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Absolute Value



Key Terms
Definition of Modulus
Special Cases
Modulus and the Conjugate
Properties of Modulus
The Triangle Inequality
Proving the Triangle Inequality
The Reverse Triangle Inequality
Applications
Summary: How |z| Behaves Under Every Operation on z



Measuring Distance in the Complex Plane

Every complex number occupies a position in the complex plane, and a natural question arises: how far does it lie from the origin? The modulus — also called absolute value — answers this question with a single non-negative real number. Beyond mere measurement, the modulus obeys algebraic laws that connect it to the conjugate, govern products and quotients, and establish fundamental inequalities used throughout analysis.

Key Terms

Modulus— the absolute value ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}
Complex Plane— where modulus measures distance
Complex Conjugate— connected via ∣z∣2=z⋅z‾|z|^2 = z \cdot \overline{z}

See All Complex Numbers Definitions →


Definition of Modulus

The modulus of a complex number z=a+biz = a + bi, written ∣z∣|z|, measures the straight-line distance from the origin to the point (a,b)(a, b) in the complex plane. This distance follows directly from the Pythagorean theorem: the point (a,b)(a, b) sits at the end of a right triangle with horizontal leg aa and vertical leg bb, so the hypotenuse has length a2+b2\sqrt{a^2 + b^2}.

The formula is thus:

Modulus
∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}
Learn more about this formula: Modulus →


An often-useful equivalent form avoids the square root:

Modulus Squared
∣z∣2=a2+b2|z|^2 = a^2 + b^2
Learn more about this formula: Modulus Squared →


The notation mirrors real number absolute value, and for good reason — modulus generalizes the familiar concept to two dimensions. Where real absolute value measures distance from zero along a line, complex modulus measures distance from zero across a plane.

Computation proceeds by squaring both components, adding, and taking the square root. For z=3+4iz = 3 + 4i, we calculate ∣z∣=32+42=9+16=25=5|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. The famous 3-4-5 right triangle appears here — the complex number 3+4i3 + 4i lies exactly 5 units from the origin. For z=−2+iz = -2 + i, the modulus is ∣z∣=(−2)2+12=4+1=5|z| = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}, an irrational distance approximately equal to 2.2362.236.

The geometric picture makes modulus tangible. Draw the point, draw the line segment connecting it to the origin, and measure that segment's length. Every complex number possesses exactly one modulus, always a non-negative real value. Numbers closer to the origin have smaller moduli; numbers farther away have larger moduli. The modulus provides the only meaningful sense of "size" for complex numbers, since C\mathbb{C} lacks the ordering that lets us compare real numbers directly.
z = a + bi a² b² a² + b² |z| = √(a² + b²)
3 + 4i 9 16 25 5  (the 3-4-5 right triangle)
−2 + i 4 1 5 √5 ≈ 2.236
1 − i 1 1 2 √2 ≈ 1.414
5  (real) 25 0 25 5  (= ordinary absolute value)
−3i  (pure imag) 0 9 9 3  (= absolute value of the coefficient)
0 0 0 0 0  (the only z with |z| = 0)

Special Cases

The modulus formula applies uniformly to all complex numbers, but certain categories yield particularly simple results. These special cases connect complex modulus back to familiar real-number concepts.

For purely real numbers — those with b=0b = 0 — the modulus reduces to ordinary absolute value. If z=az = a where aa is real, then ∣z∣=a2+02=a2=∣a∣|z| = \sqrt{a^2 + 0^2} = \sqrt{a^2} = |a|. The number 55 has modulus 55; the number −7-7 has modulus 77. Real numbers sit on the horizontal axis of the complex plane, and their distance from the origin equals their distance from zero on the real line. Complex modulus and real absolute value agree completely on their shared domain.

For pure imaginary numbers — those with a=0a = 0 — the modulus equals the absolute value of the coefficient. If z=biz = bi, then ∣z∣=02+b2=b2=∣b∣|z| = \sqrt{0^2 + b^2} = \sqrt{b^2} = |b|. The number 4i4i has modulus 44; the number −3i-3i has modulus 33. Pure imaginaries sit on the vertical axis, and their modulus simply measures how far up or down they lie from the origin, ignoring the sign.

The complex number zero stands alone with modulus zero. Since 0=0+0i0 = 0 + 0i, we have ∣0∣=02+02=0|0| = \sqrt{0^2 + 0^2} = 0. This is the only complex number with vanishing modulus — every nonzero complex number lies some positive distance from the origin. The equivalence "∣z∣=0|z| = 0 if and only if z=0z = 0" proves essential in analysis, allowing us to test whether an expression equals zero by checking whether its modulus vanishes.

Modulus and the Conjugate

A beautiful identity connects the modulus to the complex conjugate: for any complex number zz, the product of zz and its conjugate equals the square of the modulus.

Conjugate Times Number
z⋅z‾=∣z∣2=a2+b2z \cdot \overline{z} = |z|^2 = a^2 + b^2
Learn more about this formula: Conjugate Times Number →


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

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


The cross terms cancel, and since i2=−1i^2 = -1, the imaginary square becomes positive. The result a2+b2a^2 + b^2 is precisely ∣z∣2|z|^2 by definition of modulus.

This identity appears everywhere in complex analysis. It explains why multiplying by the conjugate eliminates imaginary components — the product is guaranteed real. It provides a quick way to compute modulus: rather than taking a square root, compute z⋅zˉz \cdot \bar{z} and recognize the result as ∣z∣2|z|^2.

The identity also enables division. To compute wz\frac{w}{z}, multiply numerator and denominator by zˉ\bar{z}:

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


The denominator becomes the real number ∣z∣2|z|^2, converting the quotient to standard algebraic form. Without the identity z⋅zˉ=∣z∣2z \cdot \bar{z} = |z|^2, division of complex numbers would require far more cumbersome techniques.

Properties of Modulus

The modulus operation obeys algebraic laws that mirror and extend properties of real absolute value. These rules simplify calculations and enable estimation when exact values prove difficult.

Non-negativity holds universally: ∣z∣≥0|z| \geq 0 for every complex number zz. The modulus is a sum of squares under a square root, and both operations preserve non-negativity. Equality ∣z∣=0|z| = 0 occurs precisely when z=0z = 0, since only the origin lies zero distance from itself.

The conjugate preserves modulus:

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


Conjugation reflects a point across the real axis, and reflection does not change distance from the origin. The points z=3+4iz = 3 + 4i and zˉ=3−4i\bar{z} = 3 - 4i both lie exactly 5 units from zero.

Products follow a multiplicative rule:

Modulus of a Product
∣z1⋅z2∣=∣z1∣⋅∣z2∣|z_1 \cdot z_2| = |z_1| \cdot |z_2|
Learn more about this formula: Modulus of a Product →


The modulus of a product equals the product of the moduli. This property, impossible to guess from the definition, emerges from the algebra of complex multiplication and finds its natural explanation in trigonometric form, where multiplication multiplies lengths.

Division obeys the corresponding quotient rule:

Modulus of a Quotient
∣z1z2∣=∣z1∣∣z2∣\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}
Learn more about this formula: Modulus of a Quotient →


The modulus of a quotient equals the quotient of the moduli (for z2≠0z_2 \neq 0).

Powers extend the pattern:

Modulus of a Power
∣zn∣=∣z∣n|z^n| = |z|^n
Learn more about this formula: Modulus of a Power →


Raising a complex number to the nn-th power raises its modulus to the nn-th power. Combined with De Moivre's theorem, this property makes computing powers of complex numbers remarkably efficient.

These rules transform modulus from a mere distance measurement into a powerful algebraic tool. When facing a complicated product or quotient, compute each factor's modulus separately and combine using multiplication or division — often far simpler than working with the full complex expressions.
Property Formula Note
Non-negativity |z| ≥ 0 equality holds iff z = 0
Conjugate-preservation |z̄| = |z| reflection across the real axis doesn't change distance from origin
Conjugate identity z · z̄ = |z|² the cornerstone identity; powers division and many proofs
Product rule |z₁ · z₂| = |z₁| · |z₂| moduli of a product multiply
Quotient rule |z₁ / z₂| = |z₁| / |z₂|  (z₂ ≠ 0) moduli of a quotient divide
Power rule |zⁿ| = |z|ⁿ  (integer n) modulus raised to the same power; pairs naturally with De Moivre's theorem

The Triangle Inequality

Among all properties of modulus, none proves more fundamental than the triangle inequality. This bound constrains how moduli combine under addition and provides the foundation for estimation throughout complex analysis.

The statement is concise: for any two complex numbers z1z_1 and z2z_2,

Triangle Inequality
∣z1+z2∣≤∣z1∣+∣z2∣|z_1 + z_2| \leq |z_1| + |z_2|
Learn more about this formula: Triangle Inequality →


The modulus of a sum never exceeds the sum of the moduli. Equality can occur, but the left side can never surpass the right.

The name comes from geometry. In the complex plane, the numbers 00, z1z_1, and z1+z2z_1 + z_2 form vertices of a triangle (possibly degenerate). The side from 00 to z1+z2z_1 + z_2 has length ∣z1+z2∣|z_1 + z_2|. The path from 00 to z1z_1 and then from z1z_1 to z1+z2z_1 + z_2 has total length ∣z1∣+∣z2∣|z_1| + |z_2|. The direct path cannot exceed the detour — a triangle's side never surpasses the sum of the other two sides.

The vector interpretation makes this vivid. View z1z_1 and z2z_2 as arrows from the origin. Their sum z1+z2z_1 + z_2 places z2z_2 at the tip of z1z_1 and measures the resultant. Walking along both vectors sequentially covers distance ∣z1∣+∣z2∣|z_1| + |z_2|. The straight-line resultant covers at most that distance, and typically less unless the vectors point in the same direction.

The inequality extends to any finite sum: ∣z1+z2+⋯+zn∣≤∣z1∣+∣z2∣+⋯+∣zn∣|z_1 + z_2 + \cdots + z_n| \leq |z_1| + |z_2| + \cdots + |z_n|. Repeated application of the two-term inequality yields this generalization immediately.

Proving the Triangle Inequality

The geometric picture motivates the triangle inequality, but rigorous proof requires algebraic verification. The argument proceeds by squaring both sides and exploiting the identity ∣z∣2=z⋅zˉ|z|^2 = z \cdot \bar{z}.

Begin with the squared modulus of the sum:

∣z1+z2∣2=(z1+z2)⋅(z1+z2)‾=(z1+z2)(z1ˉ+z2ˉ)|z_1 + z_2|^2 = (z_1 + z_2) \cdot \overline{(z_1 + z_2)} = (z_1 + z_2)(\bar{z_1} + \bar{z_2})


Expanding the product:

=z1z1ˉ+z1z2ˉ+z2z1ˉ+z2z2ˉ=∣z1∣2+z1z2ˉ+z1z2ˉ‾+∣z2∣2= z_1\bar{z_1} + z_1\bar{z_2} + z_2\bar{z_1} + z_2\bar{z_2} = |z_1|^2 + z_1\bar{z_2} + \overline{z_1\bar{z_2}} + |z_2|^2


The middle terms z1z2ˉz_1\bar{z_2} and z1z2ˉ‾\overline{z_1\bar{z_2}} are conjugates of each other. Their sum equals 2Re(z1z2ˉ)2\text{Re}(z_1\bar{z_2}), twice the real part. Thus:

∣z1+z2∣2=∣z1∣2+2Re(z1z2ˉ)+∣z2∣2|z_1 + z_2|^2 = |z_1|^2 + 2\text{Re}(z_1\bar{z_2}) + |z_2|^2


The key observation: the real part of any complex number never exceeds its modulus, so Re(z1z2ˉ)≤∣z1z2ˉ∣=∣z1∣∣z2∣\text{Re}(z_1\bar{z_2}) \leq |z_1\bar{z_2}| = |z_1||z_2|. Substituting:

∣z1+z2∣2≤∣z1∣2+2∣z1∣∣z2∣+∣z2∣2=(∣z1∣+∣z2∣)2|z_1 + z_2|^2 \leq |z_1|^2 + 2|z_1||z_2| + |z_2|^2 = (|z_1| + |z_2|)^2


Taking square roots of both sides (valid since all quantities are non-negative) yields the triangle inequality.

Equality holds precisely when Re(z1z2ˉ)=∣z1z2ˉ∣\text{Re}(z_1\bar{z_2}) = |z_1\bar{z_2}|, which occurs when z1z2ˉz_1\bar{z_2} is a non-negative real number. This happens if and only if z1z_1 and z2z_2 point in the same direction from the origin — one is a non-negative real multiple of the other. Geometrically, the triangle degenerates to a line segment when both vectors align.

The Reverse Triangle Inequality

A companion inequality bounds differences rather than sums. The reverse triangle inequality states:

Reverse Triangle Inequality
∣∣z1∣−∣z2∣∣≤∣z1−z2∣\bigl||z_1| - |z_2|\bigr| \leq |z_1 - z_2|
Learn more about this formula: Reverse Triangle Inequality →


The absolute difference of two moduli never exceeds the modulus of the difference. This bound proves useful when estimating how much moduli can vary as complex numbers change.

The proof derives from the standard triangle inequality. Write z1=(z1−z2)+z2z_1 = (z_1 - z_2) + z_2. Applying the triangle inequality:

∣z1∣=∣(z1−z2)+z2∣≤∣z1−z2∣+∣z2∣|z_1| = |(z_1 - z_2) + z_2| \leq |z_1 - z_2| + |z_2|


Rearranging gives ∣z1∣−∣z2∣≤∣z1−z2∣|z_1| - |z_2| \leq |z_1 - z_2|. The same argument with roles reversed yields ∣z2∣−∣z1∣≤∣z2−z1∣=∣z1−z2∣|z_2| - |z_1| \leq |z_2 - z_1| = |z_1 - z_2|. Combining both inequalities:

−∣z1−z2∣≤∣z1∣−∣z2∣≤∣z1−z2∣-|z_1 - z_2| \leq |z_1| - |z_2| \leq |z_1 - z_2|


This is equivalent to ∣∣z1∣−∣z2∣∣≤∣z1−z2∣\big||z_1| - |z_2|\big| \leq |z_1 - z_2|.

The geometric interpretation: ∣z1−z2∣|z_1 - z_2| measures the distance between two points in the plane. The quantity ∣∣z1∣−∣z2∣∣\big||z_1| - |z_2|\big| measures how much their distances from the origin differ. Two points can be close together (small ∣z1−z2∣|z_1 - z_2|) even if they lie at very different distances from the origin — but the difference in their radii cannot exceed their separation. Conversely, two points far apart have moduli that may differ substantially, but never by more than the distance between them.
Inequality Statement Geometric reading Equality holds when…
Triangle |z₁ + z₂| ≤ |z₁| + |z₂| a triangle's side never exceeds the sum of the other two z₁, z₂ point in the same direction from the origin (one is a non-negative real multiple of the other)
Triangle (n terms) |z₁ + z₂ + … + zₙ| ≤ |z₁| + |z₂| + … + |zₙ| iterated detour through every intermediate point all zₖ point in the same direction from the origin
Reverse triangle ||z₁| − |z₂|| ≤ |z₁ − z₂| difference of radii never exceeds the distance between points z₁, z₂ point in the same direction from the origin

Applications

The modulus and its inequalities serve practical purposes throughout complex analysis, from routine calculations to sophisticated proofs.

Bounding provides estimates when exact computation proves difficult. Suppose we need to show that some complicated expression stays small. Rather than evaluating it precisely, apply the triangle inequality to bound its modulus by a sum of simpler terms. The product rule ∣z1z2∣=∣z1∣∣z2∣|z_1 z_2| = |z_1||z_2| often simplifies factors further. Such estimates form the backbone of convergence proofs and error analysis.

Distance measurement follows immediately from modulus. The quantity ∣z1−z2∣|z_1 - z_2| gives the Euclidean distance between points z1z_1 and z2z_2 in the complex plane. Formally:

Distance Between Complex Numbers
d(z1,z2)=∣z1−z2∣=(a1−a2)2+(b1−b2)2d(z_1, z_2) = |z_1 - z_2| = \sqrt{(a_1 - a_2)^2 + (b_1 - b_2)^2}
Learn more about this formula: Distance Between Complex Numbers →


This interpretation converts geometric problems into algebraic ones and vice versa. Finding the closest point, testing whether two regions overlap, measuring path lengths — all reduce to modulus calculations.

Circles admit elegant description through modulus. The set of all complex numbers zz satisfying ∣z−z0∣=r|z - z_0| = r forms a circle centered at z0z_0 with radius rr. The equation asks for all points lying distance exactly rr from the center z0z_0. Similarly, ∣z−z0∣<r|z - z_0| < r describes the open disk interior, and ∣z−z0∣≤r|z - z_0| \leq r describes the closed disk including its boundary. These sets appear constantly in complex analysis: convergence regions, domains of analyticity, and integration contours often take circular form.

The inequality ∣z−z0∣<r|z - z_0| < r can be rewritten as z0−r<z<z0+rz_0 - r < z < z_0 + r only when restricted to real numbers. In the full complex plane, the inequality carves out a two-dimensional region, not an interval. This shift from intervals to disks marks one of the fundamental differences between real and complex analysis.

Summary: How |z| Behaves Under Every Operation on z

The modulus is most usefully understood not in isolation but through how it behaves when complex numbers are combined or transformed. The table below classifies every standard operation on z by its effect on |z| — some operations leave the modulus exactly unchanged, some transform it in a clean multiplicative way that mirrors the operation itself, and some yield only an inequality rather than an equality.
Operation on z Effect on |z| Behavior class
z̄  (conjugate) |z̄| = |z| INVARIANT — exactly preserved
−z  (additive inverse) |−z| = |z| INVARIANT — exactly preserved
z₁ · z₂  (product) |z₁ · z₂| = |z₁| · |z₂| MULTIPLICATIVE — moduli combine via the same operation
z₁ / z₂  (quotient, z₂ ≠ 0) |z₁ / z₂| = |z₁| / |z₂| MULTIPLICATIVE — moduli combine via the same operation
zⁿ  (integer power) |zⁿ| = |z|ⁿ MULTIPLICATIVE — modulus is raised to the same exponent
z⁻¹  (multiplicative inverse, z ≠ 0) |z⁻¹| = 1 / |z| MULTIPLICATIVE — modulus reciprocates
z₁ + z₂  (sum) |z₁ + z₂| ≤ |z₁| + |z₂| BOUNDED — only an inequality (triangle inequality)
z₁ − z₂  (difference) ||z₁| − |z₂|| ≤ |z₁ − z₂| BOUNDED — only an inequality (reverse triangle inequality)