Why Does Raising a Negative Number to a Power Yield a Negative Result?

[ecoo]

How come when you do something like -3^2=-9. Does it mean that the dash is a subtraction symbol and not a negative symbol?

 

[Matterwave]

This is basically convention. When you see something like -3^2 without parenthesis, you take the exponent of the positive number 3 and then leave the negative symbol. Essentially -3^2 is shorthand for -(3^2). We put the parenthesis between the number and the exponent by convention. If you want to get the positive number 9, you would write (-3)^2=9.

Really, this is just convention, like reading an equation from left to right, or PEMDAS.

 

[ecoo]

This is basically convention. When you see something like -3^2 without parenthesis, you take the exponent of the positive number 3 and then leave the negative symbol. Essentially -3^2 is shorthand for -(3^2). We put the parenthesis between the number and the exponent by convention. If you want to get the positive number 9, you would write (-3)^2=9.

Really, this is just convention, like reading an equation from left to right, or PEMDAS.

Sure, but what I was wondering is that the dash makes the number a negative. Why would you have to put it in parenthesis? The dash is simply showing the number is negative. Why would you put 3^2 in parenthesis then multiply by negative 1? Just as if 3 was positive it would be +3. Isn't the dash just saying the number is negative?

 

[Mentallic]

And keep in mind that the reason (-3)²=9 is because

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

 

[ecoo]

And keep in mind that the reason (-3)²=9 is because

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

So a negative number is always -1 multiplied by that number? -3 = -1 * 3, but can't it as well just be 1 * -3? Negative 3 is just another number like positive 3, I thought.

 

[Matterwave]

Sure, but what I was wondering is that the dash makes the number a negative. Why would you have to put it in parenthesis? The dash is simply showing the number is negative. Why would you put 3^2 in parenthesis then multiply by negative 1? Just as if 3 was positive it would be +3. Isn't the dash just saying the number is negative?

It's just the way people decided to do things. There's nothing really wrong if you define -3^2 to equal (-3)^2, but we defined it the other way as -(3^2).

People could be completely pedantic and just demand the usage of parenthesis everywhere. So that 1+2*3 must be written 1+(2*3). But this is a hassle, so we use PEMDAS. This eliminates the need for a lot of parentheses. Exponentiation comes before all other operations (other than parenthesis) so you should read -3^2 as -(3^2).

 

[ecoo]

This is basically convention. When you see something like -3^2 without parenthesis, you take the exponent of the positive number 3 and then leave the negative symbol. Essentially -3^2 is shorthand for -(3^2). We put the parenthesis between the number and the exponent by convention. If you want to get the positive number 9, you would write (-3)^2=9.

Really, this is just convention, like reading an equation from left to right, or PEMDAS.

If it's convention wouldn't it be better to just write (-1)3^2 instead of saying just -3^2 = -9 which is a bit confusing? (Or maybe it's just me :D)

 

[ecoo]

It's just the way people decided to do things. There's nothing really wrong if you define -3^2 to equal (-3)^2, but we defined it the other way as -(3^2).

People could be completely pedantic and just demand the usage of parenthesis everywhere. So that 1+2*3 must be written 1+(2*3). But this is a hassle, so we use PEMDAS. This eliminates the need for a lot of parentheses. Exponentiation comes before all other operations (other than parenthesis) so you should read -3^2 as -(3^2).

OHHHH I see now! :D thanks man. Now it is clear!

 

[Mentallic]

So a negative number is always -1 multiplied by that number? -3 = -1 * 3, but can't it as well just be 1 * -3? Negative 3 is just another number like positive 3, I thought.

-3 IS just another number, but each number can be expressed in varying ways, and it turns out that -3 = -1*3 helps us, while -3 and 1*-3 give us no extra insight into why (-3)²=9 and not -9. If you know the rule $(ab)^n=a^n\times b^n$ then you can use the fact that $-3=-1\times 3$ to prove to yourself that it is indeed the case.

 

[ecoo]

-3 IS just another number, but each number can be expressed in varying ways, and it turns out that -3 = -1*3 helps us, while -3 and 1*-3 give us no extra insight into why (-3)²=9 and not -9. If you know the rule $(ab)^n=a^n\times b^n$ then you can use the fact that $-3=-1\times 3$ to prove to yourself that it is indeed the case.

Yes yes I see that :D but just as you can say 3 = -1 * -3. I forgot about pemdas, but now I see. I just think it makes more sense (to me) to say that -3^2 = 9 instead of -9 (but of course mathematicians follow pemdas, so yea :D)

 

[symbolipoint]

Yes yes I see that :D but just as you can say 3 = -1 * -3. I forgot about pemdas, but now I see. I just think it makes more sense (to me) to say that -3^2 = 9 instead of -9 (but of course mathematicians follow pemdas, so yea :D)

You are going backward from what you began to understand.

3^2 is 9.
+3^2 is 9. The plus sign was not needed.
-3^2 is the OPPOSITE of 3^2; therefore -3^2 is the opposite of 9.

 

[Matterwave]

Yes yes I see that :D but just as you can say 3 = -1 * -3. I forgot about pemdas, but now I see. I just think it makes more sense (to me) to say that -3^2 = 9 instead of -9 (but of course mathematicians follow pemdas, so yea :D)

Your suggestion would get really confusing in expressions like 1-3^2. We should be consistent in using PEMDAS, otherwise, one would have to always write 1-(3^2).

 

[HallsofIvy]

How come when you do something like -3^2=-9. Does it mean that the dash is a subtraction symbol and not a negative symbol?

That is a result of the "precedence" conventions for arithmetic: PEMDAS (or "Please Excuse My Dear Aunt Sally") Any calculations in Parentheses are done first, then exponents, then multiplication and division, then addition and subtraction. Here the exponentiation is done first, 3^2= 9, then the "multiplying" by -1. (It really doesn't matter whether you think of this as "negation" or "subtraction" from 0.)

 

[ecoo]

You are going backward from what you began to understand.

3^2 is 9.
+3^2 is 9. The plus sign was not needed.
-3^2 is the OPPOSITE of 3^2; therefore -3^2 is the opposite of 9.

But (-3)^2 =9. I'm wondering why you have to write it in parenthesis to get (-3) * (-3) = 9

 

[ecoo]

That is a result of the "precedence" conventions for arithmetic: PEMDAS (or "Please Excuse My Dear Aunt Sally") Any calculations in Parentheses are done first, then exponents, then multiplication and division, then addition and subtraction. Here the exponentiation is done first, 3^2= 9, then the "multiplying" by -1. (It really doesn't matter whether you think of this as "negation" or "subtraction" from 0.)

But isn't negatuve 3 just another number like positive ? So they negative sign is just telling that 3 is negatuve. Why would you have to put that in parenthesis?

-3^2 to me should still me -3 * -3

 

[jbunniii]

But isn't negatuve 3 just another number like positive ? So they negative sign is just telling that 3 is negatuve. Why would you have to put that in parenthesis?

-3^2 to me should still me -3 * -3

If we used your convention, such that negation has higher priority than exponentiation, then our notation for polynomials starts to fall apart. For example, under your convention,
$$x^2 - y^2 = x^2 + (-y)^2 = x^2 + y^2$$
or maybe, if we don't generously insert the implied $+$, we interpret it as follows:
$$x^2 - y^2 = (x^2)(-y)^2 = x^2 y^2$$
If we wanted to subtract the square of $y$ from the square of $x$ we would have to write
$$x^2 - (y^2)$$
Also, we would have the curious situation that
$$-x^2 = (-x)^2 = x^2$$
but
$$-1x^2 = (-1)(x^2) = -(x^2)$$
so now we can no longer even multiply by $1$ without potentially changing the value.

 

[Mentallic]

But isn't negatuve 3 just another number like positive ? So they negative sign is just telling that 3 is negatuve. Why would you have to put that in parenthesis?

-3^2 to me should still me -3 * -3

Like Matterwave brought up in post #12, expressions such as 1-3² would get confusing if it were calculated the way you've proposed.

 

[ecoo]

I get it now after reading this
http://mathforum.org/library/drmath/view/61523.html

So i was right in thinking the dash is actually treated as a subtraction symbol?

 

[adjacent]

I get it now after reading this
http://mathforum.org/library/drmath/view/61523.html

So i was right in thinking the dash is actually treated as a subtraction symbol?

Of course. If you have a number like :$-5$ it's actually $0-5$

 

[gopher_p]

But isn't negatuve 3 just another number like positive ? So they negative sign is just telling that 3 is negatuve. Why would you have to put that in parenthesis?

-3^2 to me should still me -3 * -3

I think maybe where you're getting confused is that you think that the grouping of sybmbols "$-3^2$" has inherent meaning. It does not. It's just a bunch of squiggles on the page until we've given it meaning.

If you believe in alternate universes, then in one of them "$-3^2$" is interpreted as meaning $(-3)\times(-3)=9$, and this is perfectly reasonable. In this universe, however, we have chosen to interpret "$-3^2$" as meaning $-(3\times3)=-9$.

Now others here have tried to persuade you by giving you reasons why the notation means what it means. I'm going to tell you that there are no reasons. It's (basically) completely arbitrary. There are two completely reasonable interpretations, and as a group we decided a long time ago to use the second. It's fine if you disagree with that decision; I have a ton of issues with mathematical notation that I think is "wrong". But disagreeing with the chosen notation does not change the fact that, to everyone that "matters", "$-3^2$" means $-(3\times 3)$.

Now to those here that argue that the notation would get confusing if we chose the alternate interpretation or a different order of operations, understand two things; (1) it only seems confusing to you because you are so used to the commonly accepted conventions and (2) the conventional notation is as confusing to your students (if not more so) than unconventional notation is to you. In a different universe, the distributive law looks like $c\times a+b=(c\times a)+(c\times b)$, and in another it looks like $c+(a\times b)=(c+a)\times(c+b)$. This isn't because the math is different in that universe; it's that the symbols used to express the mathematics are interpret differently.

 

[Matterwave]

I think maybe where you're getting confused is that you think that the grouping of sybmbols "$-3^2$" has inherent meaning. It does not. It's just a bunch of squiggles on the page until we've given it meaning.

If you believe in alternate universes, then in one of them "$-3^2$" is interpreted as meaning $(-3)\times(-3)=9$, and this is perfectly reasonable. In this universe, however, we have chosen to interpret "$-3^2$" as meaning $-(3\times3)=-9$.

Now others here have tried to persuade you by giving you reasons why the notation means what it means. I'm going to tell you that there are no reasons. It's (basically) completely arbitrary. There are two completely reasonable interpretations, and as a group we decided a long time ago to use the second. It's fine if you disagree with that decision; I have a ton of issues with mathematical notation that I think is "wrong". But disagreeing with the chosen notation does not change the fact that, to everyone that "matters", "$-3^2$" means $-(3\times 3)$.

Now to those here that argue that the notation would get confusing if we chose the alternate interpretation or a different order of operations, understand two things; (1) it only seems confusing to you because you are so used to the commonly accepted conventions and (2) the conventional notation is as confusing to your students (if not more so) than unconventional notation is to you. In a different universe, the distributive law looks like $c\times a+b=(c\times a)+(c\times b)$, and in another it looks like $c+(a\times b)=(c+a)\times(c+b)$. This isn't because the math is different in that universe; it's that the symbols used to express the mathematics are interpret differently.

Although it is true that notational convention is notational convention and is completely arbitrary, if you change only ONE piece of notational convention, without changing the others, you will most likely get unresolvable ambiguities. If the OP wants to change -3^2 to mean (-3)^2, then he has to change basically ALL of our conventions around to make them consistently interpretable. This was what I was trying to get at earlier when I said statements such as 1-3^2 would be ambiguous if we chose to keep PEMDAS. Of course, we can very well discard PEMDAS, and choose 1-3^2=1+(-3)^2 with implicit addition symbols...but...then there's other rules we have to define!

 

[gopher_p]

Although it is true that notational convention is notational convention and is completely arbitrary, if you change only ONE piece of notational convention, without changing the others, you will most likely get unresolvable ambiguities. If the OP wants to change -3^2 to mean (-3)^2, then he has to change basically ALL of our conventions around to make them consistently interpretable. This was what I was trying to get at earlier when I said statements such as 1-3^2 would be ambiguous if we chose to keep PEMDAS. Of course, we can very well discard PEMDAS, and choose 1-3^2=1+(-3)^2 with implicit addition symbols...but...then there's other rules we have to define!

If you decide that subtraction/negation takes precedence over exponentiation, as is the case with the interpretation $-3^2=(-3)^2$, then $1-3^2=(1-3)^2$.

 

[Matterwave]

If you decide that subtraction/negation takes precedence over exponentiation, as is the case with the interpretation $-3^2=(-3)^2$, then $1-3^2=(1-3)^2$.

But this is assuming that the OP takes subtraction from 0 and the symbol -x to be equivalent procedures. Is that THE ONLY possible way? It seems to me that the OP, at least initially, thought that there was a fundamental difference between seeing 0-3 and -3. If this is so, then in his new notation, there would seem to be a possibility of a difference in ordering between 1-3^2 and 1+-3^2.

Is this not an ambiguity?

 

[jbriggs444]

There is certainly a distinction that can be drawn between negation (a unary operation) and subtraction (a binary operation). It would not be unreasonable to assign different precedence to the two operators. For instance, one might decide, contrary to accepted practice, that unary negation should bind more tightly than exponentiation while still allowing binary subtraction to bind less tightly.

[I can't think of a reasonable meaning for unary negation that would be different from the result obtained by subtraction from zero, but that may be simply lack of imagination on my part]

One could even decide that the minus sign should be interpreted as part of a numeric literal such as "-3". If you are doing a first pass over an equation with a lexical analyzer looking for syntax elements, this approach might seem attractive at first glance. However it becomes problematic when using juxtaposition to indicate multiplication.

Would 1-3 then be interpreted as the multiplication of 1 and -3 or as the subtraction of 3 from 1?

 

[ecoo]

So from what I am thinking, would it be wrong to think that in the equation -3^2 = -9, then the dash in front of the 3 is a subtraction symbol. So if we have subtraction a dash symbol and negative numbers a superscript dash symbol, then the dash in front of 3 would be a regular dash and the dash in front of 9 would be superscript dash.

So it would seem that if the dash in front of 3 were superscript, then the answer would be positive 9.

 

[micromass]

So from what I am thinking, would it be wrong to think that in the equation -3^2 = -9, then the dash in front of the 3 is a subtraction symbol. So if we have subtraction a dash symbol and negative numbers a superscript dash symbol, then the dash in front of 3 would be a regular dash and the dash in front of 9 would be superscript dash.

So it would seem that if the dash in front of 3 were superscript, then the answer would be positive 9.

There aren't different types of dash symbols. You only have two types.

1) Substraction. This is a binary operation between two numbers. Like $3-9$ and $1-1$.

2) Negative of a number. This is a unary operation on one number, like $-3$.

The two are linked by $0-x=-x$ and $x-y = x+(-y)$.

So $-3^2$ can be seen as $0-3^2$ if you want, but the result is still $-9$. The only way you can get this to yield $9$ is by inserting brackets.

 

[ecoo]

There aren't different types of dash symbols. You only have two types.

1) Substraction. This is a binary operation between two numbers. Like $3-9$ and $1-1$.

2) Negative of a number. This is a unary operation on one number, like $-3$.

The two are linked by $0-x=-x$ and $x-y = x+(-y)$.

So $-3^2$ can be seen as $0-3^2$ if you want, but the result is still $-9$. The only way you can get this to yield $9$ is by inserting brackets.

Yes yes I have nothing wrong with the answer, but to get that answer the dash must represent subtraction, right? Parenthesis around dash and the number (e.g. (-3)^2) is just to clarify that the dash is the symbol for negative and not subtraction.
So -3^2 is actually 0 - (3^2), which is the subtraction of the square of a positive number.

 

[micromass]

Yes yes I have nothing wrong with the answer, but to get that answer the dash must represent subtraction, right? Parenthesis around dash and the number (e.g. (-3)^2) is just to clarify that the dash is the symbol for negative and not subtraction.

No. The dash in $-x$ just means "take the negative of $x$".
Similarly, if we write $-3^2$, then this means "take the negative of $3^2$," which is $-9$.

 

[ecoo]

No. The dash in $-x$ just means "take the negative of $x$".
Similarly, if we write $-3^2$, then this means "take the negative of $3^2$," which is $-9$.

So what you are saying is that -3^2 = -1(3^2)

So like the subtraction, the dash is not stating the 3 is negative, it is either saying subtraction or multiplying by -1 (that is a negative 1 confirmed) :D

I also added a bit more on previous post.

 

[ecoo]

OHHHHH I think I see what you guys are talking about.

http://mathforum.org/library/drmath/view/55713.html#search