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How can I convincingly and mathematically explain the reason behind difference between $(-1)^2$ and $-1^2$?

I used to add "negation" to the order of operations, in the same row as multiplication and division, so that in $-1^2$ we should first carry on the exponentiation and then the negation. Hence the answer is $-1$. But recently I've asked myself if there's a better explanation, more conceivable and convincing.

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    $\begingroup$ "and mathematically explain" You don't. This is a notation convention that arose as a compromise between readability and ambiguity, and it has been extensively discussed on the internet since the 1990s in sci.math and other places. Also many times in Stack Exchange, such as What is the accepted syntax for a negative number with an exponent? AND Is a negative number squared negative? $\endgroup$ – Dave L Renfro Feb 16 '18 at 20:41
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    $\begingroup$ Thanks for the links. By "mathematical explanation" I mean an explanation which considers and uses a general rule or principle or convention. It's in contrast to a "cookbook-type explanation" - where you just say "this thing should be handled this way" without referring to any generality. $\endgroup$ – Behzad Feb 16 '18 at 21:31
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    $\begingroup$ "I mean an explanation which considers and uses a general rule or principle or convention" I don't think there is one. When I brought up readability and ambiguity, I was thinking of things like plugging in $x = 5$ to the function $f(x) = -x^2$ is supposed to give $-25.$ With this convention, you only have to replace "$x$" with "$5$", rather than replace "$x$" with "$(5)$" (and even this requires a convention to have it be $-25,$ now that I think about it, because we still need to know to perform the exponent first in $-(5)^2).$ $\endgroup$ – Dave L Renfro Feb 16 '18 at 22:30
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    $\begingroup$ This is exactly about a notational_convention, rather than any innate mathematical facts. That is, it is only about a widely-agreed-upon choice of how to interpret otherwise-ambiguous notation. There is not necessarily any great virtue in this particular choice of disambiguation. It's more like the choice of which side of the road to drive on, left, or right? It really doesn't matter... as long as we all agree... and make clear to beginners that there is a definite convention. $\endgroup$ – paul garrett Feb 17 '18 at 0:57
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    $\begingroup$ This is about "default" precedence, so you use additional parentheses, like this: (−1)^2 not the same as -(1^2), and point out that the latter expression, (where exponentiation is performed first), is the "default" precedence. $\endgroup$ – Charles Bretana Feb 17 '18 at 13:20
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You might explain that BEDMAS is not the whole story when it comes to the order of operations. There is an operation called negation. It reverses the sign on numerical quantifies. It gives the additive inverse of any number, i.e. for all $x \in R$, we have $x + (-x) = 0$.

Unfortunately, most textbooks use the same symbol for both subtraction and negation. Some calculators use different symbols ( - for negation, $-$ for subtraction.) It should be fairly easy with a bit of practice to tell from the context whether a '$-$' symbol is being used as a subtraction operator or a negation operator.

The negation operator usually has a precedence that is less than that of exponentiation but greater than than of multiplication. So, we now have BENDMAS.

In the expression $-1^2$, we have two operations: negation and exponentiation. BENDMAS tells us to do the exponentiation first, then the negation to get a result of $-1$.

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    $\begingroup$ I understand you're idea and as a matter of fact, I explained that I've been using it during years. But there's a problem here: Why do we HAVE TO interpret $-1^2$ as "negation of $1^2$? What prevents us form interpreting it as " $-1$ to the power of $2$? $\endgroup$ – Behzad Feb 16 '18 at 20:22
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    $\begingroup$ Convention about operator precedence. $\endgroup$ – Jasper Feb 16 '18 at 21:13
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    $\begingroup$ "Unfortunately, most textbooks use the same symbol for both subtraction and negation." I'd say it's not really unfortunate. As far as I know, negation of $x$ is always equivalent to $0-x$, so there is a certain intuition behind using the same symbol. It also suggests some sort of relationship between subtraction and negative numbers; I'd be awfully surprised if the notion of negative numbers didn't arise out of thinking about subtracting one positive number from another smaller one. Last, there is value in needing to remember a reduced number of distinct symbols. $\endgroup$ – jpmc26 Feb 16 '18 at 22:43
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    $\begingroup$ @Behzad Nothing logically prevents it. It's just not the convention mathematicians have adopted. You'd do better to ask what motivates it. $\endgroup$ – jpmc26 Feb 16 '18 at 22:44
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    $\begingroup$ @Behzad: "What prevents us form interpreting it as " −1 to the power of 2?" The same thing that stops us from interpreting "2-3*5" as "(2-3)*5": "This is what everyone else agreed to do". It's purely a social agreement. There's nothing logical or mandatory about any part of notation. It's merely a notation. $\endgroup$ – Mooing Duck Feb 17 '18 at 0:04
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If your students already understand that exponents precede multiplication, and that multiplying by $-1$ is the "negation" operator, then you should be able to convince them that $$-5^2 = -1*5^2 = -1*25 = -25$$ is a reasonable way to interpret this expression.

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  • $\begingroup$ Yes, but why does she/he have to interpret $-5^2$ as $negation of $5^2$? $\endgroup$ – Behzad Feb 16 '18 at 21:32
  • $\begingroup$ It is simply a widely accepted convention (the order of operations) that exponentiation has a higher precedence than negation. If you want to square $-5$, you should write $(-5)^2$ to obtain $+25$. $\endgroup$ – Dan Christensen Feb 18 '18 at 6:04
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I tutored a student who had a hard time understanding this, and the way that helped him to understand it was this:

Any time there is a negative sign on a number, we can read it as $(-1)$ . So $-5 \equiv (-1)5$ and $-3^2 \equiv (-1)3^2 = (-1)9 \equiv -9$.

In the case you mention then, $-1^2 \equiv (-1)1^2 = (-1)1 \equiv -1$

The key to helping him understand it was always wrapping the negative 1 that the minus sign implies in parenthesis, from which he could use BEMDAS on everything.

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If you're main concern is students' writing, you could take the route of avoiding the issue altogether and tell students that $-1^2$, regardless of what it should be equal to according to BEDMAS, is just bad notation and that it can be avoided. The purpose of writing mathematics, of writing anything really, is to communicate effectively to a reader. The notation $-1^2$ is ambiguous since it could mean $(-1)^2$ or $-(1^2)$, and this ambiguity can easily be avoided by just writing one of $(-1)^2$ or $-(1^2)$ instead.

But unfortunately this $-1^2$ notation does appear in textbooks (I'm almost certain I've seen it before). For that, I would just tell my students that "the textbooks authors could have been clearer, but they are just using the general convention that ..." Students might even like that there is something we have to take as convention. It's a break from the rigidity they see in the rest of mathematics.

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I like the presentation on the NCTM Math Forum/Dr. Math website:

We don't usually list unary operators in PEMDAS because they're thought of as being implied by the rules for binary operations. You can think of the minus sign as either subtraction

 -3^2 = 0 - 3^2 = 0 - 9 = -9

or multiplication

 -3^2 = -1 * 3^2 = -1 * 9 = -9

and in either case it has lower precedence than exponentiation, so it gives the same result. I can't think of any case where it would matter which of these two latter interpretations we give, that is, any case where giving "-" an additive precedence or a multiplicative precedence will make a difference in the result; commutativity and distributivity seem to take care of that.

I would observe that the historic development of negative numbers was closely related to subtraction (i.e., finding the closure of various number sets under subtraction), so arguably the most straightforward interpretation of negation is $-x = 0 - x$. We want $-1^2$ to have the same value and order (up to the final subtraction) of $0 - 1^2$, which implies the convention.

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