Friday, January 31, 2014

14: Negatives, Squares and the Mistaken Calculator

From Kenneth Tilton via email (
If -3 is a non-zero number and a non-zero number squared is positive,
why does $-3^2 = -9$?

I went to and typed it in:

I know I've got my TI around here:

That can't be right.  Wolfram is always the definitive source, right?^2

What the heck is going on?


  1. The obvious answer is that it's not a negative squared, it's multiplication by -1 and exponents happen first. If we want a negative 3 squared, we'd need parentheses (-3)^2. I've never had kids get that far on their own, though.

  2. Although the I concur with Curmudgeon's analysis, I would further argue that -3^2 means take the opposite of 3^2. 3^2 is 9, the opposite of which is -9.

  3. When I taught Algebra II and we went over quadratics, I'd get students who would punch in these calculator errors. They would see the answer on the calculator, realize it's wrong (after all, they know themselves that squaring -3 gives you +9), but the would still use the wrong value because that's what the calculator gave them.

  4. I always say that it's "Operator Error"...the calculator is doing exactly what you asked it to do! It's giving you the opposite of 3^2... I bring this up all the time in class, and it always leads several kids to nod and say things like "Oh, that's why?!" GREAT discussion piece that should be brought up in every class because there are SO many kids that do this error!