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Additive Inverse of a Complex Number



Key Terms
Additive Inverse in Real Numbers
Additive Inverse of a Complex Number
Additive Inverse Notation
Geometric Interpretation
Properties of Additive Inverse
Additive Inverse vs. Complex Conjugate
The Zero Case
Connection to Subtraction
Uniqueness
Formula Summary
Common Mistakes
Summary: Side-by-Side Examples of −z and z̄
Additive Inverse FAQ



The Number That Cancels Through Addition

Every complex number has a counterpart that, when added to it, produces zero. This counterpart is called the additive inverse. The concept extends directly from real numbers, where 55 and −5-5 cancel each other, to complex numbers, where both the real and imaginary parts must be negated. Understanding the additive inverse clarifies the nature of subtraction and reveals a fundamental symmetry in the complex plane.

Key Terms

Additive Inverse— the number −z-z satisfying z+(−z)=0z + (-z) = 0
Complex Conjugate— a different operation that negates only the imaginary part
Complex Plane— additive inverse reflects through the origin

See All Complex Numbers Definitions →


Additive Inverse in Real Numbers

    Before examining complex numbers, consider how additive inverses work in the familiar real number system.

    For any real number aa, the additive inverse is −a-a. The defining property is:

    Additive Inverse in Real Numbers
    a+(−a)=0a + (-a) = 0

    The sum of a number and its additive inverse always equals zero.

    Examples


  • The additive inverse of 55 is −5-5, since 5+(−5)=05 + (-5) = 0

  • The additive inverse of −3-3 is 33, since −3+3=0-3 + 3 = 0

  • The additive inverse of 72\frac{7}{2} is −72-\frac{7}{2}, since 72+(−72)=0\frac{7}{2} + \left(-\frac{7}{2}\right) = 0

    • Geometric Interpretation


    On the number line, a number and its additive inverse lie at equal distances from the origin but on opposite sides. The number 55 sits five units to the right of zero; its inverse −5-5 sits five units to the left. They are mirror images across the origin.

    Uniqueness


    Every real number has exactly one additive inverse. If a+x=0a + x = 0 and a+y=0a + y = 0, then x=yx = y. No real number has two different additive inverses, and no real number lacks one.

    This structure carries over completely to complex numbers.

Additive Inverse of a Complex Number

Undoing z = −2 − 5i needs a partner that adds to 0 — and several sign rules look like they would give it.
For a real number the answer is simply "change the sign". A complex number has two signs, one on each part, so there are several ways to change them. Here are four rules that all sound reasonable.
rulepartner for −2 − 5i
flip the sign of the imaginary part−2 + 5i
flip the sign of the real part2 − 5i
make both parts negative−2 − 5i
flip the sign of both parts2 + 5i
Adding each partner to −2 − 5i: only one sum is 0.
partner(−2 − 5i) + partnerzero?
−2 + 5i−4no
2 − 5i−10ino
−2 − 5i−4 − 10ino
2 + 5i0yes
Real parts add to real parts and imaginary parts to imaginary parts, so both must cancel on their own.
Flipping only the imaginary sign cancels the −5i-5i but doubles the −2-2. Flipping only the real sign does the opposite. "Make both parts negative" sounds like flipping both, but −2−5i-2 - 5i is already negative in both parts, so it is left unchanged and the sum simply doubles.
Pitfall: Making both parts negative
Reading −z-z as the number whose two parts are negative leaves z=−2−5iz = -2 - 5i unchanged, and z+z=−4−10i≠0z + z = -4 - 10i \neq 0. Flipping each sign gives −z=2+5i-z = 2 + 5i.
The real components cancel and the imaginary components cancel, leaving zero.
The rule: replace each part by its own opposite.
Think of −2−5i-2 - 5i as a walk: 2 steps west and 5 steps south. To get back to the start you walk 2 east and 5 north — every leg reversed, not just one, and not "both legs southward".
Both the real part and the imaginary part are negated. The defining property remains unchanged:
Additive Inverse of a Complex Number
z+(−z)=0z + (-z) = 0
Verification is straightforward. Adding z=a+biz = a + bi to −z=−a−bi-z = -a - bi gives:
Additive Inverse of a Complex Number
(a+bi)+(−a−bi)=(a+(−a))+(b+(−b))i=0+0i=0(a + bi) + (-a - bi) = (a + (-a)) + (b + (-b))i = 0 + 0i = 0
The definition
The additive inverse of a complex number zz, written −z-z, is the complex number that adds to zz to give 00.
The additive inverse of a complex number follows the same principle as real numbers: find the number that sums to zero. For a complex number z=a+biz = a + bi, the additive inverse is:
Additive Inverse
−z=−a−bifor    z=a+bi-z = -a - bi \quad \text{for} \;\; z = a + bi
Learn more about this formula: Additive Inverse →
So the additive inverse of −2−5i-2 - 5i is 2+5i2 + 5i.
−z is a complex number again: for a real number it is the familiar −a from the section above.
Consider z=3+2iz = 3 + 2i. The additive inverse is −z=−3−2i-z = -3 - 2i. Adding them: (3+2i)+(−3−2i)=0(3 + 2i) + (-3 - 2i) = 0. For z=−4+5iz = -4 + 5i, the inverse is −z=4−5i-z = 4 - 5i. The signs of both components flip. For z=−1−7iz = -1 - 7i, the inverse is −z=1+7i-z = 1 + 7i. A number in the third quadrant has its inverse in the first quadrant, and vice versa.
The pattern is consistent: whatever signs the real and imaginary parts carry, the additive inverse reverses both.
The rejected rule that flips only the imaginary sign is not useless: it is the complex conjugate.
−2+5i-2 + 5i does not undo −2−5i-2 - 5i, but it is its complex conjugate, with uses of its own — sums and products with it come out real. The two operations are easy to mix up; the section additive inverse vs. complex conjugate sets them side by side.
The rest of the page covers the notation, the picture of −z-z as a half turn about the origin, the properties, the comparison with the conjugate, the zero case, subtraction and uniqueness.

Additive Inverse Notation

Notation

Additive Inverse Notation

One glyph carries this page: the minus. This section owns its unary job on a complex number, the three equivalent spellings it produces, and the boundary between minus-as-inverse and minus-as-subtraction.
zz, a+bia + bi and C\mathbb{C} come from complex numbers basics, ii from the imaginary unit; the conjugate bar is owned by its page, the modulus bars by absolute value.
−z-z
negative z; minus z
The unary minus as an operator: −z-z names the number that cancels zz, and for z=a+biz = a + bi it flips both signs — −z=−a−bi-z = -a - bi, per Additive Inverse of a Complex Number above. One mark, one point-reflection through the origin.
CasesModulus survives, direction reverses: ∣−z∣=∣z∣|-z| = |z| while arg⁡(−z)=arg⁡(z)+π\arg(-z) = \arg(z) + \pi — Geometric Interpretation below; on the polar side the sign is absorbed into the exponent, −z=rei(θ+π)-z = re^{i(\theta + \pi)}.
Do not confuseThe conjugate. The bar flips one sign, the minus flips two — reflection across the real axis versus through the origin; Additive Inverse vs. Complex Conjugate below draws both pictures.
−(a+bi)-(a + bi)
minus, quantity a plus b i
Three equivalent spellings, one number: −z-z, −(a+bi)-(a + bi), −a−bi-a - bi. Compact for abstract work, parenthesized before distributing, fully distributed when the components matter — context picks the form.
CasesNesting demands parentheses: the inverse of −3+2i-3 + 2i is −(−3+2i)=3−2i-(-3 + 2i) = 3 - 2i, and a bare −−3+2i--3 + 2i is malformed — double negation must be scoped.
Do not confuseHalf-distribution. −(a+bi)=−a+bi-(a + bi) = -a + bi is the standard slip — the minus must reach the bibi term too; Common Mistakes below collects the variants.
z−wz - w
z minus w
The glyph's second job: between two numbers it subtracts, and the jobs meet in the definition z−w=z+(−w)z - w = z + (-w) — Connection to Subtraction below. Same stroke, different arity.
CasesUnary binds to one operand (−w-w, −3i-3i); binary sits between two (z−wz - w). Calculators give them separate keys and programming languages parse them by position — mathematics never marks the difference visually.
Do not confuseA sign attached to ww. In z−wz - w the minus belongs to the operation, not the number; rewriting as z+(−w)z + (-w) moves it onto the number explicitly — which is exactly what makes the identity useful.
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Notation · Cheat sheet
Complex Number Notation at a Glance
The marks from the notation cards across the complex-number pages: how each is read and what it means. Every group links to the full card on its page.
Basics
z,  wz, \; w
z, w: complex numbers
The usual letters for complex numbers, as xx and yy are for reals.
a+bia + bi
a plus b i
Real part aa, imaginary part bb; both are real numbers.
C\mathbb{C}
C, the complex numbers
The set of all a+bia + bi; it contains R\mathbb{R}.
Imaginary numbers
ii
i, the imaginary unit
The number with i2=−1i^2 = -1.
−1\sqrt{-1}
the square root of minus one
Another name for ii; root rules for reals fail with it.
bibi
b i
A pure imaginary number: real part 00.
Algebraic form
Re⁡(z),  Im⁡(z)\operatorname{Re}(z), \; \operatorname{Im}(z)
real part, imaginary part
For z=a+biz = a + bi: aa and bb. Im⁡(z)\operatorname{Im}(z) is bb, not bibi.
a+(−b)ia + (-b)i
a plus negative b, i
Written a−bia - bi; the imaginary part is −b-b.
z=x+yiz = x + yi
z equals x plus y i
Same form with plane letters: the point (x,y)(x, y).
Complex conjugate
zˉ\bar{z}
z bar, the conjugate
For z=a+biz = a + bi, zˉ=a−bi\bar{z} = a - bi: the mirror image across the real axis.
z1+z2‾\overline{z_1 + z_2}
the conjugate of the sum
The bar covers the whole expression; it equals zˉ1+zˉ2\bar{z}_1 + \bar{z}_2.
z∗z^*
z star
The same conjugate, as written in physics and engineering.
Absolute value
∣z∣|z|
the modulus of z
Distance from 00: a2+b2\sqrt{a^2 + b^2}.
∣z∣2=zzˉ|z|^2 = z\bar{z}
mod z squared
Equals a2+b2a^2 + b^2, with no root to take.
∣z−c∣=R|z - c| = R
mod z minus c equals R
The circle of radius RR around cc.
Additive inverseThis page
−z-z
negative z
Both parts change sign: −a−bi-a - bi.
−(a+bi)-(a + bi)
minus, quantity a plus b i
The brackets carry the minus to both parts.
z−wz - w
z minus w
Adding the inverse: z+(−w)z + (-w).
Multiplicative inverse
z−1,  1zz^{-1}, \; \frac{1}{z}
z inverse; one over z
The number with z⋅z−1=1z \cdot z^{-1} = 1; it equals zˉ∣z∣2\frac{\bar{z}}{|z|^2}.
(3+2i)−1(3 + 2i)^{-1}
the inverse of three plus two i
Found by multiplying top and bottom by the conjugate: 3−2i13\frac{3 - 2i}{13}.
Trigonometric form
r,  θr, \; \theta
r and theta, the polar pair
Distance from 00 and angle from the positive real axis.
arg⁡(z)\arg(z)
the argument of z
Any angle of zz; there are infinitely many, 2π2\pi apart.
Arg⁡(z)\operatorname{Arg}(z)
the principal argument
One chosen angle, usually in (−π,π](-\pi, \pi]; some texts use [0,2π)[0, 2\pi).
cis⁡θ\operatorname{cis}\theta
sis theta
Short for cos⁡θ+isin⁡θ\cos\theta + i\sin\theta.
Exponential form
eiθe^{i\theta}
e to the i theta
Equals cis⁡θ\operatorname{cis}\theta (Euler's formula): a point on the unit circle.
reiθre^{i\theta}
r e to the i theta
The whole number zz: length rr, angle θ\theta.
ei(θ+2πk)e^{i(\theta + 2\pi k)}
theta plus two pi k
The same number for every integer kk.
De Moivre's theorem
zkz_k
z sub k, the k-th root
The roots of zn=wz^n = w, numbered k=0,…,n−1k = 0, \ldots, n - 1.
ω\omega
omega
A primitive root of unity: its powers give all nn roots of 11.
R1/nR^{1/n}
R to the one over n
The real root of the length; the complex root w1/nw^{1/n} has nn values.
Same marks, different job
∣z∣|z|The modulus here; for a real number, the absolute value; for a matrix, the determinant.
zˉ\bar{z}The conjugate here; in statistics a bar means a mean, in sets a complement.
z−1z^{-1} vs z∗z^*The reciprocal, against the conjugate: they agree only on the unit circle.
w1/nw^{1/n}nn values for a complex ww; a single real root for a positive real number.
Complex number symbols tableComplex numbers
Complex Number Notation at a Glance·/complex-numbers/basicsLearn Math Class
ReIm-5−5i-4−4i-3−3i-2−2i-1−1i11i22i33i44i55iz₁−z₂−z₂z₁z₂
−z₂ is z₂ turned through 180°

The additive inverse −z₂ is the arrow z₂ reversed: same length, opposite direction, the point reflected through the origin. Subtracting z₂ is then adding this reversed arrow, which is the diagonal z₁ − z₂ drawn in the picture. Flip the sign of any number and watch it pass through the origin on the complex addition and subtraction tool.

Reflection through the origin is the geometric meaning of a change of sign in both parts.

Geometric Interpretation

In the complex plane, a complex number z=a+biz = a + bi corresponds to the point (a,b)(a, b). Its additive inverse −z=−a−bi-z = -a - bi corresponds to the point (−a,−b)(-a, -b). The relationship between these two points reveals a clean geometric picture.

RULE
The additive inverse is the reflection of zz through the origin.

Not across the real axis, not across the imaginary axis, but through the origin itself — the single point where both axes intersect. If zz lies in the first quadrant, −z-z lies in the third. If zz lies in the second quadrant, −z-z lies in the fourth. The origin sits exactly midway between any number and its additive inverse.

The modulus remains unchanged under this reflection:

Geometric Interpretation
∣−z∣=∣z∣|-z| = |z|

Both zz and −z-z lie at the same distance from the origin. They occupy opposite ends of a diameter of the circle centered at the origin with radius ∣z∣|z|.

Pitfall: Pulling the minus out of the modulus
Writing ∣−z∣=−∣z∣|-z| = -|z| for z=3+4iz = 3 + 4i gives ∣−3−4i∣=−5|-3 - 4i| = -5, a negative distance. The modulus ignores the signs: ∣−3−4i∣=9+16=5|-3 - 4i| = \sqrt{9 + 16} = 5.

The modulus is computed from the squares a2a^2 and b2b^2, and squaring erases the sign of each part. Negating zz changes both signs at once, yet the two squares under the root, and with them the distance from the origin, stay exactly as they were.

The argument shifts by exactly π\pi radians:

Argument of Negative
arg⁡(−z)=arg⁡(z)+π\arg(-z) = \arg(z) + \pi
Learn more about this formula: Argument of Negative →


If zz points in direction θ\theta, then −z-z points in direction θ+π\theta + \pi — the exact opposite direction. A rotation of 180°180° about the origin carries zz to −z-z.

This geometric understanding unifies the algebraic definition. Negating both coordinates, preserving distance, reversing direction, and reflecting through the origin are all descriptions of the same transformation.
z = 3 + 2i and -z = -3 - 2i, a half turn apart about 0ReIm−4−4i−2−2i22i44i180°0: the midpointz = 3 + 2i−z = −3 − 2i|z| = |−z| = √13−z is z turned a half turn about 0: same distance, opposite direction
z and −z through the origin

Negating both coordinates of z = 3 + 2i gives −z = −3 − 2i, the point directly opposite through 0. The two sit at the ends of one diameter of the circle of radius |z| = √13, so |−z| = |z|, and the origin is their midpoint. The curved arrow is the half turn about 0 that carries z to −z: the argument grows by exactly 180°.

Compare the conjugate: it reflects z across the real axis instead, so it keeps the real part and flips only the sign of the imaginary part.

Properties of Additive Inverse

The additive inverse interacts predictably with other operations on complex numbers. These properties follow from the definitions and mirror the corresponding rules for real numbers.

Double negation returns the original number:

Properties of Additive Inverse
−(−z)=z-(-z) = z

Taking the additive inverse twice leaves zz unchanged. Reflecting through the origin twice brings you back to where you started.

Negation distributes over addition:

Properties of Additive Inverse
−(z1+z2)=−z1−z2-(z_1 + z_2) = -z_1 - z_2

The inverse of a sum equals the sum of the inverses. This property underlies the familiar rule for expanding expressions like −(3+2i+4−i)-(3 + 2i + 4 - i).

Negation and conjugation commute:

Properties of Additive Inverse
−z‾=−z‾-\overline{z} = \overline{-z}

It does not matter whether you conjugate first and then negate, or negate first and then conjugate.

Example
For z=3+2iz = 3 + 2i, we have z‾=3−2i\overline{z} = 3 - 2i and −z‾=−3+2i-\overline{z} = -3 + 2i. Alternatively, −z=−3−2i-z = -3 - 2i and −z‾=−3+2i\overline{-z} = -3 + 2i. Both paths arrive at the same destination.

The modulus is unchanged:

Properties of Additive Inverse
∣−z∣=∣z∣|-z| = |z|

Distance from the origin depends only on position, not direction.

The argument shifts by π\pi:

Argument of Negative
arg⁡(−z)=arg⁡(z)+π\arg(-z) = \arg(z) + \pi
Learn more about this formula: Argument of Negative →


Direction reverses completely, adding half a full rotation.
Property Formula Meaning
Defining property z + (−z) = 0 a number plus its inverse sums to zero
Double negation −(−z) = z applying inversion twice returns the original
Distribution over sum −(z₁ + z₂) = −z₁ − z₂ inverse of a sum = sum of inverses
Commutes with conjugation −z̄ = −z order of negation and conjugation doesn't matter
Modulus preservation |−z| = |z| same distance from the origin
Argument shift arg(−z) = arg(z) + π direction reverses — a 180° rotation about the origin

Additive Inverse vs. Complex Conjugate

Students frequently confuse the additive inverse with the complex conjugate. Both operations involve changing signs, but they change different signs and produce different geometric effects.

The conjugate of z=a+biz = a + bi is z‾=a−bi\overline{z} = a - bi. Only the imaginary part changes sign. The real part remains untouched.

The additive inverse of z=a+biz = a + bi is −z=−a−bi-z = -a - bi. Both parts change sign. Nothing remains untouched.

Consider z=3+2iz = 3 + 2i. The conjugate is z‾=3−2i\overline{z} = 3 - 2i — same real part, opposite imaginary part. The additive inverse is −z=−3−2i-z = -3 - 2i — both parts negated. These are different numbers occupying different locations in the complex plane.

Geometrically, conjugation reflects across the real axis. A point above the axis moves to the corresponding point below, and vice versa. The horizontal position stays fixed. Additive inversion reflects through the origin. A point moves to the diametrically opposite position, with both coordinates reversed.

For z=4+3iz = 4 + 3i: the conjugate z‾=4−3i\overline{z} = 4 - 3i lies directly below zz, reflected across the horizontal axis. The additive inverse −z=−4−3i-z = -4 - 3i lies diagonally opposite, reflected through the origin.

When are they equal? Only for pure imaginary numbers (zero included). If z=biz = bi, then z‾=−bi\overline{z} = -bi and −z=−bi-z = -bi. Both operations produce the same result. For any number with a nonzero real part, conjugate and additive inverse differ.

The most common mistake is writing −(3+2i)=−3+2i-(3 + 2i) = -3 + 2i, negating only the real part. This produces the negation of the conjugate, not the additive inverse. The correct result is −(3+2i)=−3−2i-(3 + 2i) = -3 - 2i, with both signs flipped.
Feature Additive inverse  −z Complex conjugate  z̄
Algebraic action on a + bi negates both parts: −a − bi negates only the imaginary part: a − bi
Geometric action reflection through the origin (180° rotation) reflection across the real axis
Modulus |−z| = |z| |z̄| = |z|
Argument arg(z) + π −arg(z)
Defining property the number w with z + w = 0 the reflection a − bi of a + bi
−z and z̄ coincide when…
z is pure imaginary (zero included)
Most common confusion writing −(a + bi) = −a + bi (flipping only one sign) writing z̄ = −z (which is true only for pure imaginary z, zero included)
ReIm-10−10i-8−8i-6−6i-4−4i-2−2i22i44i66i88i1010i|z| = 3.6zz̅
z̄ reflects in the real axis; −z would reflect through the origin

The conjugate shown here changes only the sign of the imaginary part, so z and z̄ are mirror images across the real axis. The additive inverse changes both signs and lands diagonally opposite, in the quadrant the conjugate never visits. Compare the two by hand on the conjugate and modulus tool.

Two sign changes versus one: that is the whole difference between the two operations.

The Zero Case

The complex number zero occupies a unique position in the additive structure. It is the only complex number that equals its own additive inverse.

For z=0=0+0iz = 0 = 0 + 0i, the additive inverse is:

The Zero Case
−0=−0−0i=0+0i=0-0 = -0 - 0i = 0 + 0i = 0

Fact: Floating point keeps a minus zero
The IEEE 754 standard stores −0-0 apart from 00: the two compare equal, but the sign stays. Its designer William Kahan kept it so complex functions know which side of a cut a value came from: in Python, the square root of −4+0i-4 + 0i is 2i2i, but that of −4−0i-4 - 0i is −2i-2i.

Negating zero returns zero. The equation 0+0=00 + 0 = 0 confirms the defining property: zero added to its inverse yields zero.

No other complex number shares this property. For any z≠0z \neq 0, the additive inverse −z-z differs from zz. If z=a+biz = a + bi with at least one of aa or bb nonzero, then −z=−a−bi-z = -a - bi cannot equal zz because either −a≠a-a \neq a or −b≠b-b \neq b (or both).

Zero also serves as the additive identity — the unique complex number satisfying z+0=zz + 0 = z for every zz. These two roles are connected. The additive identity must be its own inverse: if 0+0=00 + 0 = 0 follows from the identity property, and 0+(−0)=00 + (-0) = 0 follows from the inverse property, then 0=−00 = -0.

Geometrically, zero sits at the origin of the complex plane. Reflection through the origin maps the origin to itself. Every other point moves to a different location; the origin alone stays fixed.

Zero is the only complex number that is its own opposite, the one point a half turn about the origin leaves in place.

Connection to Subtraction

Subtraction of complex numbers is not a separate operation but rather addition in disguise. To subtract one complex number from another, add the additive inverse:

Connection to Subtraction
z1−z2=z1+(−z2)z_1 - z_2 = z_1 + (-z_2)

This definition reduces subtraction to addition, which is already well understood. All properties of subtraction follow from properties of addition combined with properties of the additive inverse.

Consider (5+3i)−(2+7i)(5 + 3i) - (2 + 7i). The additive inverse of 2+7i2 + 7i is −2−7i-2 - 7i. Adding:

Connection to Subtraction
(5+3i)+(−2−7i)=(5+(−2))+(3+(−7))i=3−4i(5 + 3i) + (-2 - 7i) = (5 + (-2)) + (3 + (-7))i = 3 - 4i

Pitfall: Dropping a sign in subtraction
Changing only the real sign of the subtracted number gives (5+3i)−(2+7i)=5+3i−2+7i=3+10i(5 + 3i) - (2 + 7i) = 5 + 3i - 2 + 7i = 3 + 10i. The inverse −2−7i-2 - 7i changes both signs, so the difference is 3−4i3 - 4i.

This matches the component-wise subtraction rule: subtract real parts and subtract imaginary parts. The rule works precisely because subtraction means adding the inverse, and the inverse negates both components.

The connection explains why subtraction inherits certain behaviors. Subtraction is not commutative: z1−z2≠z2−z1z_1 - z_2 \neq z_2 - z_1 in general, because z1+(−z2)≠z2+(−z1)z_1 + (-z_2) \neq z_2 + (-z_1) unless z1=z2z_1 = z_2. Subtraction is not associative in the usual sense, but the expression z1−z2−z3z_1 - z_2 - z_3 is unambiguous when read left to right as (z1−z2)−z3=z1+(−z2)+(−z3)(z_1 - z_2) - z_3 = z_1 + (-z_2) + (-z_3).

For detailed coverage of subtraction and other arithmetic, see operations.

Subtraction is never a new operation: subtracting a number means adding its opposite, and every rule for differences follows from that.

Uniqueness

THEOREM
Every complex number has exactly one additive inverse.

Not zero inverses, not two inverses — precisely one.

The existence is clear: for any z=a+biz = a + bi, the number −z=−a−bi-z = -a - bi satisfies the defining property z+(−z)=0z + (-z) = 0. An additive inverse always exists.

Uniqueness requires a short argument. Suppose w1w_1 and w2w_2 are both additive inverses of zz, meaning z+w1=0z + w_1 = 0 and z+w2=0z + w_2 = 0. Then:

Uniqueness
w1=w1+0=w1+(z+w2)=(w1+z)+w2=0+w2=w2w_1 = w_1 + 0 = w_1 + (z + w_2) = (w_1 + z) + w_2 = 0 + w_2 = w_2

The two supposed inverses must be the same. No complex number admits two different additive inverses.
The uniqueness chain above, split at each equals sign, with the rule used at every step.
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ProveIf z+w1=0z + w_1 = 0 and z+w2=0z + w_2 = 0, then w1=w2w_1 = w_2
1
w1=w1+0w_1 = w_1 + 0
identityZero is the additive identity
2
=w1+(z+w2)= w_1 + (z + w_2)
substituteSince z+w2=0z + w_2 = 0
3
=(w1+z)+w2= (w_1 + z) + w_2
key stepAssociativity of addition
4
=0+w2= 0 + w_2
substituteSince w1w_1 is an additive inverse of zz
5
=w2= w_2
identityZero is the additive identity
w1=w2w_1 = w_2
The two supposed inverses must be the same.
Proving the additive inverse is uniqueLearn Math Class
The gold line is associativity: regrouping w1+(z+w2)w_1 + (z + w_2) as (w1+z)+w2(w_1 + z) + w_2 is what lets the two inverses meet.
This uniqueness is not an accident but a consequence of the field axioms that govern complex arithmetic. In any field, additive inverses are unique. The argument above uses only associativity of addition and the properties of zero as the additive identity — both guaranteed by the field structure.

Uniqueness justifies the notation −z-z. If multiple inverses existed, writing −z-z would be ambiguous — which inverse do we mean? Because exactly one inverse exists, the notation refers unambiguously to that single number.

Formula Summary

This summary collects the formulas for the additive inverse of a complex number in three families, shown one at a time with the buttons above the cards. Each card gives a formula with a line on what it says and a link to the section that explains it.

The first family defines the negative, which flips the sign of both parts and adds to the number to give zero. The second places it in the plane, at the same distance from the origin and half a turn round. The last gathers the rules: a double negative, the negative of a sum, the swap with the conjugate, and subtraction as adding the negative.
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Definition2 formulas
What the negative of a complex number is.
−z=−a−bi-z = -a - bi
For z=a+biz = a + bi, both the real and the imaginary part change sign, as the additive inverse of a complex number defines it.
z+(−z)=0z + (-z) = 0
A number plus its negative gives zero, the property that makes it the inverse, from the same section.
In the plane2 formulas
Where the negative sits compared with the number.
∣−z∣=∣z∣|-z| = |z|
The negative lies at the same distance from the origin, as the geometric interpretation shows.
arg⁡(−z)=arg⁡(z)+π\arg(-z) = \arg(z) + \pi
Its angle is half a turn more, so −z-z is zz reflected through the origin (geometric interpretation).
Rules4 formulas
How the negative combines with other operations.
−(−z)=z-(-z) = z
Taking the negative twice gives the number back, from properties of additive inverse.
−(z1+z2)=−z1−z2-(z_1 + z_2) = -z_1 - z_2
The negative of a sum is the sum of the negatives, as the properties show.
−z‾=−z‾-\overline{z} = \overline{-z}
Negating and taking the conjugate give the same result in either order, from properties of additive inverse.
z1−z2=z1+(−z2)z_1 - z_2 = z_1 + (-z_2)
Subtracting a number means adding its negative, as connection to subtraction shows.
Additive inverse formulas·All complex number formulasLearn Math Class
Every formula here comes from flipping both signs. When a result looks doubtful, check it against the picture: the negative sits exactly opposite the number, through the origin and at the same distance, while the conjugate flips only the imaginary part.

Common Mistakes

Several errors appear repeatedly when students work with additive inverses of complex numbers. Recognizing these patterns helps avoid them.

The most frequent mistake is negating only the real part. Given z=3+2iz = 3 + 2i, a student writes −z=−3+2i-z = -3 + 2i instead of the correct −z=−3−2i-z = -3 - 2i. This error produces the negative of the conjugate, not the additive inverse. Both components must change sign.

The reverse error also occurs: negating only the imaginary part. Writing −z=3−2i-z = 3 - 2i instead of −z=−3−2i-z = -3 - 2i produces the conjugate, not the additive inverse. Again, both components must change sign.

Confusion between −z-z and z‾\overline{z} underlies both mistakes. The conjugate flips one sign; the additive inverse flips both. Keeping this distinction clear prevents the errors.

Sign mistakes compound in subtraction. Computing (5+2i)−(3−4i)(5 + 2i) - (3 - 4i) requires the additive inverse of 3−4i3 - 4i, which is −3+4i-3 + 4i. The full calculation:

Common Mistakes
(5+2i)+(−3+4i)=2+6i(5 + 2i) + (-3 + 4i) = 2 + 6i

A common error is writing (5+2i)−(3−4i)=5+2i−3−4i=2−2i(5 + 2i) - (3 - 4i) = 5 + 2i - 3 - 4i = 2 - 2i, forgetting that subtracting −4i-4i means adding +4i+4i. Distributing the negative sign completely — to both the real and imaginary parts of the subtracted number — eliminates this mistake.

When in doubt, write out the additive inverse explicitly before combining terms. Converting z1−z2z_1 - z_2 to z1+(−z2)z_1 + (-z_2) and computing −z2-z_2 as a separate step reduces errors.
Mistake Wrong result Correct result Why
Negating only the real part −(3 + 2i) = −3 + 2i  ✗ −3 − 2i this gives −z̄ , not −z; the additive inverse flips both signs
Negating only the imaginary part −(3 + 2i) = 3 − 2i  ✗ −3 − 2i this gives z̄ , not −z; same root cause — both parts must change sign
Dropping a sign in subtraction (5 + 2i) − (3 − 4i) = 2 − 2i  ✗ 2 + 6i subtracting −4i means adding +4i; the negative must distribute to both parts of the subtrahend
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1Making both parts negative
The minus in −z-z is read as an instruction to make both parts negative, rather than to change the sign of each part. A number whose parts are already negative is then left as it is, and a number with one negative part gets only its other part changed.
Start
For z=−2−5iz = -2 - 5i,
Wrong turn
Error here
Both parts are already negative, so −z=−2−5i-z = -2 - 5i.
Right path
−z=−(−2)−(−5)i=2+5i-z = -(-2) - (-5)i = 2 + 5i, and (−2−5i)+(2+5i)=0(-2 - 5i) + (2 + 5i) = 0.
Why it happensThe formula −z=−a−bi-z = -a - bi shows two minus signs, which suggests that the inverse always has negative parts. The minus signs act on aa and bb, and when these are negative, the parts of −z-z come out positive.
How to checkAdd the result to zz. With the wrong answer, (−2−5i)+(−2−5i)=−4−10i(-2 - 5i) + (-2 - 5i) = -4 - 10i, not 00.
2Pulling the minus out of the modulus
The minus sign of −z-z is carried outside the modulus bars, so ∣−z∣|-z| is written as −∣z∣-|z|. The result is a negative number, which cannot be a distance.
Start
For z=3+4iz = 3 + 4i,
Wrong turn
Error here
∣−z∣=−∣z∣=−9+16=−5|-z| = -|z| = -\sqrt{9 + 16} = -5.
Right path
−z=−3−4i-z = -3 - 4i, so ∣−z∣=(−3)2+(−4)2=9+16=5=∣z∣|-z| = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = 5 = |z|.
Why it happensIn ordinary algebra a factor of −1-1 is often moved to the front of an expression, and the same move is tried on the modulus. The modulus of a product is the product of the moduli, so ∣−z∣=∣−1∣⋅∣z∣=∣z∣|-z| = |-1| \cdot |z| = |z|, because ∣−1∣=1|-1| = 1, not −1-1.
How to checkA modulus is never negative, so any negative value for ∣−z∣|-z| is wrong at once.
Common pitfalls with the additive inverse·/complex-numbers/additive-inverseLearn Math Class
Both slips forget what −z-z is: the number that adds to zz to give 00. It flips the sign of both parts, whatever those signs are, and it lies at the same distance from the origin, so ∣−z∣=∣z∣|-z| = |z|. Adding the answer to zz, and remembering that a modulus is never negative, catches both.

Summary: Side-by-Side Examples of −z and z̄

The additive inverse and the conjugate both negate signs, but they negate different signs and produce different points on the complex plane. The table below presents both operations side by side for a representative selection of complex numbers, highlighting when −z and z̄ coincide and when they diverge.
z −z  (both signs flip) z̄  (only imag flips) −z = z̄ ? Geometric note
3 + 2i  (Q1) −3 − 2i  (Q3) 3 − 2i  (Q4) ✗ −z opposite quadrant; z̄ mirrored across real axis
−1 + 4i  (Q2) 1 − 4i  (Q4) −1 − 4i  (Q3) ✗ −z and z̄ are mirror images of each other across the imaginary axis
5  (real) −5 5 ✗ only −z reflects; z̄ = z since the point lies on the real axis
4i  (pure imag) −4i −4i ✓ both operations land at (0, −4); the only nonzero case where they agree
0 0 0 ✓ zero is the unique fixed point of both reflections
2 − 3i  (Q4) −2 + 3i  (Q2) 2 + 3i  (Q1) ✗ again −z and z̄ are mirror images across the imaginary axis

Additive Inverse FAQ

Does negating a complex number change its modulus?

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No. Rotating a point half a turn about the origin leaves its distance from the origin untouched, so a number and its negative always lie on the same circle. What does change is the argument, which shifts by half a turn and may then need reducing back into the principal range.Read more →

Do negation and conjugation commute?

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Yes. Negating then conjugating gives the same result as conjugating then negating, because the two operations act on different components independently. One flips both signs, the other flips only the imaginary sign, and applying them in either order flips the real part once and the imaginary part twice.Read more →

Is negative zero different from zero?

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Not as a number: −0=−0−0i=0-0 = -0 - 0i = 0, and zero is the only complex number that is its own additive inverse. Computers are the exception. The IEEE 754 floating-point standard stores −0-0 apart from 00; the two compare equal, but the sign is kept. William Kahan designed it so that complex functions can tell which side of a cut a value came from: in Python the square root of −4+0i-4 + 0i is 2i2i, but that of −4−0i-4 - 0i is −2i-2i.Read more →

Does the additive inverse always have negative parts?

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No. The minus in −z-z changes the sign of each part, whatever that sign was, so for z=−2−5iz = -2 - 5i the inverse is −z=2+5i-z = 2 + 5i, with both parts positive. Reading −z-z as the number whose parts are both negative would leave −2−5i-2 - 5i unchanged, and z+z=−4−10iz + z = -4 - 10i is not 00. Adding the answer to zz is the quick check: the sum must be exactly 00.Read more →

Is the minus in −z-z the same as the minus in z−wz - w?

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It is the same stroke doing two jobs. In −z-z it acts on one number and gives its additive inverse; in z−wz - w it sits between two numbers and subtracts. The two jobs meet in the definition z−w=z+(−w)z - w = z + (-w), which moves the minus onto ww. Calculators give the two separate keys, and programming languages tell them apart by position.Read more →