What Is the Modulus of a Complex Number?
The modulus of a complex number $z = a + bi$, written $|z|$, is the length of the line from the origin to the point $(a, b)$ in the complex plane. It is found with $|z| = \sqrt{a^2 + b^2}$, where $a$ is the real part and $b$ is the imaginary part.
The modulus is always a non-negative real number, because squaring $a$ and $b$ erases any negative signs. It answers a single question: how far is this number from zero? For a real number (where $b = 0$), the modulus $\sqrt{a^2} = |a|$ is exactly the ordinary absolute value — so the modulus is what absolute value becomes once you move from the number line to the plane.
Modulus of a Complex Number Formula
For $z = a + bi$, the modulus is:
$$|z| = \sqrt{a^2 + b^2}$$
There is a second form that is often more useful in higher algebra, written with the conjugate $\bar{z} = a - bi$:
$$|z| = \sqrt{z \cdot \bar{z}}, \qquad |z|^2 = z\bar{z}$$
Both give the same value, because $z\bar{z} = (a + bi)(a - bi) = a^2 + b^2$.
Symbol | Meaning |
|---|---|
$z$ | the complex number $a + bi$ |
$a$ | the real part of $z$ |
$b$ | the imaginary part of $z$ |
$\bar{z}$ | the conjugate, $a - bi$ |
$\lvert z\rvert$ | the modulus — a non-negative real number |
The Modulus on the Argand Plane
The formula is not arbitrary — it is the Pythagorean distance applied to the complex plane. Plot $z = a + bi$ as the point $(a, b)$ on the Argand plane: the real part is the horizontal leg, the imaginary part is the vertical leg, and the modulus is the hypotenuse from the origin.
$$|z|^2 = a^2 + b^2 \quad\Longrightarrow\quad |z| = \sqrt{a^2 + b^2}$$
That is the distance formula in disguise, which is why the modulus is a genuine length rather than just a number you compute.
Examples of the Modulus of a Complex Number
Example 1: Find the modulus of $z = 3 + 4i$
Identify the parts: $a = 3$, $b = 4$.
Apply the formula.
$|z| = \sqrt{3^2 + 4^2}$
$|z| = \sqrt{9 + 16}$
$|z| = \sqrt{25} = 5$
This is the familiar 3-4-5 right triangle drawn on the complex plane.
Example 2 (Wrong path first): Find the modulus of $z = -6 + 8i$
A first instinct is to just add the parts, or to keep the negative sign and write $|z| = -6 + 8 = 2$.
That cannot be right: a modulus is a distance, and distances are never negative or dependent on the sign of the coordinates.
The formula squares each part, which removes the sign entirely.
$|z| = \sqrt{(-6)^2 + 8^2}$
$|z| = \sqrt{36 + 64}$
$|z| = \sqrt{100} = 10$
The modulus of $-6 + 8i$ is 10 — the same as $6 + 8i$, because both points sit the same distance from the origin.
Example 3: Find the modulus of a purely imaginary number $z = 7i$
Here the real part is 0, so $a = 0$ and $b = 7$.
$|z| = \sqrt{0^2 + 7^2}$
$|z| = \sqrt{49} = 7$
For a purely imaginary number, the modulus is just the absolute value of the imaginary part.
Example 4: Find the modulus of $z = 1 - i$
Identify $a = 1$, $b = -1$.
$|z| = \sqrt{1^2 + (-1)^2}$
$|z| = \sqrt{1 + 1} = \sqrt{2}$
$|z| \approx 1.414$
The exact form $\sqrt{2}$ is preferred; the decimal is an approximation.
Example 5: Use the property $|z_1 z_2| = |z_1| \cdot |z_2|$ for $z_1 = 1 + 2i$ and $z_2 = -2 + 3i$
The modulus of a product equals the product of the moduli, so compute each separately.
$|z_1| = \sqrt{1^2 + 2^2} = \sqrt{5}$
$|z_2| = \sqrt{(-2)^2 + 3^2} = \sqrt{13}$
Multiply.
$|z_1 z_2| = \sqrt{5} \cdot \sqrt{13} = \sqrt{65}$
$|z_1 z_2| \approx 8.06$
This is faster than multiplying the complex numbers first and taking the modulus afterward.
Example 6: Find the modulus of $z = a + bi$ using its conjugate
The conjugate of $z = a + bi$ is $\bar{z} = a - bi$.
Multiply them.
$z \cdot \bar{z} = (a + bi)(a - bi) = a^2 + b^2$
So the product $z\bar{z}$ equals $a^2 + b^2$, which means:
$|z| = \sqrt{z \cdot \bar{z}}$
This is the definition most useful in higher algebra: $|z|^2 = z\bar{z}$.
Properties of the Modulus of a Complex Number
These properties hold for any complex numbers $z, z_1, z_2$. Each one follows from the fact that the modulus is a distance.
Non-negativity: $|z| \ge 0$, and $|z| = 0$ only when $z = 0$. A distance is never negative, and only the origin is zero distance from itself.
Conjugate and negation: $|z| = |\bar{z}| = |-z|$. Reflecting a point across the real axis, or through the origin, does not change how far it is from the origin.
Product rule: $|z_1 z_2| = |z_1|\,|z_2|$. The modulus of a product is the product of the moduli.
Quotient rule: $\left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|}$ for $z_2 \neq 0$.
Power rule: $|z^n| = |z|^n$. This falls straight out of the product rule applied $n$ times.
Modulus and conjugate: $z\bar{z} = |z|^2$. This is the identity that turns division of complex numbers into multiplication by the conjugate.
Triangle inequality: $|z_1 + z_2| \le |z_1| + |z_2|$. The direct distance to a point is never longer than a detour through another — the same rule that holds for vectors.
Modulus in Polar Form
Every complex number can be written in polar form using its modulus and its argument (the angle $\theta$ from the positive real axis):
$$z = r(\cos\theta + i\sin\theta), \qquad r = |z| = \sqrt{a^2 + b^2}$$
Here the modulus $r$ is the radius — how far the point sits from the origin — and the argument $\theta$ is the direction. The modulus answers "how big," the argument answers "which way." This pairing is what makes multiplication, powers, and roots of complex numbers manageable: in polar form you multiply the moduli and add the arguments, which is far simpler than expanding brackets.
Why the Modulus Matters — "How Far a Signal Strays From Silence"
The distance interpretation is what makes the modulus useful well beyond the classroom:
In electrical engineering, an alternating current or voltage is written as a complex number, and its modulus gives the amplitude — the size of the signal, independent of its phase.
In signal processing, the modulus of a Fourier coefficient tells you how strongly a given frequency is present in a sound or image.
In control systems and physics, the modulus measures the size of a quantity that carries both a magnitude and a phase, from impedance to quantum amplitudes.
In every case the modulus collapses a two-part object — real and imaginary — into a single honest measure of size. That is the same job absolute value does on the number line, extended to the plane.
Common Mistakes With the Modulus of a Complex Number
Mistake 1: Forgetting to square before adding
Where it slips in: Computing $|3 + 4i|$ as $3 + 4 = 7$.
Don't do this: Adding the parts directly. That is not the length of the hypotenuse.
The correct way: Square each part first: $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$. The squaring is what makes it a distance.
Mistake 2: Keeping the sign of a negative part
Where it slips in: Writing $|-6 + 8i| = \sqrt{-6^2 + 8^2}$ and mishandling the negative.
Don't do this: Letting $-6^2$ become $-36$; a modulus can never come out imaginary or negative. The rusher who plows through the arithmetic without brackets lands here.
The correct way: Bracket the term: $(-6)^2 = 36$. Squaring always gives a non-negative result, so $|-6 + 8i| = \sqrt{36 + 64} = 10$.
Mistake 3: Confusing modulus with argument
Where it slips in: Being asked for $|z|$ and computing the angle instead.
Don't do this: Reporting $\theta = \tan^{-1}(b/a)$ when the question wants the size. The second-guesser who half-remembers "there's an angle involved" reaches for the wrong tool.
The correct way: The modulus is the distance $\sqrt{a^2 + b^2}$; the argument is the direction $\theta$. One is a length, the other an angle — they answer different questions.
Conclusion
The modulus of a complex number $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$, its distance from the origin on the Argand plane.
It is always a non-negative real number, and squaring the parts removes any negative signs.
Its key properties — product, quotient, power, conjugate, and the triangle inequality — all follow from treating it as a distance.
In polar form the modulus is the radius and the argument is the angle, which is what makes multiplication and roots of complex numbers simple.
To take the modulus and complex numbers further with a teacher, work with an algebra tutor, find focused help with algebra, or explore math classes online.
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