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According to algebra as I know it, $-2^2 = 4$, but most calculators expand this to $-2 * 2 = -4$, which yields a different answer. This is because of the order of precedence. In traditional math, I've seen that the unary operator is applied before exponents. This is not the case for calculators.

I have two questions:

1) What caused this difference and why does it persist?
2) Is one method "more accurate" than the other and why?

kettlecrab
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  • Can you be more specific with what you mean by "traditional math"? Perhaps give an example of an instance where a unary operation is applied before exponents? – Cameron Buie Oct 27 '13 at 19:27
  • $-2^2=-4$, not $4$. (unless you meant $(-2)^2=4$) – Sujaan Kunalan Oct 27 '13 at 19:27
  • @Sujaan: If I wrote $(-2)^2$, then the statement would read "according to algebra as I know it, $(-2)^2 = 4$", which would be pointless because calculators evaluate to the same result once you remove the ambiguity. – kettlecrab Oct 27 '13 at 19:55

1 Answers1

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The point at issue, here, is whether $-2^2$ indicates the square of the opposite of $2$ (which is $4$), or whether it indicates the opposite of the square of $2$ (which is $-4$). As a rule, we indicate the former by $(-2)^2,$ and the latter by $-2^2.$ This is because of the order of operations. $-2^2=-4$ because exponentiation occurs before multiplication (by $-1$, in this case), while $(-2)^2=4$ because the multiplication inside the parentheses takes place before the exponentiation.

Why would we do this, though? It isn't just convention. If we consider the general case, with an arbitrary real $x,$ and forget all rules of precedence except parentheses, then there are two possible meanings for $-x^2,$ namely:

  • $(-x)^2$
  • $-(x^2)$

In the first case, this is simply $x^2,$ so not only was there no real need to write the parentheses, there wasn't a need to write the $-$ sign! However, we'd like to have a way to talk about a perfect square's opposite, which the first interpretation quite simply won't allow. That's why we take the second interpretation of $-x^2,$ and use parentheses when we need to square a number's opposite.

Ultimately, the order of operations is done for precisely such reasons: to let us have flexibility enough to say what we mean without needing a whole bunch of parentheses.

Cameron Buie
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    I would say it differently. There are two expressions, $-(x^2)$ and $(-x)^2$. The first is often needed, but the second is equal to $x^2$, which we would usually write instead of $(-x)^2$. Therefore we save ink (and thought) by declaring $-x^2$ to mean $-(x^2)$. – GEdgar Oct 27 '13 at 19:37
  • @GEdgar: That's a good point--and one I was working on explaining when you commented. Thanks! – Cameron Buie Oct 27 '13 at 19:45