# What is -3^2 ?

I’m blogging Maths at 1.30am. My mind is blown. I have spent the last hour in a Twitter conversation about the answer to the calculation -3^2. I need your help.

Now if you ask me the question I will confidently tell you the answer is -9. That’s obvious to me. That the fundamentals of Maths. It’s convention. It’s. BIDMAS/ BODMAS/ PIDMAS/ PODMAS wherever you are whatever mnemonic you prefer. Right? Well seemingly there are Mathematicians out there who disagree. Mathematicians whom I respect I would like to point out. No marks for guessing that they think the answer to the question is 9.

I feel I need to blog to get my head straight but more because I’m interested in what camp you fall into. If you never comment on a blog post of mine again please comment on this.

So here are my key points.

Order of Operations
In the calculation -3^2. BIDMAS tells us indices first then the multiplication (by -1) this we get the answer -9
There are no brackets in the calculation so we acts as if they aren’t there?

I then got into a separate (brief) conversation about the calculation 9 – 2 + 3 with a different user. The answer to this is 10 not 4 in my opinion. This is a fundamental problem with the mnemonic and perhaps the way we expose pupils to it. The hierarchy in my mind is:
1) Brackets
2) Indices
3) Division and Multiplication
4) Addition and Subtraction
The mnemonic would suggest that eg division comes before multiplication but in fact these have equal hierarchy and hence we calculate left to right. Am I right and do I/ we make this clear to pupils if so?

Directed Numbers
One of the main arguments for the 9 camp was that the notation was ambiguous. I see no ambiguity. To be directing a number negatively is the same as multiplication by negative 1 and so the first operation is square then direct. The 9 camp view was different.

I just cannot see this argument. Directing a number/ changing the direction of a number is multiplication by minus 1. This is why a negative multiplied by a negative is a positive? Hence it isn’t done first as indices are further up the hierarchy?

I’m not sure this post is completely coherent. It’s late. I’m tired. Above all I’m confused. In my head we have conventions such as order of operations to standardise our subject and to avoid ambiguity. To me the 9 camp are misapplying the rules but a little bit of doubt in my mind has me worried the very foundations of my Mathematical world are wrong.

Please, please comment. I’m going to try to get some sleep!

Note For those of you who know Colin I’m quite aware that he’s happy to play Devil’s advocate to get a debate going but he has messaged me to say he thinks that the notation is ambiguous. It’s this ambiguity I just can’t see.

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## 19 comments

1. · July 28, 2013

Wolfram Alpha agrees with you, as do I. However, the issue I believe comes from the lack of brackets. If the problem was specifically stated as (-3)^2, then it would be 9(obviously) but as it is written, it is actually (-1)3^2, which would be -9. If the brackets are not specifically stated, the the base is considered positive and any negative sign is considered to be (-1).

2. · July 28, 2013

I think you’re right, though it’s not very common to see Mathematicians using brackets when they need to.

3. · July 28, 2013

This persuades me you’re right, but explains why I was a bit confused: http://mathforum.org/library/drmath/view/61633.html

4. · July 28, 2013

Found back up for our reasoning!
http://mathforum.org/library/drmath/view/53240.html

5. · July 28, 2013

The ambiguity lies in your question. The answer will vary depending on whether a person, a calculator, or a compiler is evaluating the expression, and even within the latter two categories, the answer will depend on the particulars. You wrote the expression as -3^2, not -3², which leads one to believe that you’re writing computer code or formatting for LaTeX, not presenting a problem to students.

As for the mnemonics, most versions are wrong, and why teach shortcuts instead of real understanding anyway?

• · July 28, 2013

That’s my point. It shouldn’t matter who or what is doing the question. That’s why we have conventions/ rules call them what you will. Type that in a calculator you will get the answer -9, as I did, because that’s what the order of operations tell us to do.

The way the question is written is semantics and doesn’t change the calculation. Both are equivalent. I presume the original question (which wasn’t mine) was written on a keyboard or mobile with no superscript functionality. In fact, I’m not sure there is a way to do superscript in Twitter on any platform.

I’m also not sure how you can teach order of operations for understanding? It’s a convention to standardise the way we do calculations. How would you teach a student to understand that multiplication is done before addition, say?

6. · July 28, 2013

http://tutorial.math.lamar.edu/Classes/Alg/IntegerExponents.aspx
This explains why it is -9. it is all about convention, nothing to do with whether it is written with a carat or a superscript, there is no ambiguity. The exponent only affects the quantity directly to its left. in -3^2, the quantity directly to the left is 3, not -3. in (-3)^2, the quantity is (-3).

7. · July 28, 2013

Negative 3 squared is surely 9! Though after all these years I now understand why the calculator doesn’t know this!

• · July 28, 2013

I think the question is: is that what the calculation asks. To me negative three squared would be written with brackets and this says negate the result of three squared. Thanks for the comment. Really interesting to hear how differently people interpret it!

8. · July 28, 2013

I am with you on this – and i also teach students that + and – are of the same ‘importance’ – read left to right like a sentence (same for x and /)

Directed Numbers
One of the main arguments for the 9 camp was that the notation was ambiguous. I see no ambiguity. To be directing a number negatively is the same as multiplication by negative 1 and so the first operation is square then direct. The 9 camp view was different.

9. · July 28, 2013

I completely agree with you and don’t see any ambiguity. Interesting debate.

10. · July 28, 2013

Regarding 9 – 2 + 3, in my mind, which is also a bit blown, this is equivalent to 9 + (-2) + 3 which is always 10, no matter which order you do it in. Don’t we always teach kids (as I was taught anyway) that ‘subtraction is the same as adding a negative’?

• · July 28, 2013

Yes. I agree with that. 🙂

11. · July 28, 2013

Agree with you as mentioned last night.
I find it interesting that another person (above) wrote it is 9! when clearly as mathematicians we know ! is Factorial and so we shouldn’t use !
When I ask questions I usually leave a space and use ?? instead of ? so I suppose I should use ‘proper’ grammar and punctuation but find I am often corrected by people that then use CAPS to write: LO: To Be Able to find… IMO should be LO: To be able to find… or they write LO: To understand how to CONSTRUCT angle BISECTOR
I see the need for keywords but find that as Maths teachers some of us (myself included) often mix the notation we use and sometimes even the order we process operations (for example where we have missed brackets) for -3^2 = 9 or -9 depending on what we consider the answer to be. Thoughts ??

12. · July 28, 2013

-3² = ±9
*runs away*

• · July 28, 2013

*Sigh*

• · July 28, 2013

At least they ran away…

13. · July 30, 2013

Although it’s really just a convention, the consensus view in the mathematical community is that -3^2 means -(3^2), or -9. Colin Graham seemed to claim that -3^2 and -x^2 should follow different rules, but this seems unworkable to me.

However, it is interesting that -3^2 evaluates to 9 in Microsoft Excel, and has done so since the first version. This bug (if it is one) can’t be fixed because it would break compatibility with older versions of Excel.

14. · June 10, 2014

When learning how to do indices I was taught that a^2=a x a right?
I was also taught that a negative multiplied by a negative is always a positive?
So therefore doesn’t -3^2 mean that it actually means -3 x -3? Which means it’s +9?
Sincerely,
Edee, A very confused gcse student :’)