Math CentralQuandaries & Queries


Question from Christine, a parent:

On a math test, it said "What is -3 squared?" It did NOT say "What's is (-3)squared"

The teacher's explanation is that if there are no brackets or parenthesis, you ALWAYS square the number first then do the negative, so the answer should be -9, but I can't find anywhere that confirms this.

Help please.

When we evaluate -x² we square first. This convention is probably more a matter of utility than anything else: (-x)² is the same thing as x² so we don't need to write the first form often. Also, it's consistent with the way we use the minus sign as a two-argument operation in (eg) y² - x² . Unary operations are always done before two-argument operations unless parentheses say to do it differently. (A big square-root sign with a bar acts as its own parenthesis, of course.)

Conventions about the order of unary operations are complicated and mixed up with the different ways they are written. It is clear what sin(x²) means, and we understand sin x² to mean the same thing. The formual (sin x)² is clear, though in speech we have to say "sine of x all squared" or "sine of x [pause] squared". We simetimes write sin² x ("sine squared of x") for that, too.

HOWEVER, I would argue that if it was written exactly as you put it - with "squared" written out in letters - that the written-out operation should be performed after all the "formula stuff" was done. So for instance:

"The square of x+y is x² + 2xy + y² "

Good hunting!

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