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Complex Conjugate



Key Terms
Definition of the Conjugate
Geometric Interpretation
Fundamental Properties
Conjugate and Modulus
Classification Theorems
Useful Identities
Applications to Division
Conjugate Pairs in Polynomials
Summary: What z̄ Tells You About z



The Mirror Operation


The Mirror Operation

Key Terms

Complex Conjugate— the reflection z‾=a−bi\overline{z} = a - bi
Modulus— satisfies ∣z∣2=z⋅z‾|z|^2 = z \cdot \overline{z}
Multiplicative Inverse— computed using the conjugate
Complex Plane— conjugation reflects across the real axis

See All Complex Numbers Definitions →


Definition of the Conjugate

For any complex number z=a+biz = a + bi, the conjugate is defined as:

Complex Conjugate
z‾=a−bifor    z=a+bi\overline{z} = a - bi \quad \text{for} \;\; z = a + bi
Learn more about this formula: Complex Conjugate →


The operation preserves the real part aa exactly while reversing the sign of the imaginary part. Where the original number has +bi+bi, the conjugate has −bi-bi. Where the original has −bi-bi, the conjugate has +bi+bi. Only the vertical component changes; the horizontal component remains untouched.

Concrete examples illustrate the pattern. The conjugate of 3+2i3 + 2i is 3−2i3 - 2i. The conjugate of −1−4i-1 - 4i is −1+4i-1 + 4i — the negative imaginary part becomes positive. The conjugate of 55 (a real number, written as 5+0i5 + 0i) is simply 55 — with no imaginary part to flip, nothing changes. The conjugate of 7i7i (pure imaginary, written as 0+7i0 + 7i) is −7i-7i — the entire number negates because the real part contributes nothing.

Two notations appear in mathematical literature. The overline zˉ\bar{z} dominates pure mathematics and most textbooks. The asterisk z∗z^* appears frequently in physics and engineering, particularly in quantum mechanics and signal processing. Both symbols denote the identical operation. This text uses the overline convention, but readers should recognize both forms as equivalent.

The conjugate is not the same as negation. The negative of 3+2i3 + 2i is −3−2i-3 - 2i, changing both signs. The conjugate 3−2i3 - 2i changes only the imaginary sign. Confusing these operations leads to errors, particularly when manipulating equations involving both.
z Category z̄ Note
3 + 2i general complex (Q1) 3 − 2i imaginary part flips sign
−1 − 4i general complex (Q3) −1 + 4i negative imaginary becomes positive
5 real 5 unchanged — z̄ = z for every real z
7i pure imaginary −7i z̄ = −z for every pure imaginary z
0 zero (both real and pure imag.) 0 the unique fixed point of every reflection
(1⁄2) + (√3⁄2)i on the unit circle (|z| = 1) (1⁄2) − (√3⁄2)i here z̄ also equals z⁻¹

Geometric Interpretation

The complex plane transforms the conjugate from an algebraic rule into a visible action. If z=a+biz = a + bi corresponds to the point (a,b)(a, b), then zˉ=a−bi\bar{z} = a - bi corresponds to (a,−b)(a, -b). The horizontal coordinate stays fixed while the vertical coordinate negates. This is reflection across the real axis.

Picture the real axis as a horizontal mirror. Every point above the axis has a mirror image below; every point below has an image above. The conjugate operation sends each complex number to its reflection. The point 2+3i2 + 3i at coordinates (2,3)(2, 3) reflects to 2−3i2 - 3i at (2,−3)(2, -3). The point −1−4i-1 - 4i at (−1,−4)(-1, -4) reflects to −1+4i-1 + 4i at (−1,4)(-1, 4).

Points lying directly on the mirror remain stationary. These are precisely the real numbers — they have no vertical displacement to reverse, so conjugation leaves them unchanged. The number 55 sits at (5,0)(5, 0) on the real axis, and its reflection lands at the same spot. This geometric fact corresponds to the algebraic observation that aˉ=a\bar{a} = a for any real aa.

Points on the imaginary axis exhibit different behavior. They lie perpendicular to the mirror, equidistant above and below the real axis. Reflection sends each to the opposite side of the origin. The number 4i4i at (0,4)(0, 4) reflects to −4i-4i at (0,−4)(0, -4). For pure imaginary numbers, conjugation equals negation: zˉ=−z\bar{z} = -z.

The reflection interpretation explains why conjugating twice returns the original number. Reflect a point across a line, then reflect again across the same line — the point returns home. Two mirror operations cancel completely, regardless of where the point started.

Fundamental Properties

The conjugate operation interacts predictably with arithmetic. Five properties govern how conjugation passes through sums, products, and powers, enabling simplification of complicated expressions.

The involution property states that conjugating twice recovers the original:

Conjugate of a Conjugate
z‾‾=z\overline{\overline{z}} = z
Learn more about this formula: Conjugate of a Conjugate →


Apply the definition twice: if z=a+biz = a + bi, then zˉ=a−bi\bar{z} = a - bi, and zˉ‾=a−(−b)i=a+bi=z\overline{\bar{z}} = a - (-b)i = a + bi = z. The double sign flip restores the original imaginary part. Geometrically, reflecting twice across the same axis brings every point back to its starting position.

Additivity allows conjugation to distribute over sums:

Conjugate of a Sum
z1+z2‾=z1‾+z2‾\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}
Learn more about this formula: Conjugate of a Sum →


The conjugate of a sum equals the sum of the conjugates. Proof: let z1=a+biz_1 = a + bi and z2=c+diz_2 = c + di. Then z1+z2=(a+c)+(b+d)iz_1 + z_2 = (a + c) + (b + d)i, so z1+z2‾=(a+c)−(b+d)i=(a−bi)+(c−di)=z1ˉ+z2ˉ\overline{z_1 + z_2} = (a + c) - (b + d)i = (a - bi) + (c - di) = \bar{z_1} + \bar{z_2}. The same property extends to subtraction: z1−z2‾=z1ˉ−z2ˉ\overline{z_1 - z_2} = \bar{z_1} - \bar{z_2}.

Multiplicativity states that conjugation distributes over products:

Conjugate of a Product
z1⋅z2‾=z1‾⋅z2‾\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}
Learn more about this formula: Conjugate of a Product →


The conjugate of a product equals the product of the conjugates. Verification requires expanding both sides and comparing — the algebra confirms equality. This property proves invaluable when simplifying products or verifying identities.

Division follows the same pattern:

Conjugate of a Quotient
(z1z2)‾=z1‾z2‾\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}
Learn more about this formula: Conjugate of a Quotient →


valid for z2≠0z_2 \neq 0. Conjugation passes through quotients just as it passes through products.

Powers inherit the multiplicative property:

Conjugate of a Power
zn‾=(z‾)n\overline{z^n} = (\overline{z})^n
Learn more about this formula: Conjugate of a Power →


for any integer nn. Repeated application of the product rule establishes this for positive integers, and the quotient rule extends it to negative integers. Conjugating a power equals powering the conjugate.
Property Formula In words
Involution z̄ = z conjugating twice returns the original
Additivity z₁ + z₂ = z̄₁ + z̄₂ conjugate of a sum = sum of conjugates
Subtraction z₁ − z₂ = z̄₁ − z̄₂ conjugate of a difference = difference of conjugates
Multiplicativity z₁ · z₂ = z̄₁ · z̄₂ conjugate of a product = product of conjugates
Quotient z₁ ⁄ z₂ = z̄₁ ⁄ z̄₂  (z₂ ≠ 0) conjugate of a quotient = quotient of conjugates
Powers zⁿ = (z̄)ⁿ  (integer n) conjugate of a power = power of the conjugate

Conjugate and Modulus

A fundamental identity connects the conjugate to the modulus:

z⋅zˉ=∣z∣2z \cdot \bar{z} = |z|^2


The product of any complex number with its conjugate yields the square of its modulus — a real, non-negative value. This relationship lies at the heart of complex arithmetic.

The proof follows from direct expansion. Let z=a+biz = a + bi, so zˉ=a−bi\bar{z} = a - bi. The product becomes:

z⋅zˉ=(a+bi)(a−bi)=a2−abi+abi−b2i2=a2−b2(−1)=a2+b2z \cdot \bar{z} = (a + bi)(a - bi) = a^2 - abi + abi - b^2i^2 = a^2 - b^2(-1) = a^2 + b^2


The middle terms cancel, and i2=−1i^2 = -1 converts the final term to positive. The result a2+b2a^2 + b^2 matches the definition ∣z∣2=a2+b2|z|^2 = a^2 + b^2 exactly.

This identity explains why z⋅zˉz \cdot \bar{z} always produces a real number. The imaginary parts eliminate each other through the cancellation of −abi-abi and +abi+abi. No matter how complicated zz appears, multiplying by its conjugate guarantees a real outcome.

A related fact: conjugation preserves modulus. The numbers zz and zˉ\bar{z} lie at equal distances from the origin, as reflection across a line through the origin does not change radial distance. Algebraically, ∣zˉ∣=∣a−bi∣=a2+(−b)2=a2+b2=∣z∣|\bar{z}| = |a - bi| = \sqrt{a^2 + (-b)^2} = \sqrt{a^2 + b^2} = |z|. The identity ∣zˉ∣=∣z∣|\bar{z}| = |z| holds universally.

Taking square roots of z⋅zˉ=∣z∣2z \cdot \bar{z} = |z|^2 gives ∣z∣=z⋅zˉ|z| = \sqrt{z \cdot \bar{z}}, an alternative formula for modulus that sometimes proves more convenient than a2+b2\sqrt{a^2 + b^2}.

Classification Theorems

The conjugate provides algebraic tests for determining whether a complex number belongs to special subcategories. Two classification theorems identify real numbers and pure imaginary numbers through their relationship with their conjugates.

The first theorem characterizes real numbers via the conjugate:

Real Number Test
z∈R  ⟺  z=z‾z \in \mathbb{R} \iff z = \overline{z}
Learn more about this formula: Real Number Test →


For the forward direction, suppose zz is real, meaning z=a+0i=az = a + 0i = a for some a∈Ra \in \mathbb{R}. Then zˉ=a−0i=a=z\bar{z} = a - 0i = a = z. The conjugate equals the original.

For the reverse direction, suppose z=zˉz = \bar{z}. Writing z=a+biz = a + bi, the equation becomes a+bi=a−bia + bi = a - bi. Comparing imaginary parts: b=−bb = -b, which forces 2b=02b = 0 and thus b=0b = 0. With zero imaginary part, z=az = a is real.

Geometrically, real numbers sit on the real axis — the mirror line for conjugation. Points on a mirror remain fixed under reflection, so z=zˉz = \bar{z} characterizes exactly those points.

The second theorem characterizes pure imaginary numbers:

Pure Imaginary Test
z  is pure imaginary  ⟺  z‾=−zz \;\text{is pure imaginary} \iff \overline{z} = -z
Learn more about this formula: Pure Imaginary Test →


Forward: if z=biz = bi for real bb, then zˉ=−bi=−(bi)=−z\bar{z} = -bi = -(bi) = -z.

Reverse: if zˉ=−z\bar{z} = -z, then a−bi=−a−bia - bi = -a - bi. Comparing real parts: a=−aa = -a, so 2a=02a = 0 and a=0a = 0. The number has no real part and is pure imaginary.

Geometrically, pure imaginaries sit on the imaginary axis, perpendicular to the mirror. Reflection through the real axis sends each such point to its opposite through the origin, making zˉ=−z\bar{z} = -z.
Category of z Algebraic test Geometric meaning Example
Real (b = 0) z̄ = z lies on the real axis — the mirror itself, fixed by reflection z = 5: z̄ = 5 = z  ✓
Pure imaginary (a = 0) z̄ = −z lies on the imaginary axis — reflection sends to opposite point through origin z = 4i: z̄ = −4i = −z  ✓

Useful Identities

Three identities involving zz and zˉ\bar{z} appear constantly in calculations. Each extracts specific information from a complex number or produces a value with guaranteed properties.

The sum of a number and its conjugate isolates the real part:

z+zˉ=(a+bi)+(a−bi)=2az + \bar{z} = (a + bi) + (a - bi) = 2a


The imaginary terms cancel, leaving twice the real part. Rearranging provides a formula: Re(z)=z+zˉ2Re(z) = \frac{z + \bar{z}}{2}. This identity guarantees that z+zˉz + \bar{z} is always real, regardless of the original number. Encountering this sum in any calculation signals that the result lies on the real axis.

The difference between a number and its conjugate isolates the imaginary part:

z−zˉ=(a+bi)−(a−bi)=2biz - \bar{z} = (a + bi) - (a - bi) = 2bi


The real terms cancel, leaving twice the imaginary term. Rearranging: Im(z)=z−zˉ2iIm(z) = \frac{z - \bar{z}}{2i}. This identity guarantees that z−zˉz - \bar{z} is always pure imaginary. The result necessarily sits on the vertical axis.

The product of a number and its conjugate yields the squared modulus:

z⋅zˉ=a2+b2=∣z∣2z \cdot \bar{z} = a^2 + b^2 = |z|^2


This result is always real and always non-negative. It equals zero only when z=0z = 0. The identity underlies division, modulus computation, and countless proofs.

These three identities — sum, difference, and product with the conjugate — form a toolkit for manipulating complex expressions. Recognizing when they apply often transforms an intimidating calculation into straightforward algebra.

Applications to Division

Division of complex numbers requires expressing the quotient in standard algebraic form a+bia + bi. A complex denominator violates this requirement — the conjugate provides the remedy.

Consider the division wz\frac{w}{z} where both ww and zz are complex. The denominator z=c+diz = c + di contains an imaginary part, preventing direct interpretation as a number in standard form. The strategy: multiply both numerator and denominator by zˉ\bar{z}, the conjugate of the denominator.

wz=wz⋅zˉzˉ=w⋅zˉz⋅zˉ=w⋅zˉ∣z∣2\frac{w}{z} = \frac{w}{z} \cdot \frac{\bar{z}}{\bar{z}} = \frac{w \cdot \bar{z}}{z \cdot \bar{z}} = \frac{w \cdot \bar{z}}{|z|^2}


The denominator becomes z⋅zˉ=∣z∣2z \cdot \bar{z} = |z|^2, a real number. The numerator w⋅zˉw \cdot \bar{z} is some complex number that can be computed by standard multiplication. Dividing a complex number by a real number simply scales both components, yielding standard form.

A complete example: compute 3+2i1−4i\frac{3 + 2i}{1 - 4i}.

The denominator is 1−4i1 - 4i, so its conjugate is 1+4i1 + 4i. Multiply numerator and denominator:

3+2i1−4i⋅1+4i1+4i=(3+2i)(1+4i)(1−4i)(1+4i)\frac{3 + 2i}{1 - 4i} \cdot \frac{1 + 4i}{1 + 4i} = \frac{(3 + 2i)(1 + 4i)}{(1 - 4i)(1 + 4i)}


The denominator: (1−4i)(1+4i)=1−16i2=1+16=17(1 - 4i)(1 + 4i) = 1 - 16i^2 = 1 + 16 = 17.

The numerator: (3+2i)(1+4i)=3+12i+2i+8i2=3+14i−8=−5+14i(3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i^2 = 3 + 14i - 8 = -5 + 14i.

The quotient: −5+14i17=−517+1417i\frac{-5 + 14i}{17} = -\frac{5}{17} + \frac{14}{17}i.

Without the conjugate, no systematic method converts complex quotients to standard form. The identity z⋅zˉ=∣z∣2z \cdot \bar{z} = |z|^2 makes the technique work — it guarantees the denominator becomes real.

Conjugate Pairs in Polynomials

Polynomials with real coefficients exhibit remarkable structure in their complex roots: non-real roots always appear in conjugate pairs. If z0z_0 solves the equation, so does z0ˉ\bar{z_0}.

The theorem states: let p(z)=anzn+an−1zn−1+⋯+a1z+a0p(z) = a_nz^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0 be a polynomial with all coefficients aka_k real. If p(z0)=0p(z_0) = 0 for some complex number z0z_0, then p(z0ˉ)=0p(\bar{z_0}) = 0 as well.

The proof exploits how conjugation interacts with polynomial evaluation. Since conjugation distributes over sums and products, and since conjugating a real number leaves it unchanged:

p(z0)‾=anz0n+⋯+a0‾=anz0n‾+⋯+a0=an(z0ˉ)n+⋯+a0=p(z0ˉ)\overline{p(z_0)} = \overline{a_nz_0^n + \cdots + a_0} = a_n\overline{z_0^n} + \cdots + a_0 = a_n(\bar{z_0})^n + \cdots + a_0 = p(\bar{z_0})


The key step uses ak‾=ak\overline{a_k} = a_k because each coefficient is real. If p(z0)=0p(z_0) = 0, then p(z0)‾=0ˉ=0\overline{p(z_0)} = \bar{0} = 0, so p(z0ˉ)=0p(\bar{z_0}) = 0.

Consequences flow immediately. A real quadratic with no real roots must have two complex conjugate roots — if 2+3i2 + 3i solves it, so does 2−3i2 - 3i. A real cubic always has at least one real root, since complex roots pair off and an odd number of roots cannot all be paired. A real polynomial of degree 4 might have four real roots, two real and two complex conjugates, or two pairs of complex conjugates — but never three real and one complex.

Conjugate pairs multiply to give real quadratic factors. If z0=a+biz_0 = a + bi is a root, then (z−z0)(z−z0ˉ)=z2−2az+(a2+b2)(z - z_0)(z - \bar{z_0}) = z^2 - 2az + (a^2 + b^2), a quadratic with real coefficients. This factorization explains why every real polynomial factors completely into real linear and real quadratic terms — the Fundamental Theorem of Algebra guarantees complex roots exist, and conjugate pairing ensures they combine into real factors.

Summary: What z̄ Tells You About z

The conjugate is most useful not for its own sake but for what it reveals when combined with z itself. The table below collects the identities and tests in which z̄ extracts a specific quantity from z, signals a real or non-negative result, or characterizes z as belonging to a special subset (the real axis, the imaginary axis, or the unit circle).
Expression involving z̄ Result Always… What it gives you
z + z̄ 2 · Re(z) = 2a real isolates the real part of z
z − z̄ 2i · Im(z) = 2bi pure imaginary isolates the imaginary part of z
z · z̄ |z|² = a² + b² real and non-negative the squared modulus; underpins division
|z̄| |z| equal to |z| confirms that reflection preserves distance from origin
z̄ = z condition for… true iff z is real classification test for membership in ℝ
z̄ = −z condition for… true iff z is pure imaginary classification test for membership in iℝ
z̄ = z⁻¹ condition for… true iff |z| = 1 classification test for points on the unit circle