One sign flipped — and five identities that follow, each computed on your own number.
Who this is for: Why the conjugate matters, not just how to write it.
The conjugate, and what it does
Five identities, computed on your number rather than quoted.
Write i or j. Fractions are kept exact.
z = 3 + 4i
z̄ = 3 − 4i
Same real part, imaginary part negated.
What the conjugate does
Each conjugate identity with its value and the reason it comes out that way
Expression
Value
Kind
Why
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.
|z|
5
|z̄| is 5 — the same, as reflection requires
arg z
53.1301°
arg z̄ is −53.1301° — negated
Real quadratic with roots z and z̄
x² − 6x + 25
sum 6, product 25
1 ÷ z
3/25 − (4/25)i
denominator becomes 25, a real number
Reflection in the real axis
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°.
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.
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.
z · z̄ = |z|² is the identity everything else rests on: it is always real, never negative, and it is what makes complex division possible.
|z| and |z̄| are computed separately and compared — reflection preserves distance from the origin.
What this tool shows
Five identities computed on your own number — and the one that makes complex division possible at all.
z + z̄, z − z̄, z·z̄, z ÷ z̄ and the double conjugate, all computed
The reflection in the real axis, with the modulus and argument beside it
The reciprocal worked through the conjugate
The real quadratic with z and z̄ as its roots
Five identities, computed The reflection geometry The conjugate root theorem
Updated 6 September 2026 · Works in any browser, no installation
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.
At a glance
Formula shown
z̄ = a − bi. Then z + z̄ = 2a, z − z̄ = 2bi, and z·z̄ = a² + b² = |z|².
Scenario support
Any complex number with rational real and imaginary parts.
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.
Related calculators
Where this goes next:
Complex NumberAdd, subtract, multiply, divide and raise complex numbers to powers exactly — (1+i)^8 is 16, not 15.999999999999996 — with the conjugate trick shown as the working for division.
Complex RootAll n roots of any complex number by De Moivre, exact wherever a root is a Gaussian rational, arranged as the regular n-gon they form, with the sum and product checked.
Quadratic FormulaSolve any quadratic with exact roots — surds stay surds and a negative discriminant gives the complex pair — plus the vertex, the factored form and every step of the working.
DiscriminantEvaluate b² − 4ac, the cubic and quartic discriminants, or D = f_xx·f_yy − (f_xy)² for the second-derivative test, in exact arithmetic so the sign is never a rounding artefact.
Cubic EquationSolve any cubic exactly when it has a rational root — deflate and finish with the quadratic formula — and by the trigonometric form when it does not, with the discriminant saying which case you are in.
QuaternionExact quaternion arithmetic showing both pq and qp every time, both quotients rather than one, and the rotation a unit quaternion represents with axis, angle, matrix and the gimbal-lock case named.
The conjugate is what makes complex division work: multiplying the top and bottom of a quotient by it turns the denominator into a real number, which is the whole trick and the reason the operation has a name at all. For where conjugate pairs come from in the first place, Which Method Should You Use to Solve a Quadratic? covers the negative-discriminant case, where the two roots of a real quadratic are always each other’s conjugate — which is why they arrive in pairs rather than singly.
Educational use disclaimer
This calculator computes the complex conjugate and the identities that follow from it, exactly, for numbers with rational real and imaginary parts. The modulus is given as a surd where it is irrational, and the argument as a decimal because it is an arctangent. The conjugate root theorem stated here requires REAL polynomial coefficients — the page says so rather than leaving the condition implied.
Published the complex conjugate page: the sum, difference, product, quotient and double-conjugate identities computed on the reader's own number.
Explains the reflection geometry — same modulus, negated argument — and why a real number is its own conjugate as the general rule rather than a special case.
Derives the real quadratic with z and z̄ as its roots, which is the mechanism behind conjugate root pairs rather than a rule to remember.
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