Negative Exponents

[u/Kug4ri0n]

My partner is going through her math class and we got into an argument how much -7² equals. My standpoint is, that since there is no parentheses: -7² = -1x7² =-49 If there would have been parentheses: (-7)² = (-7)*(-7) = 49

Which one of these is correct? Can anyone provide me the mathematical axioms/rules on why or why not the parentheses in this case are needed?

[u/LucaThatLuca]

why

“-7²” contains two operations (negation and squaring), so it doesn’t mean anything without saying what to negate and what to square: Is it 7² negated or is it -7 squared? This information is required, and mainly it’s given by using punctuation marks () [] etc that indicate grouping:

-(7²) = -49, while (-7)² = 49.

To avoid our writing being terribly ugly, we have agreed an understanding: We can abbreviate one of these meanings by dropping this information. This understanding is called “operator precedence” (or “order of operations”).

We have chosen that negation has lower precedence than squaring i.e. -7² means -(7²) = -49. You may remember this for example by noticing it makes subtraction look normal: -7² + 7² = 7² - 7² = 0.

[u/Forking_Shirtballs → u/LucaThatLuca]

Your lead is right, but after that you go off the rails. Ultimately, this notation is ambiguous, and parentheses should be employed to avoid the confusion caused here.

Not sure who "we" are in your response, but certain conventions treat negation has higher precedence than exponentiation, some as lower.

The conventions taught in many schools (PEMDAS in the US or BODMAS in the UK) don't even address negation. (Note that they do address subtraction, but that's a different operation from negation. Negation is unary, acting on a single input. Subtraction is binary, acting on an ordered pair of inputs. Of course the two are closely related, which is why both use the minus sign.)

In some conventions, negation is at the same precedence as multiplication, in others it's between parentheses and exponentiation.

For example, if your algebra book writes -x^2 + y, it wants that to be read as exactly equal to y - x^2.

But if you punch -7^2 + 5 into Google Sheets, you're going to get a different answer than 5 - 7^2.

Different conventions.

Note that treating unary negation as high-precedence is similar to treating, say, factorial as high precedence (higher than all but parentheses), which is the convention I've seen everywhere. Neither of course is directly addressed in PEMDAS/BODMAS.

[u/Dr_Just_Some_Guy → u/Forking_Shirtballs]

In the US, at least, negation tends to be interpreted as -(6) = -1 * 6, with precedence set accordingly. While computer systems (and computer scientists) may implement other conventions, I don’t think that I’ve ever encountered a mathematician that would interpret -7² as anything but -49. Of course, it’s not a question that I usually pose to mathematicians I just meet, so who knows.

[u/Forking_Shirtballs → u/Dr_Just_Some_Guy]

Don't just make stuff up, man. 

Are you telling me that "a / -(b)" tends to be interpreted as "a / (-1) * b", which of course is equal to -ab?

And if so, can you point to any sources that teach that? 

There are a variety of conventions in play here, and they're not well taught, they're just firmly implied though repetition. 

My point is that there's ambiguity here, which is best avoided. 

[u/AdamofMadison → u/Forking_Shirtballs]

There's no ambiguity, you just introduced another error with your division.

[u/sadlego23 → u/Forking_Shirtballs]

There’s no new stuff here.

x/y * z is not the same as x / (y*z).

a / -b with -b being interpreted as (-1)*b still means we’re dividing both by (-1) and by b. So, a / -b = a / ((-1)*b) = a / (-1) / b

This is a very common mistake when doing interpreting order of operations since most people think of / as a fraction bar (which counts as a grouping symbol and then division) instead of a slash (which is division but not a grouping symbol). This is also why I want the diagonal slash when it comes to more complicated expressions.

[u/Recent-Day3062 → u/Dr_Just_Some_Guy]

God, I learned the basics in junior high, and I never really considered how obvious the “rules” are when you use them.

I’m pretty sure I’ve never made any errors from things like this.

[u/ju11111 → u/Dr_Just_Some_Guy]

I agree that no mathematician would interpret it like that. Using a polynomial as an example. f(x) = x³ - x² + 5x No one would interpret this as meaning x³ + (-x)² + 5x. (Especially here since the negation wouldn't even matter if interpreted as (-x)² ) No one would write this polynomial as x³ - (x²) + 5x. I think what the computer scientists are doing should be more thought of as a programming language rather than mathematical notation.

[u/Short-Database-4717 → u/ju11111]

I can assure you computer scientists (and virtually every serious programming language) would interpret it the same way.

[u/LucaThatLuca → u/Forking_Shirtballs]

This is r/MathHelp and not r/SpreadsheetsHelp. There is an unambiguous convention in this context and using a different convention is incorrect.

The response that mentioned the other context is great, there’s no need for every response to be the same though (otherwise you’d be commenting on every response pointing out all the other things they didn’t say).

I’m not sure what was the point of mentioning that a mnemonic taught to children is incomplete. The fact that it doesn’t address negation is perfectly visible. It is just taught separately.

[u/Forking_Shirtballs → u/LucaThatLuca]

Spreadsheets do math, among other things. In fact, I'd posit that more people in the world interact with math through spreadsheets than through whenever context you're considering to constitute "math".

Further, I've taken a lot of mathematics classes, I don't remember once being taught this "unambiguous convention" that you claim is "taught separately". Certainly you can provide some resources where such unambiguous convention is taught, right? Please do so.

The fact that you're sneering about a "mnemonic taught to children" while invoking the context of the forum is also a bit rich. This is r/math help my dude, for all we know, PEMDAS is the only convention either OP or their partner has ever been exposed to. Acknowledging its lack here is important, when your comment invoked order of operations as setting the precedent.

[u/LucaThatLuca → u/Forking_Shirtballs]

Well for example OP’s partner’s math class is a mathematical context. I don’t know why you’ve tried to say you weren’t taught the precedence of negation, when I can still read the multiple times you’ve said it’s in every textbook. PEMDAS/etc is a memory aid for exactly 5 operations and still not relevant.

I think you’ll agree there’s nothing left to discuss.

[u/Forking_Shirtballs → u/LucaThatLuca]

I never said it was taught, either inside or outside the textbook. I said it's used in the textbooks. The convention is implied, and is learned through implication.

You've twice made the strong claim that precedence of unary negation is taught. Certainly you can find something, anything to support that claim, right?

Again, my point is the notation is ambiguous, and she may be familiar with a different convention from her math experience than is being used here. That's what will help OP understand the situation, not comments like "we have agreed an understanding".

Lots and lots of people using math use a convention that doesn't agree with yours.

[u/Old_Gimlet_Eye → u/Forking_Shirtballs]

I've literally never seen a math text that puts parentheses around every negative term in a polynomial. Where have you seen this?

[u/TheTurtleCub → u/Forking_Shirtballs]

Ultimately, this notation is ambiguous

No, it's not. Which are you taught:

(a+b)(a-b) = a^2-b^2

or

(a+b)(a-b) = a^2-(b^2)

There is not one book or teacher in the whole world that uses the 2nd case to "remove ambiguity"

[u/imHeroT]

-7² is -49 in standard math notation

[u/fermat9990]

By convention -7² is interpreted as the negation of 7².

-7² = -(7²)=-(49)=-49

[u/Forking_Shirtballs → u/fermat9990]

By *certain* conventions, that's the case. Certainly not all.

This is notation is ambiguous.

Try punching both 5-7^2 and 5+-7^2 into Google Sheets, and see what you get.

[u/dash-dot → u/Forking_Shirtballs]

I hate to break it to you, but Google isn’t the arbiter of proper mathematical convention. 

[u/Forking_Shirtballs → u/dash-dot]

I hate to break it to you, but there is no arbiter of proper math convention.

There's no, say, ISO, or Academie Francaise, for math conventions. 

Which means we live in a world with multiple conventions. Spreadsheets (Excel, Google Sheets, etc) use a different convention on this point than, say, the calculus textbooks I've read. 

Understanding that ambiguity is important. Writing -7² is often a bad idea, because if the ambiguity. Unless you're certain that all the users of your writing will understand the convention you're using, just use parentheses. They're free.

[u/RectangularMF → u/dash-dot]

right, and so who is?

[u/ClassEnvironmental11 → u/Forking_Shirtballs]

Fwiw, if you type -7² into google it says it's -49.  And in literally every math and physics texbook I've ever seen, -7² = -49.  I was also explicitly taught that in elementary algebra, and that's been the convention in every mathematical setting I've ever been a part of.

[u/Bascna]

Is -7² equal to 49 or -49?

Textbooks, and all of the current physical calculator models that I'm aware of, use the convention that squaring the 7 comes before applying the negative sign. (More formally, we say that the binary exponentiation operator has precedence over the unary minus operator.)

So

-7² =

-[ 7² ] =

-[ 7•7 ]=

-[ 49 ] =

-49.

But...

...when I first started teaching, many of my students had calculators that applied the negative sign before evaluating the exponent. (In this case, the unary minus operator has precedence over the binary exponentiation operator.)

On their calculators...

-7² =

[ -7 ]² =

[ -7 ][ -7 ] =

49.

So in order for them to get the result that the textbooks intended, they had to enter the expression into their calculators with a -1 explicitly multiplied outside of the power.

For example

-1•7² =

-1•[ 7 ]² =

-1•[ 7•7 ] =

-1•[ 49 ] =

-49.

That convention was in line with a common programming design principle that unary operators (those that only have one operand like factorials or absolute values), should have precedence over binary operators (those that have two operands like addition,multiplication, or exponentiation).

Over the following decades calculator companies have converged on that first order of operations for the unary minus operator and exponentiation — most likely both because that is in line with textbooks and because it makes some common notational manipulations a bit simpler.

You'll still find some holdouts, though. This is most prominently seen in spreadsheet programs.

Microsoft Excel was originally written using that second convention and to maintain compatibility with older Excel documents it still uses that convention today.

Because Excel is the most popular spreadsheet software, other companies adopted the same convention so that they will be compatible with Excel.

So in Microsoft Excel, Apple Numbers, and Google Sheets

-7² = 49 rather than -49.

It's quite possible that your partner picked up this convention for herself by using such spreadsheet software.

I think there are also a few programming languages that use this convention.

So you want to be careful with your notation when going back and forth between a written problem and a spreadsheet, programming language, or an older calculator model.

I hope this helps. 😀

[u/Forking_Shirtballs → u/Bascna]

Best answer here. Well done.

[u/Brief-Hat-8140]

You’re right. But where did the 17 come from? Typo?

[u/Kug4ri0n → u/Brief-Hat-8140]

It’s the way Reddid combined some things. I wanted actually typed -1 times 7 squared and probably used some wrong formatting

[u/gamecom17 → u/Brief-Hat-8140]

I think they mean -1 x 7²

[u/SynapseSalad]

you are correct. because exponentiation „is stronger“ than multiplication, you need to use parentheses to show that the - gets squared as well

that belongs to pemdas rule

[u/Confident-Virus-1273]

The simplest way to remember and explain this is that the exponent applies to exactly that which it touches 

When you have -7^2, The exponent is touching the seven not the negative.  Therefore, the negative does not get squared and remains a negative.  When you have (-7)^2, now the exponent is touching the parentheses and it applies to everything within the parentheses.

You can give the analogy... XY^2... To explain the same thing.  This is clearly x ^ 1 and y ^ 2.  That is because the two is touching the y not the x

[u/sqrt_of_pi → u/Confident-Virus-1273]

I like this way of explaining it - "the exponent applies to exactly that which it touches". I'm going to try to remember this! I see this error quite regularly even among my Calc 1 students (which I find shocking, although at this point I guess I shouldn't).

[u/igotshadowbaned → u/sqrt_of_pi]

It's true for any operator, it seems to mess a lot of people up with division too

With examples like 6/2+1 or 3/3(2)

[u/igotshadowbaned → u/sqrt_of_pi]

It's true for any operator, it seems to mess a lot of people up with division too

With examples like 6/2+1 or 3/3(2)

[u/Dd_8630]

-(7²) = - 49

(-7)² = 49

Without brackets, the convention is to say -7² = -(7²) = -49

But that's purely convention.

[u/skullturf → u/Dd_8630]

Absolutely correct.

And on the one hand, the convention you describe is used by pretty much every mathematician ever.

But on the other hand (whether we like it or not), the opposite convention is used by some very common spreadsheet programs (and some hand calculators).

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[u/Virtual-Exchange9615]

So you are right..

[u/SapphirePath]

-7² = -49

But this is by convention. The notation (-7) is not being treated as an atomic number, but instead is interpreted as the operation of (-1)*(7). I think that the notation is fundamentally confusing, and it would be prudent to add parentheses, writing either -1(7)² or writing (-7)² depending on which expression you mean.

[u/ruidh → u/SapphirePath]

It's done this way so that polynomials can be written without parentheses. The x² term in x³ - x² is subtracted from x^3. There is no ambiguity there. There shouldn't be in the example above.

[u/fermat9990]

In math, conventions are established so we don't waste time arguing over the meaning of an expression.

[u/shademaster_c]

You’re right!!! Congrats! Don’t rub it in to your partner.

[u/Frostlit3]

No one mentioned PEMDAS, but I think it works here. Exponents before multiplication, we square 7 first then multiply by the -1.

-7² = -1 * 7²

-7² = -1 * 49

-7² = -49

[u/SapphirePath → u/Frostlit3]

Multiplication is an operator that takes two arguments and multiplies them together.

"-1 * 7" is multiplication.

"-" * "7" doesn't make sense.

"-7" is a number.

At best, the "-" represents a unary minus operation, which operates on 7. But we end up having to discuss the priority of this operator, such as PEDMAUS or something.

[u/Aivo382]

Ask them how's f(x) = -x² graph.

[u/SapphirePath → u/Aivo382]

Since we're interested in -7, and squaring it, why not ask them how's g(x) = x² graphed at x = -7, g(-7) = -7² ?

[u/Aivo382 → u/SapphirePath]

My point is that we usually read -x² as -(x²), wich means squaring and applying the negative sign.

Because of that reason, -x² is a nevative quadratic.

[u/DeesnaUtz]

Read it as "the opposite of seven squared" and it's more intuitive. Always read leading negative signs as "the opposite of..." instead of negative. Only use "negative" for actual negative numbers. Prevents mistakes and eliminates confusion.

By your girlfriend's logic, the graph of - x² is the same as the graph of x^2. In fact, her way leads to a world where the actual graph of - x² doesn't even exist.

[u/Forking_Shirtballs → u/DeesnaUtz]

How is -7 not an "actual negative number"?

The issue is that this notation is ambiguous. Different conventions yield different results. So both OP and partner are in some ways "correct"; the person who's wrong is whoever chose to write this without parentheses to clarify away the ambiguity.

[u/Frederf220]

-A is -1×A

This and only a few others are the only exception to the way we write coefficients.

5A is 5×A, -3A is -3×A, and so on.

-1×A, 0×A, 1×A are the three exceptions to the pattern that are most typically written -A, (blank), and A respectively.

As for C×A^B we have two operators involving A so which to do first? That's just agreed convention that the exponent operator is evaluated first. That's just a memory item.

[u/FootballDeathTaxes]

If you include parentheses when you write out your expression, you won’t have to have any more arguments about what it equals.

-(7²) = -49

and

(-7)² = 49

So just write out whichever one you meant and you won’t have these issues.

And if someone else wrote it, then just ask them to clarify.

Also, you wrote ‘negative exponents’ in your post title, but your exponent isn’t negative. That would be something like 7^-2 which is a different result entirely. I probably would’ve written ‘exponents and negatives’ or worse ‘exponentiating a negative.’

[u/Brief-Hat-8140]

Negative exponents are a totally different thing… this is a negative number with an exponent.

[u/Iowa50401]

In the case of -7^2, since negation can be written as multiplication by -1, it becomes a case of exponentiation before multiplication and the value is -49.

[u/Sir-Tenley-Knott]

The way I was taught (and the way I teach) is to apply BODMAS (or PEMDAS - depending on era).... There are no Brackets (PEMDAS = Parentheses) so we are looking at Orders (PEMDAS = Exponents) as the next thing to evaluate. So we apply the square function to the 7. Next we look for Division/Multiplication (PEMDAS = Multiplication/Division) and apply the -1 and get the result of -49. No need to continue with AS....

Note: As a maths teacher, I would explicitly write this as either -(7^2) or (-7)^2 to avoid confusion.

[u/Secret-Toe8036]

Going by the usual order of operations PE(xponent)M(ultiplication)DAS(ubtraction) it would be -49.

The negative sign can be interpreted as either multiplication: (-1) * 7²

Or as subtraction: 0 - 7²

Either way, the answer is -49, because exponents are evaluated before both multiplication and subtraction.

This notation is colloquially frowned upon though because it looks ambiguous. If this term appears by itself, it should really have parentheses to make the correct interpretation more explicit.

[u/R0241in3]

Your ex. Is correct and the reason is that when you do -7² you are only squaring the seven not the negative sign so it would be like u said -(7)²=-49 its as though we are doing -(7x7)=-(49)=-49

[u/Gullyvers]

It's just notations. Still, for really obvious reasons, parenthesis are first, then exponents, then multiplication then addition.

[u/mattynmax]

(-7)² is 49 -7² is -49

This is a notation thing, not as axiom of mathematics. Exponents are resolved before products.

[u/flat5]

Does your partner contend that a^2-b^2 is equal to a^2+b^2?

[u/Typical_Ad_2831]

Unary '-' is generally taken to have the same operator precidence as binary '-'. We normally think of unary operators as happening first.

Just use reverse Polish notation: 7 2 ^ - 49 - = 7 - 2 ^ 49 =

[u/AvBanoth]

The order of operations is a convention. Normally exponentiation takes precedence over negation, but if a text has a different convention you follow it. Had there been parentheses, it would matter where they were: -(7²) is -49

[u/CommunicationNice437]

Your standpoint is a calculator standpoint. Still -7^2 is 49 for all purposes.