# Evaluating -3^2 **URL:** \ **Category:** Factual Questions\ **Created:** [January 8, 2003, 4:21am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249 "2003-01-08T04:21:58Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Phlint](https://avatars.discourse-cdn.com/v4/letter/p/c89c15/32.png) [@Phlint](https://boards.straightdope.com/u/Phlint)\ **Post date:** [January 8, 2003, 4:21am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/1 "2003-01-08T04:21:58Z")
Earlier today, I sat down to help my younger brother with his 8th grade math homework, and ran across the expression -3^2. I’ve always thought that this evaluated to the value 9, but the book claims it is -9. I know that (-3)^2 = 9 and -(3^2) = -9, but what is the rule when there are no parenthesis? ---
**Author:** ![jbird3000](https://avatars.discourse-cdn.com/v4/letter/j/aeb1de/32.png) [@jbird3000](https://boards.straightdope.com/u/jbird3000)\ **Post date:** [January 8, 2003, 4:27am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/2 "2003-01-08T04:27:24Z")
Exponents happen before multiplications. So 3^2 is evaluated first, then the -. ---
**Author:** ![zev\_steinhardt](https://avatars.discourse-cdn.com/v4/letter/z/97f17d/32.png) [@zev\_steinhardt](https://boards.straightdope.com/u/zev_steinhardt)\ **Post date:** [January 8, 2003, 4:27am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/3 "2003-01-08T04:27:24Z")
The square of a number can never be a negative number, so (-3)^2 can never be -9. When there are no parenthesis, the order of operators takes over. Since the only operator in -3^2 is the exponation, you treat it as -3 \* -3. Zev Steinhardt ---
**Author:** ![Phlint](https://avatars.discourse-cdn.com/v4/letter/p/c89c15/32.png) [@Phlint](https://boards.straightdope.com/u/Phlint)\ **Post date:** [January 8, 2003, 4:32am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/4 "2003-01-08T04:32:06Z")
Well, I’d have to say that the first two responses at least make me feel better about being confused 🙂 And thanks for the responses! ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [January 8, 2003, 4:34am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/5 "2003-01-08T04:34:57Z")
As has been said, you follow order of operations. I’m having trouble finding a good tutorial, but here’s the basic structure of order: 1. Parentheses 2. Exponents 3. Multiplication and Division 4. Addition and Subtraction Although it is sometimes notationally convenient to place negation between steps 1 and 2, as **zev\_steinhardt** has, the book seems to want it placed in step 4, as **jbird3000** has. ---
**Author:** ![jbird3000](https://avatars.discourse-cdn.com/v4/letter/j/aeb1de/32.png) [@jbird3000](https://boards.straightdope.com/u/jbird3000)\ **Post date:** [January 8, 2003, 4:37am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/6 "2003-01-08T04:37:37Z")
Another helpful reply! 😃 ---
**Author:** ![Dr\_Paprika](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr_paprika/32/3042_2.png) [@Dr\_Paprika](https://boards.straightdope.com/u/Dr_Paprika)\ **Post date:** [January 8, 2003, 4:45am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/7 "2003-01-08T04:45:12Z")
The correct answer is -9 for the reasons stated; there are no parentheses, so the exponent 3^2 is done before multiplying the whole thing by -1. -3^2=-(3^2)=-9, this is different from (-3)^2. Dr\_Paprika (who in eleven years of university took 19 friggin’ math courses) ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [January 8, 2003, 4:48am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/8 "2003-01-08T04:48:26Z")
> [@](#): > > \*Originally posted by Achernar \* > \*\*As has been said, you follow order of operations. I’m having trouble finding a good tutorial, but here’s the basic structure of order: > > 1. Parentheses > 2. Exponents > 3. Multiplication and Division > 4. Addition and Subtraction > > Although it is sometimes notationally convenient to place negation between steps 1 and 2, as **zev\_steinhardt** has, the book seems to want it placed in step 4, as **jbird3000** has. \*\* Yes but, the exponent 2 just means multiply the number by itself. The number is -3 which multiplied by itself is 9. As far as the order of operations goes, an exponent is just a shorthand way of indicating multiplication. So if you have -3[sup]3[/sup]_5 what you really have is -3_-3\*-3_5 = -27_5. In writing the general term for an alternating series all the math books I’ve seen use the term -1[sup]n+1[/sup] to get the signs, n starting at 1 and going up. So the signs are -1[sup]2[/sup] = +1, -1[sup]3[/sup] = -1, etc. ---
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [January 8, 2003, 4:54am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/9 "2003-01-08T04:54:54Z")
If the expression was -x[sup]2[/sup], I think it would be clearer that one exponentiates first. What is the value of 2x[sup]2[/sup] - x[sup]2[/sup] for x = 3? ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [January 8, 2003, 5:04am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/10 "2003-01-08T05:04:25Z")
> [@](#): > > \*Originally posted by KarlGauss \* > \*\*If the expression was -x[sup]2[/sup], I think it would be clearer that one exponentiates first. > > What is the value of 2x[sup]2[/sup] - x[sup]2[/sup] for x = 3? \*\* Well it would be x[sup]2[/sup] irrespective of the value of x. x[sup]2[/sup] is x\*x which is 9 for x = either 3 or -3. ---
**Author:** ![AntaresJB](https://avatars.discourse-cdn.com/v4/letter/a/c2a13f/32.png) [@AntaresJB](https://boards.straightdope.com/u/AntaresJB)\ **Post date:** [January 8, 2003, 5:09am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/11 "2003-01-08T05:09:15Z")
Reiterating what some other people said – IANAMathematician, but this was something my college algebra prof emphasized at the beginning of this past semester. It’s -9. A negative sign on a number is the equivalent of multiplying that number by -1. If there are no parentheses to indicate (-3)^2, then the order of operations that **Achernar** listed applies, exponents before multiplication, therefore you get -9. ---
**Author:** ![matt\_mcl](https://avatars.discourse-cdn.com/v4/letter/m/7ab992/32.png) [@matt\_mcl](https://boards.straightdope.com/u/matt_mcl)\ **Post date:** [January 8, 2003, 5:17am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/12 "2003-01-08T05:17:24Z")
But a negative number is just that: a negative number. It’s not a positive number multiplied by -1. If that were the case, what would -1 be? It would be 1\*-1. But -1 is 1\*-1, so it would be 1_1_-1. Infinite regress – 1_1_1_1_1\*1… – and -1 would equal 1, and negative numbers would be impossible. The math I learned would have -3^2 be 9. ---
**Author:** ![moes\_lotion](https://avatars.discourse-cdn.com/v4/letter/m/7ba0ec/32.png) [@moes\_lotion](https://boards.straightdope.com/u/moes_lotion)\ **Post date:** [January 8, 2003, 5:30am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/13 "2003-01-08T05:30:56Z")
I have to agree with the -9 answer, for the reasons already mentioned by **jbird3000** and others. Not to hijack this, but **matt\_mcl** , I was taught that negative numbers are defined as the additive inverses of their corresponding positives, i.e. that for any a \> 0 there exists a number -a such that a + (-a) = 0, and further that -a = (-1)a. Thus, -3^2 = (-1)(3^2), not (-3)^2, which is why the correct answer is -9 ---
**Author:** ![TWDuke](https://avatars.discourse-cdn.com/v4/letter/t/e495f1/32.png) [@TWDuke](https://boards.straightdope.com/u/TWDuke)\ **Post date:** [January 8, 2003, 5:32am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/14 "2003-01-08T05:32:25Z")
-3^2 = -(3^2) NOT (-3)^2 The mathbook is right, or so says [http://mathforum.org/library/drmath/view/55709.html](http://mathforum.org/library/drmath/view/55709.html)[Dr. Math](http://mathforum.org/library/drmath/view/55709.html). (And AntaresJB and the rest, of course.) ---
**Author:** ![Terminus\_Est](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/terminus_est/32/3087_2.png) [@Terminus\_Est](https://boards.straightdope.com/u/Terminus_Est)\ **Post date:** [January 8, 2003, 5:32am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/15 "2003-01-08T05:32:46Z")
There are two types of “-” operator. The (binary) subtraction operator (e.g. 3-2) requires a left operand and a right operand. The unary minus operator (e.g. -3) takes only a right operand and is used to indicate negative numbers. The binary form has a lower precedence than multiplication, while the unary form has a higher precedence than multiplication or exponentiation. Which operator is being used depends on the context: ```auto -3[sup]2[/sup] = (-3) * (-3) = 9 0 - 3[sup]2[/sup] = 0 - (3 * 3) = -9 -3[sup]2[/sup] - 0 = (-3) * (-3) - 0 = 9 ``` ---
**Author:** ![viking](https://avatars.discourse-cdn.com/v4/letter/v/bc79bd/32.png) [@viking](https://boards.straightdope.com/u/viking)\ **Post date:** [January 8, 2003, 5:38am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/16 "2003-01-08T05:38:45Z")
Hmmm. Odd indeed. For what it’s worth, Perl (a computer programming language) has 24 levels of precedence of operators, rather than the paltry 4 used in high school math. The camel book (p. 77) admits that this is too many, but on the plus side, they are intuitive, unless of course you happen to be psychotic. If you are merely neurotic, the camel book suggests using paretheses. The unary + and - are at level 5, while the binary + and - are at level 8. The exponentiation operator, **, is at level 4, which binds more tightly than unary minus, so -2** 4 is -(2\*\*4), not (-2)\*\*4. Then again, I would have intuitively argued that -3^2 means (-3)^2, so I guess that makes me psychotic 🙂 There is of course a small chance that Perl syntax is not the final arbiter of correctness, but I doubt it. For those who need more details, the camel book is O’Reilly Publishing’s ``Programming Perl’’, by Larry Wall et al. ISBN 1-56592-149-6. Just so no-one yells ‘cite’ at me 🙂 ---
**Author:** ![AntaresJB](https://avatars.discourse-cdn.com/v4/letter/a/c2a13f/32.png) [@AntaresJB](https://boards.straightdope.com/u/AntaresJB)\ **Post date:** [January 8, 2003, 5:49am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/17 "2003-01-08T05:49:10Z")
The Simplify function at [QuickMath](http://www.quickmath.com) says that “-3^2” is -9. Putting in “(-3)^2” gives you 9, but plain old -3^2 says -9. The expression calculator at [AlgebraHelp.com](http://www.algebrahelp.com) says the same thing. “-3^2” is -9. (Order of Operations and Simplifying Calculator on the Calculators page.) As does the [Math.com Algebra Solver - Simplification](http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=mathcom&s1=algebra&s2=simplify&s3=basic). -3^2 = -9. Helpful? ---
**Author:** ![TWDuke](https://avatars.discourse-cdn.com/v4/letter/t/e495f1/32.png) [@TWDuke](https://boards.straightdope.com/u/TWDuke)\ **Post date:** [January 8, 2003, 5:51am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/18 "2003-01-08T05:51:00Z")
> [@](#): > > \*Originally posted by viking \* > \*\*I would have intuitively argued that -3^2 means (-3)^2, so I guess that makes me psychotic 🙂 \*\* Foolish psycho! Actually, I had written out an elaborate proof of why -3^2 had to equal 9 and my finger was trembling over the “submit” button when something told me I’d better check some math sites before posting, and sure enough I was wrong, which just goes to show you, always listen to imaginary voices in your head. Psycho fool! ---
**Author:** ![Terminus\_Est](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/terminus_est/32/3087_2.png) [@Terminus\_Est](https://boards.straightdope.com/u/Terminus_Est)\ **Post date:** [January 8, 2003, 5:58am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/19 "2003-01-08T05:58:02Z")
:smack: Ach, the heck with counter-intuitive orders of precedence! [Barbie] Math is hard! [/Barbie] ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 8, 2003, 6:08am UTC](https://boards.straightdope.com/t/evaluating-3-2/147249/20 "2003-01-08T06:08:11Z")
> [@](#): > > \*Originally posted by matt\_mcl \* > **But a negative number is just that: a negative number. It’s not a positive number multiplied by -1.** And the difference is? [Next page](https://boards.straightdope.com/t/evaluating-3-2/147249.md?page=2)