Difference between -3² and (-3)² ?

[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.

 

[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.

I think I understand what you think you meant, but that's different from what you actually wrote. No calculator will display either -n or n if the input is -n^2. IOW, squaring a number won't produce that number except if the number is 1.

*sigh* Because of PEMDAS.

Parentheses
Exponents
Multiplication and Division
Addition and Subtraction

PEMDAS doesn't really cover it, as the minus sign is really the (unary) negation operator rather than the (binary) subtraction operator.

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.

Programming languages generally do a much better job of specifying the order of operator precedence than does mathematics. In addition, many languages also specify the associativity; i.e., how operators at the same precedence level are evaluated, whether left-to-right or right-to-left.

For example, in Python, exponentiation (denoted as **) is of higher precedence than negation (see https://www.geeksforgeeks.org/precedence-and-associativity-of-operators-in-python/). For an expression such as -3**2, 3 is squared before the result is negated. Because of the precedence of the two operators, -3**2 has exactly the same value as -(3**2).

 

[DaveC426913]

PEMDAS doesn't really cover it, as the minus sign is really the (unary) negation operator rather than the (binary) subtraction operator.

I think PEMDAS covers it implicitly, if not explicitly.
IOW, if one were to find a concise description of PEMDAS it might cover negation operators under subtraction operators.

Programming languages generally do a much better job of specifying the order of operator precedence than does mathematics.

Mathematics manages with its convention. You gotta know how to math.

 

[Mark44]

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 ???

This was answered in posts 4, 14, 19, 41, plus a couple more that I might have missed. Please go back and reread them.

I think PEMDAS covers it implicitly, if not explicitly.
IOW, if one were to find a concise description of PEMDAS it might cover negation operators under subtraction operators.

Not good enough if it isn't explicit. That's why I said that programming language do a much better job of explaining how expressions are evaluated. All of the descriptions of PEMDAS (or BIDMAS) I've ever seen were concise, which is the problem. And none of the letters in either acronym covers negation.

Mathematics manages with its convention. You gotta know how to math.

Yeah, I think I know "how to math."

 

[DrClaude]

This was answered in posts 4, 14, 19, 41, plus a couple more that I might have missed. Please go back and reread them.

One last time: $-3^2$ means "take the negative of the result of 3 squared".

There is nothing more to say, so this thread will now be closed.