjugate of 3 + 4i is 3 − 4i. That part is one line. What makes it worth a page of its own is the thing that happens when you multiply the two together.
The Definition, and the Notation
For z = a + bi, the conjugate is a − bi. Flip the sign in the middle; leave the real part exactly as it was.
It is written z̄ — said "z bar" — and in physics and engineering texts often z*. Both mean the same thing.
Drawn on the complex plane it is a mirror image across the real axis: the horizontal position is unchanged, the vertical one flips. Since reflecting a point does not move it nearer or further from the origin, a number and its conjugate are always the same distance out.
One consequence is worth noticing early: a real number is its own conjugate, because there is no imaginary part to flip. And the reverse holds too — if z̄ = z then b must be 0, so z is real. Some books use exactly that as the test for whether a quantity is real.
Why the Product Comes Out Real
Here is the fact everything else rests on. Multiply a number by its conjugate:
(3 + 4i)(3 − 4i) = 9 − 12i + 12i + 16 = 25
Four terms. The two middle ones are equal and opposite, so they vanish. The last one carries an i², which turns −16i² into +16. Nothing with an i in it survives.
In general:
z·z̄ = a² + b²
Two squares added, so it can never be negative — and it is zero only when the number itself is zero. That is why the conjugate is the reliable way to turn a complex quantity into an ordinary one, and why it turns up wherever an i has to be cleared out of somewhere awkward.
This product is also |z|² — the square of the distance from the origin, the quantity usually called the modulus. Notice that the square is always an ordinary rational number, while the distance itself usually is not: for 3 + 4i it comes out at a tidy 5, but for 1 + i the product is 2 and the distance is √2. This page shows the square root only when it is genuinely whole and says nothing when it is not, rather than offering a decimal that is not the answer.
The Sum and the Difference
The other two things a conjugate pair gives you are easy to overlook and quietly useful:
z + z̄ = 2a — adding cancels the imaginary parts and doubles the real one.
z − z̄ = 2bi — subtracting does the opposite.
So a conjugate pair lets you pull out either part on its own. If you need the real part of some tangled expression, add its conjugate and halve. That is the standard way it is done, and it is why Re(z) = (z + z̄)/2 appears in so many derivations.
The Reciprocal
The commonest actual use of a conjugate is to make a division possible, and the simplest case of that is 1 ÷ z.
You cannot divide by something with an i in it, so multiply top and bottom by the conjugate. The bottom becomes a² + b², an ordinary number, and what is left is:
1/z = z̄ ÷ (a² + b²)
For 3 + 4i that is (3 − 4i) ÷ 25 = 3/25 − 4/25 i. Fractions, and they are left as fractions here, because a rounded decimal is an approximation of that answer rather than the answer itself.
Zero is the one number with no reciprocal, exactly as in ordinary arithmetic. If you want to divide one complex number by another rather than divide into 1, that is a different question and it has its own tool: the complex number calculator handles all four operations on a pair.
Why Complex Roots Come in Pairs
This is the property with the longest reach, and the two numbers this page prints are exactly the ones it needs.
Take a polynomial whose coefficients are all real. If z is a root, then z̄ is a root as well — always. Conjugating an equation with real coefficients leaves the coefficients alone and turns each root into its conjugate, so the roots must come in matched pairs.
You can see it built from the outputs above. A quadratic with roots z and z̄ is:
x² − (z + z̄)x + z·z̄ = 0
For 3 + 4i, the sum is 6 and the product is 25, giving x² − 6x + 25 = 0 — a perfectly ordinary quadratic with whole-number coefficients whose roots are 3 ± 4i. Both quantities on this page are sitting in it.
That is also why a real quadratic never has exactly one non-real root: they arrive together or not at all.
How to Use This Complex Conjugate Calculator
Two boxes and one button.
Type the number as two parts. The real part in the first box, the multiple of i in the second. An empty box counts as 0, so a lone 5 is the real number 5 and a lone 3 in the second box is 3i. Fractions like 3/4 and decimals like 0.25 both work.
| real | The ordinary part. Empty means 0 |
| imaginary (× i) | How many i s. Empty means 0 |
| ← → ↑ ↓ | Move between the two boxes |
| ± | Flips the sign of the box you are in |
| ⁄ and . | Fraction bar and decimal point |
| ⌫ / AC | Delete one character, or clear both boxes |
| = | Works it out |
Open the working and you get the whole picture rather than one flipped sign: the multiplication laid out so the middle terms can be seen cancelling, the sum and the difference, the reciprocal with its check, and — only when it happens to be whole — the distance from the origin.
Everything stays exact at every stage. That matters more here than it might look: a² + b² is always a plain rational number, so there is never any reason to round on this page, and nothing on it is rounded.
Complex Conjugate FAQ
What is the complex conjugate of a number?
Why is z times its conjugate always a real number?
What is the conjugate of a real number?
What is z plus its conjugate?
How do you find the reciprocal of a complex number?
Why do complex roots of polynomials come in conjugate pairs?
One sign, flipped. Everything on this page — the real product, the two halves it separates, the reciprocal, the paired roots — comes out of that single change.