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Taking the Square of the Norm

The Prompt

“Compute the square of the norm of the complex number $z$, where $$ z = 2 \,+ \, 3i\, \text{.”} $$


Oftentimes, when students are asked what the “square of the norm” of a number is, they can't answer. However, this is often true because they have never heard this operation, particularly for complex numbers, referred to as “taking the square of the norm.” This activity refreshes students' memories for taking the square of the norm of complex numbers, as well as how to find the complex conjugate of a complex number.

This SWBQ can be easily included in any lecture that features complex numbers, and is particularly valuable when used in a 'just-in-time approach' to the complex-valued linear algebra present in entry-level quantum mechanics courses.

Wrap Up

Some students simply “square” the complex number and consequently respond $|z|^2 = -5 +6i$. It may be helpful to remind students that the “square of the norm” as an operation returns the magnitude of the complex number, and therefore must be a real quantity. Though students know that $i^2 = -1$, many forget to use this information while taking the square of the norm. Mentioning to students that this error is pervasive enough to cause significant loss of points on exams may encourage them to commit the lesson to memory.
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