# Is the expression $-3^2$ ambiguous?

I wouldn't be asking this question if I had not found this YouTube video in which many people on the comments insist that the result must be $$9$$ or just ambiguous:

The PEMDAS/BODMAS argument does not work for them because their reasoning is that the $$-3$$ should not be seen as an operation but instead as the real number $$-3$$, that is which is later squared.

Now, my thought is that as long as we don't have an easy way to distinguish between the unary minus and the binary one, as occurs in hand writing, then it is just a matter of convenience to always make $$-3^2$$ interchangeable with $$0 - 3^2$$ or just with $$(-1)*3^2$$, first because if we see the following two expressions that only differ in a sign: $$3^2$$ and $$-3^2$$, it is natural to guess that they have opposite values instead of the same one.

$$4^2 - 3^2 = 4^2 + f(x)$$

it is natural to interpret the left side as $$(4^2) - (3^2)$$. But we could realize that we can subtract $$4^2$$ in both and get:

$$-3^2 = f(x)$$

If we want to keep the reasoning that $$-3^2$$ is viewed as $$(-3)^2$$, then the equation above wouldn't be equivalent to the previous one. We would be forced to add new parentheses or put a zero to keep the same original meaning:

$$-(3^2) = f(x)$$ $$\text{or}$$ $$0 - 3^2 = f(x)$$

So I don't think this falls into the same category of ambiguity as $$6/2(2+1)$$, but I may be wrong.

I want to know if there is any serious publication that would interpret $$-3^2$$ as $$(-3)^2$$. I was surprised to find that if we write "$$-3^2$$" in a cell of Microsoft Excel, it evaluates it in $$9$$.

• No to the title question. The squaring operation binds more tightly than the negation operation, giving $-9$ as the result of $-3^2$. Commented May 5, 2023 at 4:49
• It's just a matter of convention. In most CAS and most writings by mathematicians, squaring is evaluated first. Commented May 5, 2023 at 5:20
• In at least $99.9$% of the times I have seen $-3^2,$ it has meant $-9,$ not $9.$ In Excel, when you type $-3$ into a cell and then square the number in the cell, surely you are doing $(-3)^2,$ which is $9?$ Commented May 5, 2023 at 5:39
• No one in his right mind would think $-3^2=9$. However, many people are not in their right mind, so be sure to write $-(3^2)$ so they'll know you mean $-9$. Commented May 5, 2023 at 6:47
• If you write $f(x)=-x^2$, then $f(3)$ surely shouldn't mean anything else than $-9$, right? If you claim that $f(3)=9$, then you're saying that $-x^2$ and $x^2$ are the same polynomials, which would be senseless. (By the way, it's “subtract”, not “substract”.) Commented May 5, 2023 at 6:48

This is a great Example highlighting the Case where 2 notations interfere to give ambiguity.

Short Summary : Mathematical Convention is almost universal : $$-3^2=-9$$ : Everything else is minority View-Point.

My Interpretation/View-Points/thoughts :

(A) The Integers greater than 0 ( Natural Numbers ) have names "Positive One" , "Positive Two" , "Positive Three". The shorter alternates names are +1 , +2 , +3.
The Prefixes "Positive" & "+" are generally not used because we generally want Positive numbers Eg year = 2023 & month = 5.
Thus the names are "One" , "Two" , "Three" & 1 , 2 , 3.

The Integers lesser than 0 have the names "Negative One" , "Negative Two" , "Negative Three". The shorter alternates names are -1 , -2 , -3 where we have to use a Prefix.

(B) Addition Operation (+) has the Inverse Subtraction Operation (-) which unfortunately look like the name Prefixes.
When we want Addition , we want $$X+Y$$.
To indicate Subtraction , we want $$X-Y$$.
Occasionally , $$X=0$$ , where we can leave that out to get $$+Y$$ & $$-Y$$ , which interferes without our naming Prefix.

(C) When ambiguity occurs , the notation we use must clarify what we want. That is why we have BODMAS & PEMDAS.
(C1) In general , $$-3^2$$ means $$0-3^2$$ : Evaluate the $$3^2$$ , then take the Inverse.
That is Equivalent to : We take $$-$$ to mean the Subtraction Operator , which wiki claims is almost universal in Contemporary Mathematics & very Common in most Programing languages.
(C2) Microsoft Excel takes the other stance : $$-3^2$$ means : take the number "Negative 3" , then Evaluate the $$(-3)^2$$. This is a minority View-Point.

(D) Entirely avoiding that ambiguity without Parenthesis can be achieved via alternate notation.

(D1) Unicode has alternate negation & unary minus.

(D2) Prefixes & Operators can be given Different Symbols. Historical Examples :
(Eg D2A) Prefix gets a little higher.
(Eg D2B) Bar on top of number.
(Eg D2C) Arrow before number.
(Eg D2D) Arrow on top of number.

Historical Examples were taken from Math Doctors & Wiki :

These have not become Popular , rather we go with the convention (C1) that $$-3^2=-9$$

The references :
Math Doctors : https://www.themathdoctors.org/negative-vs-minus-two-words-one-symbol/
Wiki : https://en.wikipedia.org/wiki/Plus_and_minus_signs
These two have more references to follow !

Remember that "$$-$$" is abbreviation of "$$-1\cdot$$". Hence $$-3^2=-1\cdot3^2.$$