Complex Norms: Why z² Isn’t What You Want
A small but persistent confusion in complex arithmetic is how to correctly compute the norm-square (or absolute square) of a complex number z, often written as . It's tempting to treat this like squaring a real number and expect z² to do the job. It doesn't. The correct calculation requires the complex conjugate:
Where z* denotes the complex conjugate of z. To see why this matters, it helps to step back and think about what a norm actually is.
Norm as a Generalization of Length
For a real number r, its norm is simply its absolute value — the distance from zero on the number line. Complex numbers extend this idea to a two-dimensional plane. If you plot a complex number in Cartesian coordinates (with its real part on one axis and its imaginary part on the other), its norm is the length of the line from the origin to that point. In polar form (z = r e^(iθ)), this length is exactly r, which is why the modulus is such a natural fit for a norm.
A Formal Definition, Simplified
The general definition of a norm applies to vector spaces; complex numbers form a one-dimensional vector space over the reals, so we can adapt the definition directly. A norm on the complex plane is a real-valued function satisfying:
- Triangle inequality: for all z, w, the norm of
z + wis at most the sum of their norms. - Absolute homogeneity: scaling z by a real scalar s scales its norm by |s|.
- Positive definiteness: a zero norm implies the number itself is zero.
- Non-negativity: norms are never negative.
For complex numbers, taking the absolute value of z satisfies all four properties intuitively. Adding complex numbers works like adding vectors, so the triangle inequality holds; scaling in polar coordinates simply multiplies the radius; and a positive-definite, non-negative measure of distance is exactly what we want from a "length."
The Problem With z²
If we naively square a general complex number z = x + iy, we get a result with both real and imaginary parts:
That's not a real-valued quantity, so it cannot serve as a norm-square. Even in polar form, multiplying z by itself doubles the angle — you get the squared magnitude r² but at an angle of 2θ, not on the real axis where a "length" should live. Geometrically, it's a vector pointing away from the real line, which is why z² can't represent a squared distance.
The Conjugate Fixes the Angle
The entire issue boils down to that leftover angle. When you multiply two complex numbers, their magnitudes multiply but their angles add. To eliminate the angle and land on the real axis, you need a factor whose angle is the negative of z's angle. That's precisely what the complex conjugate z* is: for z = re^(iθ), the conjugate is z* = re^(−iθ).
Multiplying the two together yields:
That result is real and exactly equal to r², the squared "length." In Cartesian coordinates, the same logic works out: multiplying x + iy by x − iy cancels the imaginary cross-terms and leaves x² + y². So:
makes sense not just algebraically but geometrically.
Working With Complex Expressions
The same principle extends to expressions. For a sum like z + w, computing the norm-square means multiplying the sum by its own conjugate:
Expanding that product, the imaginary terms cancel and you're left with a real sum that can be rearranged into a cleaner form.
The lesson holds across all such calculations: when a norm-square is needed, always multiply by the conjugate, never by the number itself. z² might look right at first glance, but only zz* delivers a real, positive quantity that behaves as a true squared distance.
