Is |z|² the same as z² for a complex number?

Only when z is real. In general |z|² = z times its conjugate z-bar, which equals x² + y² for z = x + iy. This is a real, non-negative number, the squared distance from the origin. But z² = z times z equals x² - y² + 2xyi, a point in the plane that can be negative, imaginary, or complex. They coincide exactly when the imaginary part y is zero, that is when z is real. For z = i, |i|² = 1 but i² = -1.

Why can a complex square be negative when a real square cannot?

Because squaring a complex number is a rotation-and-scaling, not a size measurement. A number with argument theta has a square with argument 2theta, so a purely imaginary number (argument 90 degrees) squares to something with argument 180 degrees, which points along the negative real axis. That is why (2i)² = -4. The modulus |z|² ignores direction and only records size, so it is never negative.

How do I compute the sum of squared moduli of a polynomial's roots?

Add |alpha_i|² for each root, where each term is non-negative, so the total is at least zero. Do not use Vieta's sum of squares for this: Vieta gives the sum of the squares alpha_i², which for non-real roots differs from the sum of squared moduli. For the quartic x⁴ + 3x² - 4 with roots ±1 and ±2i, the sum of squared moduli is 1 + 1 + 4 + 4 = 10, while Vieta's sum of squares is 1 + 1 - 4 - 4 = -6. This is exactly the gap JEE Advanced 2024 Paper 1 Q9 is built on.

When does |z|² = z² actually hold?

Exactly when z is a real number. A real number equals its own conjugate, so z times z-bar and z times z are the same product. The moment z has a non-zero imaginary part, z² picks up the i² = -1 term and moves off the real axis, while |z|² stays a non-negative real size. So checking whether the equation holds is the same as checking whether z is real.

JEE Advanced Maths · Complex Numbers

Is |z|² the same as z²?

Only when $z$ is real. In general $|z|^2 = z\bar z$ is a real, non-negative size, the squared distance of $z$ from the origin. But $z^2 = z\cdot z$ is a point in the plane that can be negative, imaginary, or complex. The two coincide only on the real axis. The quickest counterexample is $z = i$ : $|i|^2 = 1$ while $i^2 = -1$ .

This is the most cycle-robust complex-numbers error on the JEE Advanced paper, recurring across 2020, 2023 and 2024. The clean statement to hold is: $|z|^2$ measures size and is never negative; $z^2$ is a point and can land anywhere.

Ready to fix this? The Complex Numbers lesson works through this misconception and the others in Complex Numbers, one altitude at a time.

How to spot it in your own work

You replaced $|z|^2$ with $z^2$ inside a sum over a polynomial's roots.
Your "sum of squares" came out negative and you did not treat that as a warning.
You assumed a squared quantity like $(2i)^2$ must be positive.
You wrote $z^2 = |z|^2$ for a number you never checked was real.

An exam question that triggers it

Here is a question in the style of JEE Advanced 2024 Paper 1 Q9, which tests this misconception head-on:

The polynomial $f(x) = x^4 + 3x^2 - 4$ factors as $(x^2 - 1)(x^2 + 4)$ , so its roots are $\pm 1$ and $\pm 2i$ . Find

\sum_i |\alpha_i|^2,

the sum of the squared moduli of the four roots.

The misconception answer reaches for Vieta and computes the sum of the squares: $\sum \alpha_i^2 = 1 + 1 + (-4) + (-4) = -6$ , then reports $-6$ as if a sum of squared sizes could be negative. The correct move keeps each term a genuine squared modulus:

\sum_i |\alpha_i|^2 = |1|^2 + |-1|^2 + |2i|^2 + |-2i|^2 = 1 + 1 + 4 + 4 = 10.

The real roots contribute the same either way, but each imaginary root has $|2i|^2 = 4$ against $(2i)^2 = -4$ . The genuine 2024 paper used roots $\pm 2\sqrt 2\, i$ and produced $\sum |\alpha_i|^2 = 20$ against Vieta's $\sum \alpha_i^2 = -12$ . The gap is the whole question.

Why students fall for this

The rule "a square is non-negative" is true for every real number a student has ever met, so it feels like a law of arithmetic rather than a fact about the real line. Carried into the complex plane it becomes $|z|^2 = z^2$ , which is half-right: for real $z$ the two really are equal, because a real number is its own conjugate. That partial truth is exactly why the belief survives.

Smith, Zwolak and Manogue (2015), studying student difficulties with complex numbers, document the norm-versus-square confusion directly, and Mutambara and Tsakeni (2022) record the modulus formula mutating under load. Being told $|z|^2 = z\bar z$ rarely dislodges it, because "a square cannot be negative" still feels obviously true. What dislodges it is evaluating both sides on one non-real number and watching them land in different places.

The fix: Size versus point: |z|² = z z̄, z² = z·z

$|z|^2 = z\bar z$ is a real, non-negative size. $z^2 = z \cdot z$ is a point in the plane. They are equal only when $z$ is real. For $z = x + iy$ , the conjugate product is $z\bar z = x^2 + y^2$ , always at least zero, while the plain square is $z^2 = x^2 - y^2 + 2xyi$ . These match exactly when $y = 0$ .

The practical guard on the exam: whenever you meet a sum of squared moduli over roots, add $|\alpha_i|^2$ directly and expect a non-negative total. If you instead find yourself computing Vieta's $\sum \alpha_i^2$ and it comes out negative, that negative number is the alarm that you have swapped $|\alpha|^2$ for $\alpha^2$ on the non-real roots.

Worked example

Evaluate both machines on a single number, then on a conjugate pair, so the split is unmissable. Take $z = 3 + 4i$ .

The size. $|z|^2 = 3^2 + 4^2 = 9 + 16 = 25$ , a positive real number.
The point. $z^2 = (3+4i)^2 = 9 + 24i + 16i^2 = 9 + 24i - 16 = -7 + 24i$ , a point with a negative real part, nowhere near $25$ .
Same number, two answers. One input gives $25$ as a size and $-7 + 24i$ as a square. The conjugate check confirms it: $z\bar z = (3+4i)(3-4i) = 9 + 16 = 25 = |z|^2$ .
A conjugate pair. For the roots $\pm 7i$ of $z^2 + 49 = 0$ , the sum of squared moduli is $49 + 49 = 98$ , while squaring the roots gives $(7i)^2 + (-7i)^2 = -49 - 49 = -98$ . A positive size total, a negative square total.
Conclude.
$|z|^2 = z\bar z \ge 0; \qquad z^2 = z\cdot z \text{ can be anything, equal to } |z|^2 \text{ only when } z \text{ is real.}$

The same distinction settles JEE Advanced 2024 Paper 1 Q9: the sum of squared moduli of a polynomial's roots is a non-negative total, computed root by root as $|\alpha_i|^2$ , never as Vieta's $\sum \alpha_i^2$ , which turns negative the moment a root leaves the real axis.

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Common questions

Can $z^2$ really be negative?: Yes, whenever $z$ is purely imaginary. For $z = bi$ , $z^2 = -b^2$ , a negative real number, while $|z|^2 = b^2$ stays positive. Squaring doubles the argument, so a number pointing up the imaginary axis squares to one pointing along the negative real axis.
How is $|z|^2$ related to the conjugate?: $|z|^2 = z\bar z$ exactly. Writing $z = x + iy$ gives $z\bar z = (x+iy)(x-iy) = x^2 + y^2$ , which is the squared distance from the origin and is always non-negative. This identity is the reason a squared modulus can never be negative.
Why does the "sum of squares" go negative in exam questions?: Because a student applies Vieta's $\sum \alpha_i^2$ and treats it as a sum of sizes. For non-real roots $\alpha_i^2$ is negative, so the total can drop below zero. The genuine sum of squared moduli $\sum |\alpha_i|^2$ is built from non-negative terms and is always at least zero. A negative result is a signal you have mixed the two up.
Is $|z^2| = |z|^2$ wrong too?: No, that one is genuinely true: $|z^2| = |z|\,|z| = |z|^2$ , because the modulus is multiplicative. The mistake is dropping the outer modulus bars and writing $z^2 = |z|^2$ . Keep the bars: $|z^2|$ (a size) equals $|z|^2$ (a size), but $z^2$ (a point) does not.

Related misconceptions

Realness is an equation, not a lookAn expression w is real exactly when w = w-bar; that is the condition you impose and solve, not something you read off by inspection.
Modulus and arg conditions are lociAn equation in z carves out a whole curve, not a few points; the Apollonius circle from |z - a| = k|z - b| is not centred between the two points.

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