Difference between -3² and (-3)² ?

[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

 

[anuttarasammyak]

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

 

[WWGD]

Or, similarly,
$-3^2=-(3)(3)=-9$;
$(-3)^2=(-3)(-3)=3^2=9$

 

[jedishrfu]

$-3^2 = (-1) 3^2 = (-1) 9 = -9$ // 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)

 

[gary350]

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

 

[robphy]

My TI-83+ says

What is the make and model number of your calculator?

 

[topsquark]

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
A - Addition
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

 

[PeroK]

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.

 

[PeroK]

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.

 

[gary350]

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.

 

[PeroK]

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?

 

[pasmith]

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).

 

[Mark44]

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.

 

[FactChecker]

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 $-9$.

 

[jedishrfu]

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.

 

[Mark44]

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."

 

[gary350]

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.

I asked the wrong question on this thread. I should have ask what is, -3²= ?

 

[PeroK]

I asked the wrong question on this thread. I should have ask what is, -3²= ?

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.

 

[Dale]

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 $(-3)^2$

 

[Ibix]

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?

 

[jedishrfu]

What is your calculator?

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.

 

[FactChecker]

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.

I asked the wrong question on this thread. I should have ask what is, -3²= ?

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.

 

[Mark44]

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]

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.

 

[DaveC426913]

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.

 

[Math100]

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

 

[jedishrfu]

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.

 

[Mark44]

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

Yes, that's what we've been saying throughout this thread.

 

[Rive]

My calculator shows -3² = 9

Know your calculator

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...

 

[anuttarasammyak]

We may share how windows10/11 calculator works.

 

[gmax137]

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.

 

[DeBangis21]

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.

 

[Nugatory]

What is your calculator?

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.

 

[DaveC426913]

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.

 

[symbolipoint]

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.

 

[Rive]

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.

The first time I met with RPN was when I found a really cool looking, 'vintage, for collectors' calculator (with that classic LED display: the kind with the magnifying lenses, and with the numbers scurrying and rolling while it's sweating with the job) and noticed that I cannot do any calculations with it...

Kind of:

 

[Nugatory]

That TI calculator has an “=“ key, which identifies it as a non-RPN calculator, as we’d expect from a TI calculator of the era. The RPN calculators were made by HP and had an “enter” button that did somethint completely different. The presence or absence of an ”=“ key was a big piece of the advertising wars between the two companies.

 

[Rive]

Thanks. It was some 30+ years ago (when cheap LCD with '=' was already common) and the LEDs made the most impact - and that I could not figure it out.
By next day the battery was down and with that the thing got forgotten. I could connect the dots only when some years later at the university somebody explained the RPN properly.

 

[jtbell]

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

Gee, I never knew that! I haven't used that calculator in a loooong time, ever since I found out about the RLM-11CX calculator which emulates my old HP-11C. It of course doesn't have parentheses because it uses RPN input.

 

[gary350]

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.

I want to learn why -3²=9 ???

-x-=+

Square means a number times itself.

-3² should be -3 x-3= +9

I need to understand why the correct answer is -9 ???

 

[symbolipoint]

I want to learn why -3²=9 ???

-x-=+

Square means a number times itself.

-3² should be -3 x-3= +9

I need to understand why the correct answer is -9 ???

Was this already settled several posts ago at least a few times?

-3^2
The negative sign is NOT attached to the 3.
The meaning is -(3)^2 or same as -(3^2).
the exponent 2 is attached to the 3.

 

[FactChecker]

I want to learn why -3²=9 ???

-x-=+

Square means a number times itself.

-3² should be -3 x-3= +9

I need to understand why the correct answer is -9 ???

PEMDAS, You must learn it and make it second nature before proceeding to anything else in math.

 

[DaveC426913]

I want to learn why -3²=9 ???

*sigh* Because of PEMDAS.

Parentheses
Exponents
Multiplication and Division
Addition and Subtraction

By mathematical convention, as taught in all modern mathematics, that is the order in which such an equation is solved.

So: when you see this: -3², you solve this ( 3² ) first.
The minus sign ( - ) in front is an operator, just like addition, multiplication or exponentiation. It operates on what is immediately to its right (i.e. 3² AKA +9 ) to turn it into its operative inverse. That is what that minus sign means. And so, it gets processed in PEMDAS order - i.e. last.

 

[Rive]

The minus sign ( - ) in front is an operator

Erm.
Depending on the exact input sequence a '-' can be either an operator or mean a negative value.

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

Also, if the '-3' is produced by a calculation then it'll mean a negative value and will be processed so.

Entering the negative values by starting the input sequence with '-' is IMHO a mistake and should be avoided.

 

[FactChecker]

Erm.
Depending on the exact input sequence a '-' can be either an operator or mean a negative value.

Negative value of what? Nothing should be evaluated until the input is complete. Apparently, some calculators do not allow that. IMO, PEMDAS makes $-3^2$ unambiguous no matter how it is entered.
If there is any confusion, parentheses should be used to remove any doubt about the order of the calculations.

 

[Ibix]

I want to learn why -3²=9 ???

-x-=+

Square means a number times itself.

-3² should be -3 x-3= +9

I need to understand why the correct answer is -9 ???

Because, as mentioned several times, the convention is that $-3^2$ is to be read as minus the square of three and not as the square of minus three. As others have commented this is part of a more general system often called PEMDAS (you may have learned BIDMAS or BODMAS and some other mnemonic, but they're all the same) which is a convention for the order in which operations are done.

So it's basically the same reason as why 2×3 + 4×5 is 26 and not 50 - because the order of operations is what it is, not just reading left-to-right.

 

[Rive]

IMO, PEMDAS makes unambiguous no matter how it is entered.

Basic calculators has their own logic (due lack of parentheses), and if you don't understand them then it'll be ambiguous no matter how many PEMDAS got referenced.

This Windows-thing is fortunately able to display the actual operation so you can check what happens
Sequence of 'minus, three, square' => 0-sqr(3), and that's -9, proper
Sequence of 'three, +/-, square' => sqr(-3), and that's 9, proper
Basic calculators do the same operations but without allowing it to be checked

Thus: 'know your calculator'.

 

[DaveC426913]

Erm.
Depending on the exact input sequence a '-' can be either an operator or mean a negative value.

1. I'm not talking about calculators; I'm answering gary's question. Calculator behavior is a side-quest in this discussion.
2. The problem with this discussion has to do with the terminology "a negative value" (as FC pointed out: "of what?"). I'm clarifying, by pointing out that - while the minus sign does apply to the value to its right - it is an operation (it is not simply intrinsic), and that operation has to wait for order of precedence.

 

[Rive]

I'm answering gary's question.

He was the one bringing in calculators and expressing his confusion about them:

My calculator shows -3² = 9

My calculator also shows 3²=9

Both make sense because -x-=+

-x-=- makes no sense.

Guess not attending his calculator properly might be the reason he does not feel even this many answers satisfactory. I don't know.

I'm clarifying, by pointing out that - while the minus sign does apply to the value to its right - it is an operation (it is not simply intrinsic), and that operation has to wait for order of precedence.

If there is nothing left on the left side of zero but only the right side with a mirror then I'm out of this 'precalculus mathematics homework help' topic.

 

[DaveC426913]

He was the one bringing in calculators and expressing his confusion about them:

Perhaps. But 'calculators' is not 'the reason why'; it is still a side quest, even for the OP.

Guess not attending his calculator properly might be the reason he does not feel even this many answers satisfactory. I don't know.

Possibly. It is a bad carpenter that blames his tools.

If there is nothing left on the left side of zero but only the right side with a mirror then I'm out of this 'precalculus mathematics homework help' topic.

?

My take on this is to disabuse the OP of the notion that the minus sign is there "from the start". as if it is a "property" of the value to its right.

We're trying to get him to perform the operations in order and to do that, we need to demonstrate that the operations are discreet and in-series.