# Basic Math Question **URL:** \ **Category:** Factual Questions\ **Created:** [February 10, 2010, 4:00pm UTC](https://boards.straightdope.com/t/basic-math-question/528431 "2010-02-10T16:00:38Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [February 10, 2010, 5:33pm UTC](https://boards.straightdope.com/t/basic-math-question/528431/21 "2010-02-10T17:33:04Z")
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 –x_x. 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 –3_3. So in order for everything to work consistently, –3[sup]2[/sup] has to be interpreted as –3\*3. ---
**Author:** ![Stealth\_Potato](https://avatars.discourse-cdn.com/v4/letter/s/d78d45/32.png) [@Stealth\_Potato](https://boards.straightdope.com/u/Stealth_Potato)\ **Post date:** [February 10, 2010, 5:59pm UTC](https://boards.straightdope.com/t/basic-math-question/528431/22 "2010-02-10T17:59:43Z")
> [@Jinx](#): > > First, I must say I was never taught the math jargon “unary”. Next, for math to be consistent with itself, there is no such quantity of -3 as said above it is a combination. Therefore, the only correct way to express a negative quanity, such as negatives on a numberline, would be (-3). And since -3^2 is treated differently than (-3)^2, then one can only conclude that -3 \<\> (-3). Now, the math folks will argue to the contrary, but rather I’d say their notation is flawed! They have to make up their minds! If indeed -3 = (-3), then so it IS always equal forever and ever…and by that logic, therefore: -3^2 = (-3)^2 = 9. Ha! 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 2_2 must represent the same number as 2_1 + 1! But clearly it doesn’t: 2_2 is 4, but 2_1 + 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. ---
**Author:** ![Algorithm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/algorithm/32/12204_2.png) [@Algorithm](https://boards.straightdope.com/u/Algorithm)\ **Post date:** [February 10, 2010, 6:07pm UTC](https://boards.straightdope.com/t/basic-math-question/528431/23 "2010-02-10T18:07:10Z")
> [@pulykamell](#): > > It seems logical enough to me. > > -x[sup]2[/sup] = -4 > x[sup]2[/sup] = 4 (multiply both by -1) > x = 2 > -2[sup]2[/sup] = -4 > > I suppose you can make two different rules, depending on whether you’re squaring a variable or a literal, but why? x = -2 ---
**Author:** ![ragerdude](https://avatars.discourse-cdn.com/v4/letter/r/d07c76/32.png) [@ragerdude](https://boards.straightdope.com/u/ragerdude)\ **Post date:** [February 10, 2010, 6:16pm UTC](https://boards.straightdope.com/t/basic-math-question/528431/24 "2010-02-10T18:16:59Z")
> [@Stealth\_Potato](#): > > 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 2_2 must represent the same number as 2_1 + 1! But clearly it doesn’t: 2_2 is 4, but 2_1 + 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. Wow, that is brilliant!! ---
**Author:** ![arseNal](https://avatars.discourse-cdn.com/v4/letter/a/ecae2f/32.png) [@arseNal](https://boards.straightdope.com/u/arseNal)\ **Post date:** [February 10, 2010, 6:25pm UTC](https://boards.straightdope.com/t/basic-math-question/528431/25 "2010-02-10T18:25:47Z")
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](http://mathforum.org/library/drmath/view/53194.html) ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [February 10, 2010, 6:28pm UTC](https://boards.straightdope.com/t/basic-math-question/528431/26 "2010-02-10T18:28:52Z")
> [@Algorithm](#): > > x = -2 Wow. Do I feel silly. Really, total brain fart there. ---
**Author:** ![Ruminator](https://avatars.discourse-cdn.com/v4/letter/r/b9bd4f/32.png) [@Ruminator](https://boards.straightdope.com/u/Ruminator)\ **Post date:** [February 10, 2010, 6:32pm UTC](https://boards.straightdope.com/t/basic-math-question/528431/27 "2010-02-10T18:32:52Z")
> [@arseNal](#): > > 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](http://mathforum.org/library/drmath/view/53194.html) I like the extra notes about Microsoft Excel at the bottom of that page: > [@](#): > > _You may be interested in this page I ran across, on the order of operations in Microsoft Excel. It says that they evaluate unary minus before exponentiation, and will not change it though they acknowledge that this is different from the normal order:_ > > [http://support.microsoft.com/support/kb/articles/q132/6/86.asp](http://support.microsoft.com/support/kb/articles/q132/6/86.asp) So there you have it, Microsoft Excel is the root cause of math notation illiteracy. 😃 ---
**Author:** ![Noone\_Special](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/noone_special/32/2863_2.png) [@Noone\_Special](https://boards.straightdope.com/u/Noone_Special)\ **Post date:** [February 10, 2010, 7:10pm UTC](https://boards.straightdope.com/t/basic-math-question/528431/28 "2010-02-10T19:10:03Z")
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? ---
**Author:** ![Tim\_T-Bonham.net](https://avatars.discourse-cdn.com/v4/letter/t/46a35a/32.png) [@Tim\_T-Bonham.net](https://boards.straightdope.com/u/Tim_T-Bonham.net)\ **Post date:** [February 11, 2010, 1:02am UTC](https://boards.straightdope.com/t/basic-math-question/528431/29 "2010-02-11T01:02:46Z")
> [@BrandonR](#): > > The problem with the problem is that it’s ambiguous to say the least. The book may be technically correct, but most people would simply use -(3^2) to indicate an answer of -9. To me, it’s way too easy to assume (-3^2) means (-3)^2 and hence give an answer of 9. And teachers wonder why so many kids hate math… 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. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [February 11, 2010, 3:48am UTC](https://boards.straightdope.com/t/basic-math-question/528431/30 "2010-02-11T03:48:48Z")
> [@Algorithm](#): > > x = -2 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. ---
**Author:** ![JWWJ](https://avatars.discourse-cdn.com/v4/letter/j/f07891/32.png) [@JWWJ](https://boards.straightdope.com/u/JWWJ)\ **Post date:** [February 11, 2010, 4:35am UTC](https://boards.straightdope.com/t/basic-math-question/528431/31 "2010-02-11T04:35:03Z")
> [@pulykamell](#): > > 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 ---
**Author:** ![JWWJ](https://avatars.discourse-cdn.com/v4/letter/j/f07891/32.png) [@JWWJ](https://boards.straightdope.com/u/JWWJ)\ **Post date:** [February 11, 2010, 4:43am UTC](https://boards.straightdope.com/t/basic-math-question/528431/32 "2010-02-11T04:43:31Z")
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. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [February 11, 2010, 4:51am UTC](https://boards.straightdope.com/t/basic-math-question/528431/33 "2010-02-11T04:51:11Z")
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. ---
**Author:** ![ragerdude](https://avatars.discourse-cdn.com/v4/letter/r/d07c76/32.png) [@ragerdude](https://boards.straightdope.com/u/ragerdude)\ **Post date:** [February 11, 2010, 4:56am UTC](https://boards.straightdope.com/t/basic-math-question/528431/34 "2010-02-11T04:56:22Z")
> [@JWWJ](#): > > 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 Nope. If x = -2, then -x[sup]2[/sup] = -1 \* (-2)[sup]2[/sup] = -4 ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [February 11, 2010, 5:15am UTC](https://boards.straightdope.com/t/basic-math-question/528431/35 "2010-02-11T05:15:07Z")
> [@JWWJ](#): > > 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 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. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [February 11, 2010, 5:19am UTC](https://boards.straightdope.com/t/basic-math-question/528431/36 "2010-02-11T05:19:24Z")
> [@JWWJ](#): > > 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. 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. ---
**Author:** ![JWWJ](https://avatars.discourse-cdn.com/v4/letter/j/f07891/32.png) [@JWWJ](https://boards.straightdope.com/u/JWWJ)\ **Post date:** [February 11, 2010, 6:09am UTC](https://boards.straightdope.com/t/basic-math-question/528431/37 "2010-02-11T06:09:12Z")
> [@Indistinguishable](#): > > 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. ---
**Author:** ![JWWJ](https://avatars.discourse-cdn.com/v4/letter/j/f07891/32.png) [@JWWJ](https://boards.straightdope.com/u/JWWJ)\ **Post date:** [February 11, 2010, 6:14am UTC](https://boards.straightdope.com/t/basic-math-question/528431/38 "2010-02-11T06:14:41Z")
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. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [February 11, 2010, 6:29am UTC](https://boards.straightdope.com/t/basic-math-question/528431/39 "2010-02-11T06:29:52Z")
> [@JWWJ](#): > > The exact order that I’m punching the buttons is… “2”, then “+/-” to make it negative, and it displays -2 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. > [@](#): > > (It doesn’t let me add the “-” before adding a number). 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? ---
**Author:** ![JWWJ](https://avatars.discourse-cdn.com/v4/letter/j/f07891/32.png) [@JWWJ](https://boards.straightdope.com/u/JWWJ)\ **Post date:** [February 11, 2010, 6:38am UTC](https://boards.straightdope.com/t/basic-math-question/528431/40 "2010-02-11T06:38:55Z")
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: [Previous page](https://boards.straightdope.com/t/basic-math-question/528431.md?page=1) [Next page](https://boards.straightdope.com/t/basic-math-question/528431.md?page=3)