The conjugate flips the sign of the imaginary part — and the reason it matters is that z · z̄ is always real. For z = 3 + 4i, z̄ = 3 − 4i, and their product is 25, which is |z|².
The five things the conjugate does
Each one is a different way of extracting something real from something complex:
- z + z̄ = 6 — real. The imaginary parts are 4 and −4 — equal and opposite — so they cancel exactly, leaving 2a = 6. A number plus its conjugate is always real.
- z − z̄ = 8i — imaginary. The real parts are identical, so they cancel, leaving 2bi = 8i. A number minus its conjugate is always purely imaginary.
- z · z̄ = 25 — real. (a + bi)(a − bi) is a difference of squares with i² = −1, so it comes to a² + b² = 9 + 16 = 25. That is |z|², always real and never negative — which is the property everything else here rests on.
- z ÷ z̄ = −7/25 + (24/25)i — complex. Numerator and denominator have the same modulus, so the quotient has modulus exactly 1 — it is a pure rotation, through twice the argument of z.
- z̄̄ = 3 + 4i — complex. Conjugating twice flips the sign of the imaginary part and then flips it back, so it returns z unchanged — as it does here. Reflection is its own inverse.
The first two are how you isolate the real and imaginary parts using only arithmetic — Re(z) = (z + z̄)/2 and Im(z) = (z − z̄)/2i — which matters in contexts where “take the real part” is not an available operation.
z · z̄ is why complex division exists
(a + bi)(a − bi) is a difference of squares with i² = −1, so it comes to a² + b². Always real, never negative, and zero only when z is.
Multiplying top and bottom by z̄ turns the denominator into 25, a real number, and dividing by a real number is something you already know how to do. This is not a shortcut for division; it IS complex division. Every method reduces to it.
1 / (3 + 4i) × (3 − 4i) / (3 − 4i) gives a denominator of 25 and a reciprocal of 3/25 − (4/25)i. That construction is available for every non-zero complex number, which is exactly what makes the complex numbers a field rather than merely a ring.
|z|² = 13/36 is exact; |z| itself is √13/6, an irrational number, so it is given as a surd rather than only as a decimal.
A reflection, not a rule to memorise
Conjugation reflects z in the real axis: the point moves from the first quadrant to the fourth quadrant, the same distance from the origin. The modulus is unchanged at 5 and the argument is negated, from 53.1301° to −53.1301°.
Two consequences follow immediately and need no separate proof. |z̄| = |z|, because reflection does not move a point closer to or further from the origin — the tool computes both and compares them. And arg z̄ = −arg z, because reflecting across the horizontal axis negates the angle.
It also explains why conjugating twice returns you to z: reflecting in the same line twice is the identity. Conjugating twice flips the sign of the imaginary part and then flips it back, so it returns z unchanged — as it does here. Reflection is its own inverse.
Conjugate root pairs, and the condition people drop
z + z̄ = 6 and z · z̄ = 25, and both are real — so the quadratic with these two roots has real coefficients. That is the mechanism behind "complex roots come in conjugate pairs": a real polynomial cannot have one without the other, because the pair is exactly what makes the coefficients come out real. The condition is REAL coefficients — drop it and the pairing goes away, since x − (3 + 4i) is a perfectly good polynomial with one complex root and no partner.
Look at what makes it work: z + z̄ and z·z̄ are both real, so the quadratic x² − (z + z̄)x + z·z̄ has real coefficients automatically. For z = 3 + 4i that is x² − 6x + 25.
The condition — REAL coefficients — is the part that gets dropped, and dropping it makes the theorem false. x − (3 + 4i) is a perfectly good polynomial with one complex root and no partner anywhere. The pairing is not a property of complex numbers; it is a property of real polynomials.
This is why the quadratic formula always produces a conjugate pair when the discriminant is negative: the ±√Δ differs only in the sign of the imaginary part.
Real numbers are their own conjugate, and that is the general rule
z sits on the real axis, and reflecting the real axis in itself changes nothing — so a real number is its own conjugate. That is not a special case, it is the general rule with b = 0.
It is worth seeing this as the ordinary case with b = 0 rather than as an exception. There is nothing to flip, so nothing changes, and every identity above still holds: z + z̄ = 10, z − z̄ = 0, z · z̄ = 25.
The opposite extreme is a purely imaginary number, where conjugating is the same as negating: for 7i the conjugate is −7i, and z + z̄ = 0. Those two cases bracket everything in between.
Where the conjugate turns up outside algebra
Wherever a complex quantity has to produce a real one. In signal processing the power of a signal is z·z̄; in quantum mechanics a probability is ψ·ψ̄; in electrical engineering the conjugate appears in the formula for real power drawn from an AC source.
All three are the same fact: the modulus squared is the only bilinear way to get a non-negative real number out of a complex one, and it needs the conjugate to be written down.
|z|² = 5 is exact; |z| itself is √5, an irrational number, so it is given as a surd rather than only as a decimal.
Sources and methodology
The conjugate and its identities are standard; the reference below carries the canonical statements.
Method. Every figure on this page comes from src/lib/complex-conjugate.ts over the exact Gaussian-rational arithmetic in src/lib/algebra/complex.ts. Each identity is computed from the parsed number rather than templated, and |z| and |z̄| are computed separately and compared so the reflection claim is checked rather than asserted. That engine is verified on every change against 59 hand-written assertions, including that |z| and |z̄| always agree, and that the constructed quadratic has real coefficients and z as a root. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The conjugate is what makes complex division work, and the Complex Number Calculator uses it on every quotient. For where conjugate pairs come from in the first place, Which Method Should You Use to Solve a Quadratic? covers the negative-discriminant case.