Modulus of Complex Number

A real number has a size: its absolute value. The absolute value of -7 is 7. The absolute value of 3 is 3. It measures how far the number is from zero on the number line.

A complex number lives on a plane, not a line. The number 3 + 4i has two components — a real part 3 and an imaginary part 4 — and its position on the complex plane is the point (3, 4). The distance from this point to the origin (0, 0) is the modulus of 3 + 4i.

That distance is \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.

The modulus is the complex-number version of the absolute value. It answers the question: how far is this number from zero?

The definition

The formula is the Pythagorean theorem applied to the right triangle with legs x and y on the complex plane. The real part x is the horizontal distance, the imaginary part y is the vertical distance, and |z| is the hypotenuse.

A few immediate examples:

That last one confirms: for a real number a, the modulus |a + 0i| = \sqrt{a^2} = |a|. The complex modulus reduces to the ordinary absolute value when the imaginary part is zero.

Connection to the conjugate

You saw in Algebra of Complex Numbers that z \cdot \bar{z} = x^2 + y^2 for z = x + yi. That product is the square of the modulus:

This identity is one of the most useful in all of complex number theory. It connects three ideas — the modulus, the conjugate, and multiplication — in a single equation. You have already used it to make division work (the denominator in Division of Complex Numbers is |z_2|^2 = z_2 \bar{z_2}), and you used it to find square roots (the modulus equation in the method of comparison is x^2 + y^2 = |z|).

Properties of the modulus

The modulus satisfies a set of clean algebraic properties. Each one has both an algebraic proof (compute with x and y) and a geometric interpretation (think about distances and directions on the complex plane).

1. Non-negativity. |z| \geq 0, and |z| = 0 if and only if z = 0.

This is immediate: x^2 + y^2 \geq 0, and it equals zero only when both x = 0 and y = 0.

2. Conjugate symmetry. |\bar{z}| = |z|.

Since \bar{z} = x - yi, |\bar{z}| = \sqrt{x^2 + (-y)^2} = \sqrt{x^2 + y^2} = |z|. Geometrically, reflection across the real axis does not change the distance from the origin.

3. Multiplicativity. |z_1 z_2| = |z_1| \cdot |z_2|.

This is the key algebraic property. The modulus of a product is the product of the moduli. Here is the proof. Let z_1 = a + bi, z_2 = c + di.

Taking square roots of both sides: |z_1 z_2| = |z_1||z_2|.

Geometrically: multiplying by z_2 scales distances by a factor of |z_2|. So the distance of z_1 z_2 from the origin is |z_1| scaled by |z_2|.

4. Quotient rule. \left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|} (when z_2 \neq 0).

This follows from multiplicativity. Write z_1 = (z_1/z_2) \cdot z_2, so |z_1| = |z_1/z_2| \cdot |z_2|, giving |z_1/z_2| = |z_1|/|z_2|.

5. Power rule. |z^n| = |z|^n for any positive integer n.

Apply multiplicativity repeatedly: |z^n| = |z \cdot z \cdot \dots \cdot z| = |z|^n.

6. Modulus of the real and imaginary parts.

Since |z|^2 = x^2 + y^2 \geq x^2, taking square roots gives |z| \geq |x|. Similarly |z| \geq |y|. The modulus is at least as large as either component — the hypotenuse is never shorter than a leg.

The triangle inequality

The most important inequality in complex analysis is:

The name comes from geometry. If you draw arrows from the origin to z_1, from z_1 to z_1 + z_2, and from the origin to z_1 + z_2, the three arrows form a triangle. The inequality says: the length of the third side (the direct path from origin to z_1 + z_2) is at most the sum of the other two sides (|z_1| + |z_2|, the detour through z_1).

Proof of the triangle inequality

The proof is algebraic and uses the identity |z|^2 = z\bar{z}.

Start with |z_1 + z_2|^2. Expand it:

The middle two terms are z_1\bar{z_2} and its conjugate \overline{z_1\bar{z_2}}. A number plus its conjugate is twice its real part:

Now use the fact that the real part of any complex number is at most its modulus: \operatorname{Re}(w) \leq |w|. Therefore:

Putting it all together:

Taking square roots (both sides are non-negative):

Equality holds when \operatorname{Re}(z_1\bar{z_2}) = |z_1\bar{z_2}|, which means z_1\bar{z_2} is a non-negative real number — equivalently, z_1 is a non-negative real multiple of z_2 (they point in the same direction).

The reverse triangle inequality

There is a companion result that bounds |z_1 + z_2| from below:

The proof: apply the triangle inequality to (z_1 + z_2) + (-z_2) = z_1 to get |z_1| \leq |z_1 + z_2| + |z_2|, so |z_1 + z_2| \geq |z_1| - |z_2|. Swapping the roles gives |z_1 + z_2| \geq |z_2| - |z_1|. Combining: |z_1 + z_2| \geq \bigl||z_1| - |z_2|\bigr|.

Together, the two inequalities give a two-sided bound:

The modulus of a sum is squeezed between the difference and the sum of the individual moduli.

The modulus as a circle

All complex numbers with the same modulus lie on a circle centred at the origin. The equation |z| = r describes a circle of radius r. In Cartesian terms, x^2 + y^2 = r^2.

More generally, |z - z_0| = r describes a circle of radius r centred at z_0. This is because |z - z_0| is the distance from z to z_0, and the set of all points at a fixed distance from a centre is a circle.

Worked examples

Common confusions

• "|z|^2 = z^2." No. |z|^2 = z\bar{z} = x^2 + y^2, a real number. But z^2 = (x+yi)^2 = x^2 - y^2 + 2xyi, which is generally complex. The modulus squared uses the conjugate; the ordinary square does not.

• "|z_1 + z_2| = |z_1| + |z_2|." The triangle inequality is an inequality, not an equality. Equality holds only when z_1 and z_2 point in the same direction. In general, |z_1 + z_2| < |z_1| + |z_2| (strict inequality).

• "The modulus can be negative." Never. |z| = \sqrt{x^2 + y^2} \geq 0 always. The modulus is a distance, and distances are non-negative.

• "|z_1 z_2| = |z_1| + |z_2|." The modulus is multiplicative, not additive: |z_1 z_2| = |z_1| \cdot |z_2|. Distances multiply under complex multiplication.

• "If |z| = 1, then z = 1." The modulus |z| = 1 means z lies on the unit circle. There are infinitely many such numbers: 1, -1, i, -i, \frac{1+i}{\sqrt{2}}, and every \cos\theta + i\sin\theta for any angle \theta.

• "|z_1 - z_2| measures the difference of the moduli." No. |z_1 - z_2| is the distance between z_1 and z_2 on the complex plane. The difference of moduli is ||z_1| - |z_2||, which is a different and generally smaller quantity (by the reverse triangle inequality).

Going deeper

The modulus as a norm

In linear algebra, a norm is a function that assigns a non-negative "size" to each vector in a vector space, satisfying three axioms:

1. \|v\| \geq 0, with equality only for v = 0 (positive definiteness).
2. \|\alpha v\| = |\alpha| \cdot \|v\| for any scalar \alpha (absolute homogeneity).
3. \|u + v\| \leq \|u\| + \|v\| (triangle inequality).

The complex modulus satisfies all three, with \alpha being a real scalar and v being a complex number viewed as a vector in \mathbb{R}^2. So the modulus is a norm on the complex plane — it is the Euclidean norm, the same one you use to measure distances in ordinary two-dimensional space.

The modulus and Brahmagupta's identity

The multiplicativity |z_1 z_2| = |z_1||z_2| has a surprising consequence when written in terms of real and imaginary parts. If z_1 = a + bi and z_2 = c + di, then:

This is Brahmagupta's identity: the product of two sums of two squares is itself a sum of two squares. Brahmagupta stated this result in the 7th century, and it follows naturally from the multiplicativity of the complex modulus — though complex numbers would not be formalised for another millennium. The identity is a beautiful example of a fact about real numbers that is most easily proved by passing through the complex numbers.

The unit circle and multiplication

The complex numbers with |z| = 1 form a group under multiplication. If |z_1| = 1 and |z_2| = 1, then |z_1 z_2| = |z_1||z_2| = 1, so the product also has modulus 1. Every such number can be written as \cos\theta + i\sin\theta for some angle \theta, and multiplication corresponds to adding the angles. This is the starting point for the polar form of complex numbers and for understanding complex multiplication as rotation.

Where this leads next

The modulus gives you the "size" of a complex number. The next steps are about direction and the full geometric picture.

• Argument of Complex Number — the angle that z makes with the positive real axis, completing the polar description of z.
• Polar Form of Complex Numbers — writing z = r(\cos\theta + i\sin\theta), where r = |z| and \theta is the argument.
• Algebra of Complex Numbers — the arithmetic operations whose modulus properties you studied on this page.
• Complex Numbers — Introduction — the definition of i and the complex number system, the starting point for everything here.
• Operations and Properties — the field axioms that \mathbb{C} satisfies, underpinning the algebraic proofs in this article.