Modulus: Absolute Value & Triangle Inequality

Definition of Modulus

For a real number, the absolute value is its distance from 0 on the number line.

|-3| = 3

because

-3

is three steps from

0

;

|3| = 3

for the same reason. In practice the rule is simple: drop the minus sign.

real number	distance from 0	absolute value
−3	3 steps left	3
3	3 steps right	3
0	no steps	0

For 3 + 4i that rule has nothing to work with.

3 + 4i

has no minus sign to drop, and it is not on the number line at all, so "steps from 0 along the line" means nothing. Other quick rules give answers, but none of them is a distance.

rule tried on 3 + 4i	result	what goes wrong
drop the minus sign	3 + 4i	no sign to drop; still not a size
add the parts	7	not a distance from 0
take the larger part	4	ignores the other part

The definition

The modulus of

z = a + bi

, written

|z|

, is the distance from the origin to the point

(a, b)

in the complex plane — measured straight, like a ruler laid from

0

to the point.

The formula is thus:

Modulus

|z| = \sqrt{a^2 + b^2}

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An often-useful equivalent form avoids the square root:

Modulus Squared

|z|^2 = a^2 + b^2

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So

|3 + 4i| = \sqrt{9 + 16} = 5

.

The modulus is a length: one real number, never negative.

The modulus of a complex number

z = a + bi

, written

|z|

, measures the straight-line distance from the origin to the point

(a, b)

in the complex plane. This distance follows directly from the Pythagorean theorem: the point

(a, b)

sits at the end of a right triangle with horizontal leg

a

and vertical leg

b

, so the hypotenuse has length

\sqrt{a^2 + b^2}

.

Computation proceeds by squaring both components, adding, and taking the square root. For

z = 3 + 4i

, we calculate

|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 + 4i

lies exactly 5 units from the origin. For

z = -2 + i

, the modulus is

|z| = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}

, an irrational distance approximately equal to

2.236

.

Pitfall: Squaring the i with the imaginary part

Squaring

4i

instead of

4

in

|3 + 4i|

gives

\sqrt{9 + (4i)^2} = \sqrt{9 - 16} = \sqrt{-7}

, which is not even a real number. The imaginary part is the real number

b = 4

, so

|3 + 4i| = \sqrt{9 + 16} = 5

.

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)

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

\mathbb{C}

lacks the ordering that lets us compare real numbers directly.

|z| = 3.6, the length of the arrow

The modulus is the straight-line distance from the origin to the point, the hypotenuse of the right triangle whose legs are the real and imaginary parts, which is why it is computed by the Pythagorean formula. The conjugate below the axis has exactly the same length. Drag z and watch the readout on the conjugate and modulus tool.

Nothing was lost on the real line: for a real number, the modulus is the old absolute value.

A real number

a

is the point

(a, 0)

, so its modulus is

\sqrt{a^2 + 0^2} = \sqrt{a^2} = |a|

— the same number as before.

real number a	as a + 0i	modulus √(a² + 0²)	absolute value \|a\|
−3	−3 + 0i	√9 = 3	3
5	5 + 0i	√25 = 5	5
0	0 + 0i	√0 = 0	0

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.

What the modulus does not keep is direction.

On the line, a size of

3

leaves only two numbers,

3

and

-3

. In the plane, a modulus of

5

fits

3 + 4i

,

-5

,

5i

,

4 - 3i

and every other point on the circle of radius

5

around the origin. The modulus says how far, never which way; the direction is a separate number, the argument.

Everything below is about how this one length behaves.

The page goes on to its notation, the special cases, its link to the conjugate, its properties and the triangle inequality.

Modulus Notation

Notation

Modulus Notation

Three inherited strips promised the bars would be explained here. This section owns the mark on a complex number, its root-free square, and the locus idiom that writes circles as single equations.

z

,

a + bi

and

\mathbb{C}

come from complex numbers basics,

i

from the imaginary unit, the conjugate bar from its page; the polar letter

r

lives with the trigonometric form.

|z|

the modulus of z; mod z

The same bars as the real absolute value, promoted to two dimensions:

|z| = \sqrt{a^2 + b^2}

, the distance from the origin — Definition of Modulus above. "Modulus" is the traditional name; "absolute value" and the spoken "mod

z

" are equally current.

CasesOn the axes it collapses to the real mark:

|a|

for reals,

|b|

for pure imaginaries — Special Cases below. In polar work it answers to the letter

r

of the trigonometric form.

Do not confuseThe arithmetic "mod". Spoken aloud, "mod

z

" collides with modular arithmetic's remainder operation — unrelated; the bars never mean remainder.

Same glyph elsewhereThe bar family scales by tenant:

\|\mathbf{v}\|

for vector length,

|A|

for determinants,

|S|

for set cardinality — same fence each time.

|z|^2

mod z squared

The root-free spelling:

|z|^2 = a^2 + b^2 = z\bar{z}

— Modulus and the Conjugate below. Analysis reaches for the square precisely to avoid dragging

\sqrt{\phantom{x}}

through a calculation; the mathematical keyboard supplies the radical with its slot already open when you do want it.

CasesDivision writes it in the denominator —

\frac{w}{z} = \frac{w\bar{z}}{|z|^2}

; quantum mechanics reads

|\psi|^2

as a probability density, the same mark on a wavefunction.

Do not confuse

z^2

. For real numbers

|x|^2 = x^2

; for complex ones

|i|^2 = 1

while

i^2 = -1

— dropping the bars changes the mathematics, not just the typography.

|z - c| = R

mod z minus c equals R

Geometry written in modulus:

|z - c|

is the distance between

z

and

c

, so the equation names the circle of radius

R

centered at

c

— one mark replacing two coordinate equations.

CasesInequalities trade the circle for regions:

|z - c| < R

is the open disk,

|z - c| > R

its outside, and

|z - z_1| = |z - z_2|

the perpendicular bisector — the locus dialect of complex analysis.

Do not confuseA directed quantity.

|z_1 - z_2| = |z_2 - z_1|

always — distance forgets direction, though the difference

z_1 - z_2

itself does not.

All complex number symbols →↑ Complex Numbers

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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, \; w

z, w: complex numbers

The usual letters for complex numbers, as

x

and

y

are for reals.

a + bi

a plus b i

Real part

a

, imaginary part

b

; both are real numbers.

\mathbb{C}

C, the complex numbers

The set of all

a + bi

; it contains

\mathbb{R}

.

Imaginary numbers

i

i, the imaginary unit

The number with

i^2 = -1

.

\sqrt{-1}

the square root of minus one

Another name for

i

; root rules for reals fail with it.

bi

b i

A pure imaginary number: real part

0

.

Algebraic form

\operatorname{Re}(z), \; \operatorname{Im}(z)

real part, imaginary part

For

z = a + bi

:

a

and

b

.

\operatorname{Im}(z)

is

b

, not

bi

.

a + (-b)i

a plus negative b, i

Written

a - bi

; the imaginary part is

-b

.

z = x + yi

z equals x plus y i

Same form with plane letters: the point

(x, y)

.

Complex conjugate

\bar{z}

z bar, the conjugate

For

z = a + bi

,

\bar{z} = a - bi

: the mirror image across the real axis.

\overline{z_1 + z_2}

the conjugate of the sum

The bar covers the whole expression; it equals

\bar{z}_1 + \bar{z}_2

.

z^*

z star

The same conjugate, as written in physics and engineering.

Absolute valueThis page

|z|

the modulus of z

Distance from

0

:

\sqrt{a^2 + b^2}

.

|z|^2 = z\bar{z}

mod z squared

Equals

a^2 + b^2

, with no root to take.

|z - c| = R

mod z minus c equals R

The circle of radius

R

around

c

.

Additive inverse

-z

negative z

Both parts change sign:

-a - bi

.

-(a + bi)

minus, quantity a plus b i

The brackets carry the minus to both parts.

z - w

z minus w

Adding the inverse:

z + (-w)

.

Multiplicative inverse

z^{-1}, \; \frac{1}{z}

z inverse; one over z

The number with

z \cdot z^{-1} = 1

; it equals

\frac{\bar{z}}{|z|^2}

.

(3 + 2i)^{-1}

the inverse of three plus two i

Found by multiplying top and bottom by the conjugate:

\frac{3 - 2i}{13}

.

Trigonometric form

r, \; \theta

r and theta, the polar pair

Distance from

0

and angle from the positive real axis.

\arg(z)

the argument of z

Any angle of

z

; there are infinitely many,

2\pi

apart.

\operatorname{Arg}(z)

the principal argument

One chosen angle, usually in

(-\pi, \pi]

; some texts use

[0, 2\pi)

.

\operatorname{cis}\theta

sis theta

Short for

\cos\theta + i\sin\theta

.

Exponential form

e^{i\theta}

e to the i theta

Equals

\operatorname{cis}\theta

(Euler's formula): a point on the unit circle.

re^{i\theta}

r e to the i theta

The whole number

z

: length

r

, angle

\theta

.

e^{i(\theta + 2\pi k)}

theta plus two pi k

The same number for every integer

k

.

De Moivre's theorem

z_k

z sub k, the k-th root

The roots of

z^n = w

, numbered

k = 0, \ldots, n - 1

.

\omega

omega

A primitive root of unity: its powers give all

n

roots of

1

.

R^{1/n}

R to the one over n

The real root of the length; the complex root

w^{1/n}

has

n

values.

Same marks, different job

|z|

The modulus here; for a real number, the absolute value; for a matrix, the determinant.

\bar{z}

The conjugate here; in statistics a bar means a mean, in sets a complement.

z^{-1}

vs

z^*

The reciprocal, against the conjugate: they agree only on the unit circle.

w^{1/n}

n

values for a complex

w

; a single real root for a positive real number.

Complex number symbols table Complex numbers

Complex Number Notation at a Glance·/complex-numbers/basicsLearn Math Class

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 = 0

— the modulus reduces to ordinary absolute value. If

z = a

where

a

is real, then

|z| = \sqrt{a^2 + 0^2} = \sqrt{a^2} = |a|

. The number

5

has modulus

5

; the number

-7

has modulus

7

. 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 = 0

— the modulus equals the absolute value of the coefficient. If

z = bi

, then

|z| = \sqrt{0^2 + b^2} = \sqrt{b^2} = |b|

. The number

4i

has modulus

4

; the number

-3i

has modulus

3

. 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 + 0i

, we have

|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

if and only if

z = 0

" proves essential in analysis, allowing us to test whether an expression equals zero by checking whether its modulus vanishes.

A pure imaginary number and its modulus

z = 3i sits on the imaginary axis, three units above the origin, so |z| = √(0² + 3²) = 3. Its conjugate −3i sits three units below and has the same modulus: on the imaginary axis, as on the real axis, the modulus is the distance from 0 and ignores the sign. Move z onto either axis, or to 0, on the complex conjugate and modulus explorer.

Only z = 0 has modulus 0: every other point, on an axis or not, lies some positive distance from the origin.

Modulus and the Conjugate

A beautiful identity connects the modulus to the complex conjugate: for any complex number

z

, the product of

z

and its conjugate equals the square of the modulus.

Conjugate Times Number

z \cdot \overline{z} = |z|^2 = a^2 + b^2

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The proof follows from direct computation. Let

z = a + bi

, so

\bar{z} = a - bi

. Their product expands as:

Modulus and the Conjugate

z \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

i^2 = -1

, the imaginary square becomes positive. The result

a^2 + b^2

is precisely

|z|^2

by definition of modulus.

Pitfall: Multiplying z by itself

Computing

|z|^2

for

z = 3 + 4i

as

z \cdot z

gives

(3 + 4i)^2 = -7 + 24i

, which is not even real. The conjugate gives

(3 + 4i)(3 - 4i) = 9 + 16 = 25 = |z|^2

.

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 \cdot \bar{z}

and recognize the result as

|z|^2

.

The identity also enables division. To compute

\frac{w}{z}

, multiply numerator and denominator by

\bar{z}

:

Modulus and the Conjugate

\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

, converting the quotient to standard algebraic form. Without the identity

z \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| \geq 0

for every complex number

z

. The modulus is a sum of squares under a square root, and both operations preserve non-negativity. Equality

|z| = 0

occurs precisely when

z = 0

, since only the origin lies zero distance from itself.

The conjugate preserves modulus:

Modulus of Conjugate

|\overline{z}| = |z|

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Conjugation reflects a point across the real axis, and reflection does not change distance from the origin. The points

z = 3 + 4i

and

\bar{z} = 3 - 4i

both lie exactly 5 units from zero.

Products follow a multiplicative rule:

Modulus of a Product

|z_1 \cdot z_2| = |z_1| \cdot |z_2|

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Fact: Sums of two squares multiply

5 = 1^2 + 2^2

and

13 = 2^2 + 3^2

, so

65

is one too:

(1 + 2i)(2 + 3i) = -4 + 7i

gives

65 = 4^2 + 7^2

, and

(1 + 2i)(2 - 3i) = 8 + i

gives

65 = 8^2 + 1^2

. Diophantus noted both ways of writing

65

in the 3rd century, long before complex numbers.

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.

Written out in coordinates, the rule becomes a statement about ordinary real numbers. With

z_1 = a + bi

and

z_2 = c + di

, the product is

z_1 z_2 = (ac - bd) + (ad + bc)i

, and squaring both sides of the product rule gives

(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2

. No

i

is left in this identity. Expanding both sides confirms it, but the product rule explains why it has to hold.

Division obeys the corresponding quotient rule:

Modulus of a Quotient

\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}

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The modulus of a quotient equals the quotient of the moduli (for

z_2 \neq 0

).

Powers extend the pattern:

Modulus of a Power

|z^n| = |z|^n

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Raising a complex number to the

n

-th power raises its modulus to the

n

-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 modulus respects multiplication exactly: when complex numbers multiply, divide or take powers, their lengths do the same.

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

z_1

and

z_2

,

Triangle Inequality

|z_1 + z_2| \leq |z_1| + |z_2|

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

Pitfall: Splitting the modulus of a sum

Writing

|3 + 4i| = |3| + |4i| = 7

treats the inequality as an equation. The direct distance is

|3 + 4i| = \sqrt{9 + 16} = 5

, shorter than the two legs together.

The name comes from geometry. In the complex plane, the numbers

0

,

z_1

, and

z_1 + z_2

form vertices of a triangle (possibly degenerate). The side from

0

to

z_1 + z_2

has length

|z_1 + z_2|

. The path from

0

to

z_1

and then from

z_1

to

z_1 + z_2

has total length

|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

z_1

and

z_2

as arrows from the origin. Their sum

z_1 + z_2

places

z_2

at the tip of

z_1

and measures the resultant. Walking along both vectors sequentially covers distance

|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:

|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.

z₁ + z₂: the diagonal is shorter than the two sides together

The sum is the diagonal of the parallelogram, and the diagonal of a parallelogram is never longer than the two sides that lead to it: |z₁ + z₂| ≤ |z₁| + |z₂|, the triangle inequality as a picture. Equality happens only when the two arrows point the same way. Line the arrows up and watch the inequality become an equation on the complex addition and subtraction tool.

The algebraic proof in the next section verifies what the parallelogram makes obvious.

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 \cdot \bar{z}

.

Begin with the squared modulus of the sum:

Proving the Triangle Inequality

|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:

Proving the Triangle Inequality

= 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

z_1\bar{z_2}

and

\overline{z_1\bar{z_2}}

are conjugates of each other. Their sum equals

2\text{Re}(z_1\bar{z_2})

, twice the real part. Thus:

Proving the Triangle Inequality

|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

\text{Re}(z_1\bar{z_2}) \leq |z_1\bar{z_2}| = |z_1||z_2|

. Substituting:

Proving the Triangle Inequality

|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

\text{Re}(z_1\bar{z_2}) = |z_1\bar{z_2}|

, which occurs when

z_1\bar{z_2}

is a non-negative real number. This happens if and only if

z_1

and

z_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 whole proof rests on one fact: the real part of a complex number never exceeds its modulus.

The Reverse Triangle Inequality

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

Reverse Triangle Inequality

\bigl||z_1| - |z_2|\bigr| \leq |z_1 - z_2|

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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

z_1 = (z_1 - z_2) + z_2

. Applying the triangle inequality:

The Reverse Triangle Inequality

|z_1| = |(z_1 - z_2) + z_2| \leq |z_1 - z_2| + |z_2|

Rearranging gives

|z_1| - |z_2| \leq |z_1 - z_2|

. The same argument with roles reversed yields

|z_2| - |z_1| \leq |z_2 - z_1| = |z_1 - z_2|

. Combining both inequalities:

The Reverse Triangle Inequality

-|z_1 - z_2| \leq |z_1| - |z_2| \leq |z_1 - z_2|

This is equivalent to

\big||z_1| - |z_2|\big| \leq |z_1 - z_2|

.

The geometric interpretation:

|z_1 - z_2|

measures the distance between two points in the plane. The quantity

\big||z_1| - |z_2|\big|

measures how much their distances from the origin differ. Two points can be close together (small

|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.

Fact: Ptolemy's theorem in one line

The triangle inequality applied to the four-point identity below gives

|a - c||b - d| \leq |a - b||c - d| + |a - d||b - c|

. Equality holds exactly when the points lie in this order on a circle or a line; the circle case is Ptolemy's theorem, which he used around 150 AD for his table of chords.

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

|z_1 z_2| = |z_1||z_2|

often simplifies factors further. Such estimates form the backbone of convergence proofs and error analysis.

The same two rules prove facts of plane geometry once points are written as complex numbers. For any four points

a

,

b

,

c

,

d

, multiplying out the brackets shows that

(a - b)(c - d) + (a - d)(b - c) = (a - c)(b - d)

. By the product rule, the modulus of each of these products is a product of two distances between the points.

Distance measurement follows immediately from modulus. The quantity

|z_1 - z_2|

gives the Euclidean distance between points

z_1

and

z_2

in the complex plane. Formally:

Distance Between Complex Numbers

d(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 →

Browse all complex numbers formulas →

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

z

satisfying

|z - z_0| = r

forms a circle centered at

z_0

with radius

r

. The equation asks for all points lying distance exactly

r

from the center

z_0

. Similarly,

|z - z_0| < r

describes the open disk interior, and

|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.

Pitfall: Reading the centre with the wrong sign

Taking

2 - i

as the centre of

|z + 2 - i| = 3

keeps the signs as they stand inside the bars. Written as

|z - (-2 + i)| = 3

, the circle has centre

-2 + i

and radius

3

.

In practice the equation rarely arrives in this tidy shape. A problem may give

|z + 2 - i| = 3

or

|z - 1 + 4i| \leq 2

, with the centre mixed in among the other terms inside the bars. Finding the centre and the radius is then the first step of the problem, and everything drawn afterwards depends on it.

The inequality

|z - z_0| < r

can be rewritten as

z_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.

Distance as the modulus of a difference

For z₁ = −2 + i and z₂ = 3 + 3i the difference z₂ − z₁ has real part 5 and imaginary part 2, the legs of the right triangle drawn between the points. Its modulus is the hypotenuse: |z₁ − z₂| = √(5² + 2²) = √29 ≈ 5.39. Drag either point on the distance and midpoint visualizer.

Fix z₀ and let z vary with |z − z₀| = r: the points traced out form the circle of radius r about z₀.

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)

Common Pitfalls

Four mistakes account for many of the wrong answers in modulus work. Each is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

Squaring the i with the imaginary part

In

|a + bi| = \sqrt{a^2 + b^2}

, the letter

b

stands for the imaginary part, which is a real number. Squaring the whole term

bi

instead brings in

i^2 = -1

, and the plus under the root turns into a minus.

Wrong working: for

z = 3 + 4i

,

|z| = \sqrt{3^2 + (4i)^2} = \sqrt{9 + 16i^2} = \sqrt{9 - 16} = \sqrt{-7}

.

Right working: the real part is

a = 3

and the imaginary part is

b = 4

, so

|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

.

Why it happens: the term

4i

is what appears in the number, and it is squared as it stands. The imaginary part is

4

; the

i

only marks the axis it lies along.

How to check: a modulus is a distance, so it is a real number and never negative. A negative number under the root, or an

i

in the answer, means the

i

was squared.

Multiplying z by itself

The identity

z \cdot \bar{z} = |z|^2

is replaced by

z \cdot z

, as if

|z|^2

and

z^2

were the same thing. For real numbers they are, since

|x|^2 = x^2

, but for complex numbers they are not.

Wrong working: for

z = 3 + 4i

,

|z|^2 = z^2 = (3 + 4i)^2 = 9 + 24i + 16i^2 = -7 + 24i

.

Right working:

z \cdot \bar{z} = (3 + 4i)(3 - 4i) = 9 - 12i + 12i - 16i^2 = 9 + 16 = 25

, so

|z|^2 = 25

and

|z| = 5

.

Why it happens: the square of a real absolute value is the square of the number itself, and that habit carries over. The conjugate in the identity is then easy to drop.

How to check:

|z|^2

is a non-negative real number. If the result still has an imaginary part, or comes out negative, the conjugate is missing. For

z = i

,

z^2 = -1

while

|i|^2 = i \cdot (-i) = 1

.

Splitting the modulus of a sum

The product rule

|z_1 z_2| = |z_1||z_2|

is copied onto sums, and

|z_1 + z_2|

is replaced by

|z_1| + |z_2|

. The triangle inequality only says that the first is at most the second.

Wrong working:

|3 + 4i| = |3| + |4i| = 3 + 4 = 7

.

Right working:

|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

. Here

|3| + |4i| = 7

, and

5 < 7

, as the triangle inequality allows.

Why it happens: the modulus behaves so well with products, quotients and powers that the same kind of rule is expected for sums.

How to check: equality needs the two numbers to point in the same direction from the origin.

3

lies on the real axis and

4i

on the imaginary axis, so the triangle does not collapse and the sum of the moduli is strictly larger.

Reading the centre with the wrong sign

In

|z - z_0| = r

the centre is the number subtracted from

z

. When the equation has a plus sign or several terms inside the bars, the centre is read off with the signs exactly as they appear.

Wrong working:

|z + 2 - i| = 3

is taken as the circle with centre

2 - i

and radius

3

.

Right working:

z + 2 - i = z - (-2 + i)

, so the equation is

|z - (-2 + i)| = 3

, the circle with centre

-2 + i

and radius

3

.

Why it happens: the centre stands inside the bars with its signs reversed, and the minus in

z - z_0

is easy to overlook when the expression is not written in that form.

How to check: substitute the supposed centre. At the true centre

z = -2 + i

the left side is

|0| = 0

. At

z = 2 - i

it is

|4 - 2i| = \sqrt{20}

, not

0

, so

2 - i

is not the centre.

Modulus FAQ

What is the modulus of a real number or a pure imaginary number?

+

For a real number it is the ordinary absolute value, since the vertical component is zero and only the sign is discarded. For a pure imaginary number it is the size of the coefficient, so the modulus of −5i is 5. In both cases the modulus measures distance from the origin along one axis.Read more →

Does taking the conjugate change the modulus?

+

No. Conjugating reflects a point across the horizontal axis, and reflection does not move it any closer to or further from the origin. Since the modulus depends on the squares of both components, flipping the sign of one leaves it untouched. A number and its conjugate always lie on the same circle.Read more →

Is the modulus of a product equal to the product of the moduli?

+

Yes. Multiplying two complex numbers multiplies their distances from the origin, so the modulus of a product is the product of the moduli. Division behaves the same way with a quotient. This is one of the reasons polar form makes multiplication and powers so much easier than rectangular form does.Read more →

When does equality hold in the triangle inequality?

+

Only when the two numbers point in the same direction from the origin, meaning one is a non-negative real multiple of the other. Geometrically the triangle collapses into a straight line and the two paths coincide. In every other case the direct route is genuinely shorter and the inequality is strict.Read more →

How do you find the distance between two complex numbers?

+

Take the modulus of their difference. Subtracting gives the vector running from one point to the other, and the modulus of that vector is its length. This is the same distance formula used for points in the plane, which is why equations like |z − c| = r describe circles. See the distance and midpoint tool.Read more →

Is $|z|^2$ the same as $z^2$ ?

+

No, unless

z

is real.

|z|^2 = z\bar{z} = a^2 + b^2

is always a non-negative real number, while

z^2

is usually complex: for

z = 3 + 4i

,

|z|^2 = 25

but

z^2 = -7 + 24i

. Even

i

shows the gap, with

|i|^2 = 1

and

i^2 = -1

. For real numbers the two agree, which is why the bars are so easy to drop out of habit.Read more →

Why is a product of two sums of two squares again a sum of two squares?

+

Because moduli multiply. If

m = a^2 + b^2 = |a + bi|^2

and

n = c^2 + d^2 = |c + di|^2

, then

mn = |(a + bi)(c + di)|^2 = (ac - bd)^2 + (ad + bc)^2

. With

5 = 1^2 + 2^2

and

13 = 2^2 + 3^2

, the product

(1 + 2i)(2 + 3i) = -4 + 7i

gives

65 = 4^2 + 7^2

, and using

2 - 3i

instead gives

(1 + 2i)(2 - 3i) = 8 + i

, so

65 = 8^2 + 1^2

.Read more →

How do complex numbers prove Ptolemy's theorem?

+

With one identity and the triangle inequality. For any four points,

(a - c)(b - d) = (a - b)(c - d) + (a - d)(b - c)

, and the product rule turns the modulus of each product into a product of two distances. The triangle inequality then gives

|a - c||b - d| \leq |a - b||c - d| + |a - d||b - c|

. Equality holds exactly when the points lie in this order on a circle or a line; the circle case is Ptolemy's theorem.Read more →

Absolute Value

Measuring Distance in the Complex Plane

Key Terms

Definition of Modulus

Modulus Notation

Modulus Notation

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

Common Pitfalls

Modulus FAQ

What is the modulus of a real number or a pure imaginary number?

Does taking the conjugate change the modulus?

Is the modulus of a product equal to the product of the moduli?

When does equality hold in the triangle inequality?

How do you find the distance between two complex numbers?

Is $|z|^2$ the same as $z^2$ ?

Why is a product of two sums of two squares again a sum of two squares?

How do complex numbers prove Ptolemy's theorem?

Absolute Value

Measuring Distance in the Complex Plane

Key Terms

Definition of Modulus

Modulus Notation

Modulus Notation

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

Common Pitfalls

Modulus FAQ

What is the modulus of a real number or a pure imaginary number?

Does taking the conjugate change the modulus?

Is the modulus of a product equal to the product of the moduli?

When does equality hold in the triangle inequality?

How do you find the distance between two complex numbers?

Is ∣z∣2|z|^2∣z∣2 the same as z2z^2z2?

Why is a product of two sums of two squares again a sum of two squares?

How do complex numbers prove Ptolemy's theorem?

Is $|z|^2$ the same as $z^2$ ?