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Absolute Square — Definition, Formula & Examples

The absolute square of a complex number is the square of its absolute value (modulus). For a complex number z=a+biz = a + bi, the absolute square equals a2+b2a^2 + b^2, which is always a non-negative real number.

For z∈Cz \in \mathbb{C}, the absolute square is defined as ∣z∣2=z⋅zˉ|z|^2 = z \cdot \bar{z}, where zˉ\bar{z} denotes the complex conjugate of zz. If z=a+biz = a + bi with a,b∈Ra, b \in \mathbb{R}, then ∣z∣2=a2+b2|z|^2 = a^2 + b^2.

Key Formula

∣z∣2=z⋅zˉ=a2+b2|z|^2 = z \cdot \bar{z} = a^2 + b^2
Where:
  • zz = A complex number $a + bi$
  • zˉ\bar{z} = The complex conjugate $a - bi$
  • aa = The real part of $z$
  • bb = The imaginary part of $z$

How It Works

To compute the absolute square of a complex number, multiply it by its complex conjugate. If z=a+biz = a + bi, its conjugate is zˉ=a−bi\bar{z} = a - bi, and the product z⋅zˉ=(a+bi)(a−bi)=a2+b2z \cdot \bar{z} = (a + bi)(a - bi) = a^2 + b^2. This result is always real and non-negative, which makes the absolute square useful for avoiding square roots in calculations. Note that for a purely real number xx, the absolute square reduces to ∣x∣2=x2|x|^2 = x^2.

Worked Example

Problem: Find the absolute square of z=3+4iz = 3 + 4i.
Step 1: Identify the complex conjugate of zz.
zˉ=3−4i\bar{z} = 3 - 4i
Step 2: Multiply zz by its conjugate.
z⋅zˉ=(3+4i)(3−4i)=9−12i+12i−16i2=9+16=25z \cdot \bar{z} = (3 + 4i)(3 - 4i) = 9 - 12i + 12i - 16i^2 = 9 + 16 = 25
Step 3: Verify using the component formula.
∣z∣2=32+42=9+16=25|z|^2 = 3^2 + 4^2 = 9 + 16 = 25
Answer: ∣3+4i∣2=25|3 + 4i|^2 = 25

Why It Matters

The absolute square appears throughout quantum mechanics, where ∣ψ∣2|\psi|^2 gives probability densities, and in signal processing, where it represents power spectral density. In complex analysis courses, it provides a convenient way to simplify expressions involving moduli without introducing square roots.

Common Mistakes

Mistake: Computing ∣z∣2|z|^2 as z2z^2 instead of z⋅zˉz \cdot \bar{z}.
Correction: Squaring a complex number z2=(a+bi)2=a2−b2+2abiz^2 = (a+bi)^2 = a^2 - b^2 + 2abi gives another complex number, not a real value. The absolute square requires multiplication by the conjugate: ∣z∣2=z⋅zˉ=a2+b2|z|^2 = z \cdot \bar{z} = a^2 + b^2.