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Modulus of a Complex Number — Formula and Examples

TL;DR

The modulus of a complex number z=a+bi is ∣z∣=\sqrt{a^2 + b^2}, the distance from the origin to the point (a,b) on the complex plane. This article defines the modulus, works through examples, covers its key properties, and shows where students go wrong.

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 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∣=z⋅\bar{z}, ∣z∣^2=z\bar{z}

Both give the same value, because z\bar{z}=(a+bi)(a−bi)=a^2+b^2.

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⟹∣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

Example 2 (Wrong path first): Find the modulus of z=−6+8i

First instinct is to just add the parts. That cannot be right: a modulus is a distance.

∣z∣=\sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = 10

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} = 7

Example 4: Find the modulus of z=1−i

Identify a=1, b=−1.

∣z∣=\sqrt{1^2 + (-1)^2} = \sqrt{2}

Example 5: Use the property ∣z1z2∣=∣z1∣⋅∣z2∣ for z1=1+2i and z2=−2+3i

The modulus of a product equals the product of the moduli.

∣z1∣=\sqrt{1^2 + 2^2} = \sqrt{5}

∣z2∣=\sqrt{(-2)^2 + 3^2} = \sqrt{13}

Multiply.

∣z1z2∣=\sqrt{5} \cdot \sqrt{13} = \sqrt{65}

Example 6: Find the modulus of z=a+bi using its conjugate

The conjugate of z is \bar{z}=a−bi.

Multiply them.

z⋅\bar{z}=(a+bi)(a−bi)=a^2+b^2

Properties of the Modulus of a Complex Number

• Non-negativity: ∣z∣≥0, and ∣z∣=0 only when z=0.
• Conjugate and negation: ∣z∣=∣\bar{z}∣=∣−z∣. The modulus does not change.
• Product rule: ∣z1z2∣=∣z1∣∣z2∣. The modulus of a product is the product of the moduli.
• Quotient rule: ∣z1/z2∣=∣z1∣/∣z2∣ for z2≠0.
• Power rule: ∣z^n∣=∣z∣^n.
• Modulus and conjugate: zzˉ=∣z∣^2.
• Triangle inequality: ∣z1+z2∣≤∣z1∣+∣z2∣.

Modulus in Polar Form

Every complex number can be written in polar form using its modulus and its argument:

z=r(cos⁡θ+isin⁡θ), r=∣z∣=\sqrt{a^2 + b^2}

This pairing simplifies multiplication, powers, and roots of complex numbers.

Common Mistakes With the Modulus of a Complex Number

Mistake 1: Forgetting to square before adding

Correct: Square each part first.

Mistake 2: Keeping the sign of a negative part

Correct: Bracket the terms when squaring.

Mistake 3: Confusing modulus with argument

The modulus is a length; the argument is an angle.

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 properties follow from treating it as a distance.