# Difference between -3² and (-3)² ?

• gary350

#### gary350

What is the difference between these to. -3² and (-3)² ?

I know - x - = +

I am told -3²= -9 not +9

-3² means -3 x -3 = 9

I am told (-3)² = 9

$$3^2=9$$
$$-3^2=-9$$
$$(-3)^2=(-1)^2 3^2 = 1 \cdot 9=9$$

e_jane, nuuskur and WWGD
Or, similarly,
## -3^2=-(3)(3)=-9##;
##(-3)^2=(-3)(-3)=3^2=9 ##

e_jane and jack action
##-3^2 = (-1) 3^2 = (-1) 9 = -9## // because the power operator has higher precedence than the negative operator

in contrast:

##(-3)^2 = (-3)*(-3) = 9## the parens modifies the precedence so that (-3)^2 = (-3)*(-3)

e_jane, jack action and Ibix
My calculator shows -3² = 9

My calculator also shows 3²=9

Both make sense because -x-=+

-x-=- makes no sense.

I only know real math symbols, what are ^ & * mean?

I makes no sense to pull a -1 out of the air and put it into a problem that has no -1 in it?

-3² should = -3 x -3 = 9

My TI-83+ says

What is the make and model number of your calculator?

e_jane and Dale
I only know real math symbols, what are ^ & * mean?

I makes no sense to pull a -1 out of the air and put it into a problem that has no -1 in it?

-3² should = -3 x -3 = 9
^ and * are real Math symbols. They are "ASCII" code ("American Standard Code for Information Interchange") characters.

^ represents exponentiation: 3^5 is ##3^5##

and
* represents multiplication: 3*5 is ##3 \times 5##.

You see ASCII code all the time on forums like this, when the typist doesn't know LaTeX coding, or some other character code that gives them a more "textbooky" look.

As mentioned above, PEDMAS (or BODMAS, or whatever version of order of operations you prefer) says that exponentiation happens before mutliplication:
P - Parenthesis
E - Exponents
D - Division
M - Multiplication
S- Subtraction

When we see -a, we take that to mean ##-1 \times a##. So ##-3^2## says ##-1 \times 3^2##, so square the three first, then multiply by -1. We aren't pulling anything out of the air:
##-3^2 = -1 \times 3^2 = -1 \times 9 = -9##

-Dan

e_jane
My calculator shows -3² = 9

My calculator also shows 3²=9

Both make sense because -x-=+

-3² should = -3 x -3 = 9
By convention ##-3^2 \equiv -(3^2)##. For example, there is a clear difference (by convention) between these two quadratic expressions:
$$-x^2 + bx + c \not\equiv x^2 + bx+c$$This convention is well established, so you ought to learn it.

e_jane and topsquark
PS one of the most useful factorisations is $$x^2 - y^2 = (x +y)(x-y)$$No one is going to interpret ##x^2 - y^2## as the same as ##x^2 + y^2##. This is hard-wired into modern mathematics, so there is no point in arguing against it.

topsquark
I have been out of college 50 years. If you don't use it you loose it. I have forgotten about 90% of what I once knew, maybe more. I took every math class the college had expect, imaginary numbers. I know -x-=+ and 3²=9 and -3² should =9 also. When people add -1 to -3² I have no clue what your doing? The x & y examples don't help we under stand how -3² can be a -9 ? Why is +3² different than -3² math rules tell me - x - has to be + and 3 x 3 has to be +9 not -9.

I have been out of college 50 years. If you don't use it you loose it. I have forgotten about 90% of what I once knew, maybe more.
Right, so you'd forgotten that ##-x^2 \equiv -(x^2)##. That's something you've relearned. Good?

e_jane
My calculator shows -3² = 9

Depending on the calculator, it might be that the sequence of keys [3] [-] [x2] is interpreted as $(-3)^2$ not $-(3^2)$.

I only know real math symbols, what are ^ & * mean?

^ and * are real Math symbols.
They aren't really math symbols, at least they aren't symbols that are commonly used in math textbooks. The caret (^) was first used in BASIC, I believe, to represent exponentiation; i.e., raising a number to some power. Very few other programming languages use ^ for this purpose, however. The asterisk (*) is universally used in programming languages to represent multiplication. Both symbols are commonly used in internet forums to represent these operations.

DrClaude, topsquark and PeroK
My calculator shows -3² = 9
Just to be clear: If you type ##-3## into your calculator and hit [enter] or '=', then that part is done immediately and the calculator will store -3 somewhere. Then, when you square that, the answer will be ##(-3)^2 = 9##.
That is not the same as ##- 3^2 = - (3^2) = -9##.
A calculator where you can enter a calculation like ##-3^2## in one step, without any intermediate [enter] or '=' should give you the correct answer of ##-9##.

topsquark, Mark44 and PeroK
Since you're trying to relearn long lost math, perhaps www.mathispower4u.com would help.

Its a free website of over 5000 ten-minute videos walking you thru the steps to solve a problem presented at the start of the video.

It covers the full range of highschool thru first year college math.

topsquark
Why is +3² different than -3² math rules tell me - x - has to be + and 3 x 3 has to be +9 not -9.
Right, a negative times a negative is positive, but that's not what you have with ##-3^2##. This means literally, the negative of ##3^2##, not ##(-3) \times (-3)##.
Also "- x -" is a bit confusing. It took me a little while to get that you meant "negative times negative."

topsquark and PeroK
The original problem is, -3²= ?

There is no way to put parentheses in my calculators. I put -3 in both calculators then push the square button and the answer on both calculators is -3²=9

Everyone adds parentheses to this for some reason claiming it needs parentheses.

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The standard convention of modern mathematics is that ##-3^2 = -9##. This thread has many explanations of why.

Calculators are not infallible in this respect.

topsquark
The original problem is, -3²= ?
##-3^2=-9##

There is no way to put parentheses in my calculators. I put -3 in the calculator then push the square button and the answer on both calculators is -3²=9
Then you used your calculator wrong. You calculated ##(-3)^2## instead of ##-3^2##.

To calculate ##-3^2## on your calculator you need to put ##3## in the calculator, then push the square button, and then push the ##-## button which on some calculators may be marked ##\pm##.

If the calculator doesn't have parentheses then it requires you to provide the correct order of operations manually. By entering the ##-## first you incorrectly told the calculator that ##-## had a higher precedence than ##{}^2##. So you told it to calculate ##(-3)^2##

Algr and topsquark
As others have noted this is a convention: ##-3^2## is to be read as the negative of the square of three. There is method in the madness, though. See the pattern:$$\begin{eqnarray*} 4-3^2&=&-5\\ 3-3^2&=&-6\\ 2-3^2&=&-7\\ 1-3^2&=&-8\\ 0-3^2&=&-9 \end{eqnarray*}$$Then ask yourself if you'd want ##0-3^2## to be different from ##-3^2##. Should adding a zero make a difference?

romsofia, Dale, topsquark and 1 other person

The only one I know that has no parentheses are HP Reverse Polish Notation calculators. They use RPN style notation which means you input your equations differently than with a TI-83 calculator as an example.

topsquark
The original problem is, -3²= ?

There is no way to put parentheses in my calculators. I put -3 in both calculators then push the square button and the answer on both calculators is -3²=9

Everyone adds parentheses to this for some reason claiming it needs parentheses.

Then you will have to do the calculation in two steps in the correct order yourself and not count on the calculator. You will have to calculate ##3^2 = 9## first, and then reverse the sign to get ##-3^2 = -9##.

If you are going to get into this in a significant way, use the more advanced calculator.

topsquark
The only one I know that has no parentheses are HP Reverse Polish Notation calculators.
I could be wrong, but I seem to recall that some of the cheaper calculators don't have parentheses, but don't use RPN. Also, the Windows calculator, in Standard mode, doesn't have parentheses and doesn't do RPN. To calculate ##-3^2##, you enter 3, click ##x^2##, and then click +/-.
If you enter 3, then ##x^2##, then +/-, the result is -9.

jedishrfu and FactChecker
Yes, I think you're right. I recall having on in the 1970s that did simple arithmetic and we had to enter the calculation according to standard math rules. Sums could be stored so you could do a more complex
a*b+c*d*e+ f*g*h ....

There are articles online on how to work without parentheses:

https://study.com/skill/learn/using-the-order-of-operations-without-parentheses-explanation.html

Of course, the WIndows 10 calculator doesn't have parens

and my iPhone simple arithmetic calculator doesn't either (ie in portait mode) However switching to landscape mode and you get a full blown scientific calculator complete with parentheses.

I have been out of college 50 years. If you don't use it you loose it. I have forgotten about 90% of what I once knew, maybe more. I took every math class the college had expect, imaginary numbers. I know -x-=+ and 3²=9 and -3² should =9 also. When people add -1 to -3² I have no clue what your doing? The x & y examples don't help we under stand how -3² can be a -9 ? Why is +3² different than -3² math rules tell me - x - has to be + and 3 x 3 has to be +9 not -9.
If you have been within a metric parsec of Facebook anytime in the last MegaFortnight you have been inundated with PEMDAS riddles like this.

## -3^{2}=-9 ## but ## (-3)^2=(-3)(-3)=9 ##.

All of this commentary on calculators reminds me of a story when I was a physics undergrad at my college and calculators had just come out. We used slide rules for most of our computational tasks.

We were checking out a freshman lab, and some students were doing some sort of electrical experiment where they needed to compute the voltage across a 1.5v battery. We asked what voltage they got, and the freshman replied somewhere around 1536v.

We asked: "How did you arrive at that value ?" And the answer we got was that's what the calculator said.

Mic drop.

Time to get back to our senior studies of Quantum Mechanics, the younger generation is hopeless.

## -3^{2}=-9 ## but ## (-3)^2=(-3)(-3)=9 ##.
Yes, that's what we've been saying throughout this thread.

Gavran, Math100 and jedishrfu
My calculator shows -3² = 9

If your enter is something like 'minus, three, square' then (if it's anything decent) it'll do the square part first and the 'minus' last
If your enter is 'three, +/-, square' then it'll take it as a (minus-three) on square

The only one I know that has no parentheses are HP Reverse Polish Notation calculators.
I've seen many cheap (non-decent) ones going with in-order execution
But those rarely had x2 button...

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DeBangis21
We may share how windows10/11 calculator works.

My calculator shows -3² = 9

I have been out of college 50 years.
Put your calculator away and try ##-3^2## on your old slide rule.

I can't believe this thread has 30 posts. Oops, 31.

Math100, e_jane and Mark44
Thanks. I am getting something in here. I use to wonder and always asked myself, and others, why -n^2 in my calculator gives -n instead of n.

PeroK

The only one I know that has no parentheses are HP Reverse Polish Notation calculators. They use RPN style notation which means you input your equations differently than with a TI-83 calculator as an example.
As a historical note…. RPN calculators were developed and brought to market back in the old days when no calculators supported parentheses or understood PEMDAS, which made them far less useful for scientific and technical work. The internal logic of an RPN calculator is appreciably simpler than that of a PEMDAS-aware calculator, which mattered when designing a palm-sized device using 1970s technology.

I still prefer the RPN style, as it matches the way we would do the problem unassisted: calculate the highest-precedence intermediate results first and work out the final lowest-precedence step. The ambiguity in ##-3^2## discussed in this thread doesn’t happen in RPN.

Bystander, e_jane, Rive and 1 other person
As a historical note…. RPN calculators were developed and brought to market back in the old days when no calculators supported parentheses or understood PEMDAS, which made them far less useful for scientific and technical work. The internal logic of an RPN calculator is appreciably simpler than that of a PEMDAS-aware calculator, which mattered when designing a palm-sized device using 1970s technology.

I still prefer the RPN style, as it matches the way we would do the problem unassisted: calculate the highest-precedence intermediate results first and work out the final lowest-precedence step. The ambiguity in ##-3^2## discussed in this thread doesn’t happen in RPN.
My take on this as always been: Never trust to a machine what you can do for yourself.

Math100, Nugatory and e_jane
As if one more response would be any bit helpful after so many 34 posts --
The grouping pair symbols show what the boundary is of the expression; even if this expression just a single real number which may also be a signed number.