Squaring negative values: (-13)^2 = (-13)*(-13) = +169, but what about -13^2 ?

[sqroot]

Hi,

I am after some clarification on squaring negative values and the correct way to apply orders of operation...

I understand that $-13 * -13 = 169$ because multiplying a negative by a negative gives a positive value as the result.

However, I am confused about calculating $-13^2$, Logic tells me this should also be 169 as $-13^2$ seems to be the same as $-13 * -13 = 169$. But if I think about the order of operations BODMAS or BIDMAS then I should calculate the $13^2$ before applying the negative value, so...
Step 1: Calculate the square of 13: $13 * 13 = 169$
Step 2: Apply the negative, giving $-169$

My Casio Calculator agrees that $-13^2 = -169$ (As does Google) (thus order of operation is being applied)
Every other reference (for example cuemath.com) I have read on the internet tells me that squaring negative values always gives a positive.

Is there a hidden rule states we should treat $-13^2$ as $(-13)^2$ ?

So is it 169 or -169 that is the result of $-13^2$ ? And why?

 

[stapel]

I understand that $-13 * -13 = 169$ because multiplying a negative by a negative gives a positive value as the result.

However, I am confused about calculating $-13^2$, Logic tells me this should also be 169 as $-13^2$ seems to be the same as $-13 * -13 = 169$.

There is a big difference between $(-13)^2$ and $-(13)^2$. In the former, the power is on the "minus" as well as the "thirteen"; in the latter, the power is only on the "thirteen".

 

[BigBeachBanana]

Hi,

I am after some clarification on squaring negative values and the correct way to apply orders of operation...

I understand that $-13 * -13 = 169$ because multiplying a negative by a negative gives a positive value as the result.

However, I am confused about calculating $-13^2$, Logic tells me this should also be 169 as $-13^2$ seems to be the same as $-13 * -13 = 169$. But if I think about the order of operations BODMAS or BIDMAS then I should calculate the $13^2$ before applying the negative value, so...
Step 1: Calculate the square of 13: $13 * 13 = 169$
Step 2: Apply the negative, giving $-169$

My Casio Calculator agrees that $-13^2 = -169$ (As does Google) (thus order of operation is being applied)
Every other reference (for example cuemath.com) I have read on the internet tells me that squaring negative values always gives a positive.

Is there a hidden rule states we should treat $-13^2$ as $(-13)^2$ ?

So is it 169 or -169 that is the result of $-13^2$ ? And why?

$-13^2=-1\times 13 \times 13=-169$
$(-13)^2=(-13)\times(-13)=169$

 

[topsquark]

Hi,

I am after some clarification on squaring negative values and the correct way to apply orders of operation...

I understand that $-13 * -13 = 169$ because multiplying a negative by a negative gives a positive value as the result.

However, I am confused about calculating $-13^2$, Logic tells me this should also be 169 as $-13^2$ seems to be the same as $-13 * -13 = 169$. But if I think about the order of operations BODMAS or BIDMAS then I should calculate the $13^2$ before applying the negative value, so...
Step 1: Calculate the square of 13: $13 * 13 = 169$
Step 2: Apply the negative, giving $-169$

My Casio Calculator agrees that $-13^2 = -169$ (As does Google) (thus order of operation is being applied)
Every other reference (for example cuemath.com) I have read on the internet tells me that squaring negative values always gives a positive.

Is there a hidden rule states we should treat $-13^2$ as $(-13)^2$ ?

So is it 169 or -169 that is the result of $-13^2$ ? And why?

Apparently, this one's been going around the internet. I've seen two posts on two different forums about $-3^2$ and now this one.

The answer to your question is given in your post: Exponentiation is done before multiplication:
$-13^2 = -1 \cdot 13^2 = -1 \cdot 169 = -169$

We have to have some way to define how to read these sorts of things, in Mathematics; it's BODMAS.

-Dan

 

[stapel]

Apparently, this one's been going around the internet. I've seen two posts on two different forums about $-3^2$ and now this one.

"Facebook strikes again"?

 

[Steven G]

What if you had 200 - 13²?
Think of -13² as 0 - 13²

 

[The Highlander]

My Casio Calculator agrees that $-13^2 = -169$ (As does Google) (thus order of operation is being applied)
Every other reference (for example cuemath.com) I have read on the internet tells me that squaring negative values always gives a positive.

The important thing here is to recognize the difference between an operator and a sign.

$\displaystyle -13^2 = -169 \$ but $\displaystyle ˉ13^2=169$ just the same as $\displaystyle (-13)^2 = 169$ (because $\displaystyle (-13)$ is just a (shorthand) way of representing $\displaystyle ˉ13$ without the need to find and use the negative sign character).

(Have look at the drivel that is being posted (by Elite members of this forum!) in this thread!)

If I enter the following sequence into my scientific calculator...

[6] [-] [2] [+/-] [x²] [=]
(I am using the square brackets to indicate the buttons being pushed)
​

The answer I get is 2 because the calculator is changing the 2 into ˉ2 (negative two) before squaring it!

The $\displaystyle -$ in $\displaystyle -13^2$ is an (Arithmetic) Operator.
The $\displaystyle ˉ$ in $\displaystyle ˉ13^2$ is a (Negative) Sign.​

 

[sqroot]

Hi all,

Thanks very much for your replies and helping me with this. I think you all pretty much confirmed that I was correct in my application of the order of operation and getting the negative value (and that brackets would be required around -13 in order to get the positive value).

I think what 'The Highlander' has replied with regarding the difference between an 'operator' and a 'sign' (a unary operator?) has helped me think about the problem differently and understand how/why I was seeing two different answers to the (apparently) same problem. I guess the intention in the question I was looking at (trying to help my lad when he was querying his maths homework) was that the -13 was to be interpreted as a sign and not as an operator. I think maybe I was overthinking things or being too pedantic.

Edit: In fact I have just checked the Casio calculator manual and it does state to explicitly add brackets (parentheses) for negative values (sign, unary operator). Here is the extract...



Thanks again all!

 

[BigBeachBanana]

The important thing here is to recognize the difference between an operator and a sign.

$\displaystyle -13^2 = -169 \$ but $\displaystyle ˉ13^2=169$ just the same as $\displaystyle (-13)^2 = 169$ (because $\displaystyle (-13)$ is just a (shorthand) way of representing $\displaystyle ˉ13$ without the need to find and use the negative sign character).

(Have look at the drivel that is being posted (by Elite members of this forum!) in this thread!)

If I enter the following sequence into my scientific calculator...

[6] [-] [2] [+/-] [x²] [=]
(I am using the square brackets to indicate the buttons being pushed)
​

The answer I get is 2 because the calculator is changing the 2 into ˉ2 (negative two) before squaring it!

The $\displaystyle -$ in $\displaystyle -13^2$ is an (Arithmetic) Operator.
The $\displaystyle ˉ$ in $\displaystyle ˉ13^2$ is a (Negative) Sign.​

I would say this.
The traditional use of the +/- button on four-function calculators requires you to indicate the negative after the number, which many find counterintuitive. Many modern calculators and software applications, starting from the mid-20th century, allow you to input a negative sign before the number to indicate a negative value, which is a more intuitive and user-friendly approach. As a result, the traditional +/- button may be considered less efficient or less intuitive by today's standards. With that said your usage of the superscript as an operator is not a common standard mathematical operation (at least for me).

PS: The built-in calculator on my iPhone 11 still operates on +/-

 

[lookagain]

If I enter the following sequence into my scientific calculator...

[6] [-] [2] [+/-] [x²] [=]
(I am using the square brackets to indicate the buttons being pushed)
​

If I enter [6] [-] [2] [(-)] [2] [x squared] [enter] into my TI-30XS scientific calculator, I get the result of 10.

If I enter it the same way on a graphing calculator such as a TI-83, TI-83 Plus, TI-84, or TI-89, I also get 10.

___________________________________________________________________________________________________________________

The Highlander posted:

(Have look at the drivel that is being posted (by Elite members of this forum!) in this thread!)

You're being uncivil against forum members. If you can't stop being uncivil when you post, then you should not post,
or be made not to post.

 

[BigBeachBanana]

If I enter [6] [-] [2] [(-)] [2] [x squared] [enter] into my TI-30XS scientific calculator, I get the result of 10.

If I enter it the same way on a graphing calculator such as a TI-83, TI-83 Plus, TI-84, or TI-89, I also get 10.

The traditional button [+/-] on a four-function calculator that Highlander is referring to is not the same as the [(-)] on modern calculators. The operation s/he was referring to is actually correct as there are no parenthesis on four-function calculators. The misunderstanding was addressed above in post#9. However, the use of superscript is not standard to indicate the [+/-] operator.

 

[Dr.Peterson]

I guess the intention in the question I was looking at (trying to help my lad when he was querying his maths homework) was that the -13 was to be interpreted as a sign and not as an operator.

No, the problem uses the standard "-" sign, which is used both as a subtraction operator, as a negation (additive inverse) operator, and a negative number indicator. @The Highlander explicitly changed the problem be replacing that with the "raised minus sign", which is not standard, and (at least in his usage) is inseparable from the number. Many people, in my experience (even experienced mathematicians) have never heard of it, which is part of the reason for the discussion you've seen here.

As written, we must take the sign as the negation operator, so that -13² means, according to the order of operations as (universally?) taught, the negative of the square.

I find few references to the raised sign, one of which is here:

The minus sign, −, has three main uses in mathematics:​

1. The subtraction operator: a binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition.
2. The function whose value for any real or complex argument is the additive inverse of that argument. For example, if x = 3, then −x = −3, but if x = −3, then −x = +3. Similarly, −(−x) = x.
3. A prefix of a numeric constant. When it is placed immediately before an unsigned numeral, the combination names a negative number, the additive inverse of the positive number that the numeral would otherwise name. In this usage, '−5' names a number the same way 'semicircle' names a geometric figure, with the caveat that 'semi' does not have a separate use as a function name.

In many contexts, it does not matter whether the second or the third of these usages is intended: −5 is the same number. When it is important to distinguish them, a raised minus sign ¯ is sometimes used for negative constants, as in elementary education, the programming language APL, and some early graphing calculators.​

But this is irrelevant to your question:

Is there a hidden rule states we should treat $-13^2$ as $(-13)^2$ ?

So is it 169 or -169 that is the result of $-13^2$ ? And why?

There is no hidden rule, just the explicit one you stated, by which the answer is -169.

The answer to your question is simply that calculators differ in how they handle negation, which accounts for all the variation you found (except for some people who fail to follow the rules they are taught).

 

[Steven G]

The answer to your question is given in your post: Exponentiation is done before multiplication:
$-13^2 = -1 \cdot 13^2 = -1 \cdot 169 = -169$


-Dan

Are you sure that the - sign is for multiplication rather than addition?
I clearly see you point of view: -13²= -1*13²
I see -13² as 0-13² and think of the - sign as a subtraction symbol
I wonder what the official answer is and why.

 

[Bruce]

Are you sure that the - sign is for multiplication rather than addition?

I see -13² as 0-13² and think of the - sign as a subtraction symbol

I think both viewpoints are ok.

I myself think of negation sign (for example: -x) means factor of -1, like topsquark shows.

But, MS Windows calculator uses your viewpoint. When I enter something like -4 it changes display to 0–4.

 

[Steven G]

Yes, I know that both viewpoints are ok. I just wonder if there is an official answer.

 

[Dr.Peterson]

Are you sure that the - sign is for multiplication rather than addition?
I clearly see you point of view: -13²= -1*13²
I see -13² as 0-13² and think of the - sign as a subtraction symbol
I wonder what the official answer is and why.

Yes, I know that both viewpoints are ok. I just wonder if there is an official answer.

Of course not. It's neither. (And you know very well that there is no "official" in math.)

The "-" in $-13^2$ is neither a subtraction nor a multiplication; what it is, is a negation. That negation can be thought of as equivalent to either a subtraction or a multiplication, and both suggest the same order of operations; but both of those require rewriting the expression. They are not what it is.

But, MS Windows calculator uses your viewpoint. When I enter something like -4 it changes display to 0–4.

Can you show an image of what you are talking about? What mode are you in, and what exactly did you enter? Is this what you have in mind?

​

The calculator on my computer has a "+/-" button (postfix change-sign), not a "(-)" button (prefix negative), for negation; the "-" button is explicitly for subtraction, not negation. What I entered here, "- 4", told it to subtract 4 from whatever was already in its display, which was 0. This has nothing at all to do with what it thinks about negation.

 

[khansaheb]

(Have look at the drivel that is being posted (by Elite members of this forum!) in this thread!)

↑
This requires an assignment in the corner.....

 

[stapel]

(Have look at the drivel that is being posted (by Elite members of this forum!) in this thread!)

(Have look at the drivel that is being posted (by Elite members of this forum!) in this thread!)

You're being uncivil against forum members. If you can't stop being uncivil when you post, then you should not post,
or be made not to post.

↑
This requires an assignment in the corner.....

Two-week time-out.