How this instrument works
For a number written a + bi, the conjugate is a − bi: the real part a is left untouched, and only the sign attached to b, the coefficient of i, is reversed. Mathematicians write it z̄ or z*, and picture it geometrically as a reflection — plot a + bi in the complex plane and its conjugate is the same point flipped across the horizontal, real axis, the same distance from that axis but on the opposite side.
The reason conjugates matter is what happens when you multiply a number by its own conjugate: (a + bi)(a − bi) expands to a² − (bi)² = a² − b²i², and since i² = −1 that term becomes +b², leaving a² + b² with no i left anywhere. Squaring a real number can never reintroduce an imaginary part, so the product is guaranteed real — the exact trick used to rationalize a complex denominator, by multiplying top and bottom of a fraction by the conjugate of whatever sits underneath.
Do not confuse this with the conjugate pairs that show up as roots of a real quadratic equation. A conjugate here is a deliberate operation you can perform on any complex number at all, just by flipping one sign. A quadratic's complex roots come in conjugate pairs for a completely different reason — because the equation's own coefficients are real, that pairing is forced on the roots by the equation itself, not chosen afterward. And when b is already zero, the operation changes nothing: every real number is its own conjugate, since flipping the sign of zero leaves zero exactly where it was.
- Enter the number's real component into Real part, a.
- Enter its imaginary component into Imaginary part, b — the coefficient attached to i, not the number times i.
- Read Conjugate real part (= a): it always matches whatever is in Real part, a, unchanged.
- Read Conjugate imaginary part (= −b): it reports the negative of whatever is in Imaginary part, b.
- Try b = 0 to confirm a real number reproduces itself exactly, since it is already its own conjugate.
Worked example — conjugating 3 + 4i
Set Real part, a to 3 and Imaginary part, b to 4, describing the number 3 + 4i. Conjugate real part (= a) reads 3, untouched, while Conjugate imaginary part (= −b) reads −4, the sign on the imaginary part alone reversed. Plotted in the complex plane, 3 + 4i and its conjugate 3 − 4i sit as mirror images across the real axis, each four units from that axis on opposite sides while both sit three units out along it.
Multiply the pair together and every trace of i cancels out: (3 + 4i)(3 − 4i) = 3² − (4i)² = 9 − (−16) = 25, a plain real number with no imaginary part left at all. That 25 is also 3² + 4², exactly the a² + b² this sheet's formula predicts, and it is the same trick used to rationalize a complex fraction's denominator — multiply top and bottom by the conjugate and the bottom always lands on a real number.
Questions
What is the complex conjugate of a number?
For a number written a + bi, the conjugate is a − bi: the real part a stays exactly as it is, and only the sign of the imaginary part b flips. Nothing about the size of either part changes — only whether the imaginary term is added or subtracted.
Why does multiplying a complex number by its conjugate give a real number?
Because (a + bi)(a − bi) expands to a² − (bi)² = a² − b²i², and since i² = −1 that last term becomes +b², leaving a² + b² with no i anywhere in sight. This is exactly why 3 + 4i times its conjugate 3 − 4i equals 9 + 16 = 25, a plain real number.
How is a conjugate different from the roots of a quadratic equation?
A conjugate is something you can compute for any complex number at all, just by flipping the imaginary sign. A quadratic's complex roots only come in conjugate pairs because the equation's coefficients are real — that pairing is forced by the equation itself, not a deliberate operation performed afterward on one root.
What happens when the imaginary part is zero?
The conjugate equals the original number exactly, since flipping the sign of zero leaves zero unchanged. Every real number is its own conjugate — enter b = 0 and Conjugate imaginary part (= −b) still reads 0, matching Imaginary part, b precisely.
Why is the conjugate useful for dividing complex numbers?
Multiplying a complex denominator by its own conjugate turns that denominator into a real number, a² + b², clearing every i from underneath the fraction. That single move is the standard method for rationalizing a complex quotient, turning a division by a + bi into a division by a plain real number instead.
Does conjugating a number change its distance from zero?
No — a + bi and a − bi sit the same distance from the origin in the complex plane, since that distance is √(a² + b²) and squaring b erases its sign either way. Conjugating reflects the point across the real axis; it never moves the point closer to or farther from zero.