Basic Math Question

The problem: The expression –3[sup]2[/sup] looks like it could mean either (–3)[sup]2[/sup] (that is, –3 * –3) or –(3[sup]2[/sup]) (that is, – 3*3). Since these give different results, we need a rule specifying which of these it does mean.

The rule is that it means the latter: that the exponent comes first, before the negation. That’s the rule, and all algebra books will tell you so.

As to why that’s the rule, I think it makes more sense when you look at polynomials. In an expression like –x[sup]2[/sup] + 10 (which is equivalent to 10 – x[sup]2[/sup]), the –x[sup]2[/sup] means –xx. Like any term of any polynomial, it consists of a coefficient times a whole number of factors of x multiplied together. In this case, the coefficient is –1, and this is multiplied by two factors of x. If you “plug in” 3 for x, that part of the expression becomes –33. So in order for everything to work consistently, –3[sup]2[/sup] has to be interpreted as –3*3.

Again, you’re confusing the numbers themselves with the notation used to write them. “-3” isn’t a number in itself, it’s a notational expression – but there is a single unique number that is referred to by the expression “-3”. In much the same way, the word “earth” isn’t itself the planet on which we live, but the word refers to that planet. So you can’t treat the symbols as though their appearance must automatically translate into facts about the things they refer to. That would be like saying “the word ‘earth’ is about an inch long on my screen, therefore the planet earth must only be an inch long!”

In other words, it does not follow that if -3 = (-3), then -3^2 = (-3)^2. You could also claim that since 2 = 1 + 1, then 22 must represent the same number as 21 + 1! But clearly it doesn’t: 22 is 4, but 21 + 1 is 3.

In order for the expression “1 + 1” to work in the same way that the expression “2” does, in all scenarios, it is necessary to enclose it in parentheses. “(1 + 1)” is a different expression than “1 + 1”, but they both represent the same value. The difference in the expressions becomes apparent when you try to substitute one for the other inside another, larger expression. 2*(1 + 1) does not equal 2*1 + 1, even though (1 + 1) = 1 + 1. This is not a flaw in the notation; it is in fact working precisely as intended.

Would you really want a system in which you’re forced to say that either (1 + 1) and 1 + 1 represent different numbers, or that 2*(1 + 1) represents the same number as 2*1 + 1? It wouldn’t make any sense at all.

x = -2

Wow, that is brilliant!!

Strangely enough, here is the same question, with the same exact numbers, but posed over 10 years ago:

http://mathforum.org/library/drmath/view/53194.html

Wow. Do I feel silly. Really, total brain fart there.

I like the extra notes about Microsoft Excel at the bottom of that page:

So there you have it, Microsoft Excel is the root cause of math notation illiteracy. :smiley:

Another way to look at it… -3^2 = 0-3^2. I think you’ll agree the latter expression is unambiguously equal to (-9)

I think that in all (or at least almost all) cases, Unary Negation can be cleared up by turning it into a simple subtraction problem… -anything == 0-anything. Where “anything” can be a number, a variable, an expression…

Does that make things look any clearer?

Certainly ambiguous, and an example of a bad textbook question (all too common).

Back when I was supervising computer programmers, I used to throw code like this back at the programmer, telling them to make the order of operations explicit by putting in parenthesis. This despite computer languages having very specific rules about order of evaluation. Making it clear was a requirement of our coding standards.

Actually, I just realized that doesn’t really make much a difference to my analogy, as substituting back in you get either:

-2[sup]2[/sup] = -4

or

-(-2)[sup]2[/sup] = -4

Though I suppose it’s not as clear as other explanations in this thread.

Actually I think you were right the first time. x = 2

-2[sup]2[/sup] = -1 * 2[sup]2[/sup] = -4

if X were -2 then

-X[sup]2[/sup] = -1 * X[sup]2[/sup] = -1 * -1 * 2[sup]2[/sup] = 4

Another thing I just noticed. Out of curiosity I punched -2[sup]2[/sup] into some calculators. Real calculators like TI-84 or TI-30XS seem to know the order of operations and gives -4. But, the microsoft calculator found in accessories gives an answer of 4.

This is very discouraging.

You probably typed a caret (that is, the character ‘^’) into it. But the caret isn’t interpreted as exponentiation by Microsoft Calculator, but rather as bitwise xor (e.g., observe the value of “5^3”). If you use the exponentiation button, you get the desired result.

Nope. If x = -2, then -x[sup]2[/sup] = -1 * (-2)[sup]2[/sup] = -4

No, 2 and -2 are both the correct solution to the problem. But I was trying to analogize using variable and literals and realized that, while the analogy may work, it’s probably more confusing than illuminating.

Actually, the caret thing doesn’t explain this, since you’d get 0 instead. Still, if I type in “-”, then “2”, then hit the square button, then hit enter, I get -4. If I type in “-”, then “2”, then hit the exponentiation button, then hit enter, I get -4 as well. You may have thought it gave an answer of four because it displays intermediate results before you hit enter.

Thanks. I still can’t make it work though. I’m using the calculator that comes with windows XP. The only button I can find to give the square function is labeled “x^2”. There is a button labeled “Exp”, but it gives me scientific notation. Typing “2” then “Exp” just gives me 2e+0, with the option to change the zero. Changing the 0 to 2 and pressing “=” gives 200.

The exact order that I’m punching the buttons is… “2”, then “+/-” to make it negative, and it displays -2 (It doesn’t let me add the “-” before adding a number). Then press “x^2” and it gives me 4, even after I press “=”.

I don’t see a square button other than “x^2”. I know there must be something simple I’m missing, and I’m calculator illiterate, but I can’t find it.

My hand held calculators work fine. I just can’t get the windows calculator to work.

I understand that anything substituting the X in X^2 is treated as if in parenthesis, which was how I messed up the answer in my response a few posts up. But still I don’t see another way on this calculator, besides making the number negative later when you need it to be.

Oh. Well, that “+/-” doesn’t correspond to the prefix negation symbol, and so has its own conventions for operator precedence (always taking effect immediately, I believe). There’s nothing disappointing about that, IMHO.

I’m on XP too, and I have no problem clicking, e.g., “-”, “2”, “x^2”, and then “=”, in that order, receiving -4 as output. What goes wrong for you when you attempt to use “-” as a unary prefix function symbol?

Holy cow it worked. You were right about not hitting “=” after using “-” as a prefix, and assuming it wasn’t there. I got stuck on wanting to use the “+/-”. Sorry about that.:smack: