Is Zero Raised to the Power of Zero Equal to One?

[matt grime]

Very much false.

log(0) is not defined. Indeed 0 is an essential singularity of log. In laymans terms, not only does |log(z)| tend to infinity as z tends to zero along the real axis, but it does so in a very very nasty way.

 

[whozum]

Sorry you guys didnt pick it up, there was a heavy tone of sarcasm in my last post.

I compeltely understand what you guys are saying.

 

[The Reverend BigBoa]

An interesting discussion indeed. I thought at least SOMEWHERE there might be something a bit more "conclusive", geez, let's discuss what THAT means too! Hah,,,,

Anyways, here is another aspect, especially what the author brings up regarding gamma functions,,,,

Food for thought anyways, and I'm presuming that's the intent of this thread,,,,,,

http://home.att.net/~numericana/answer/algebra.htm#zeroth

 

[The Reverend BigBoa]

Perhaps this says that everybody's right. At least, given the situation. This guy does agree though, that the majority view is that 0^0 = 1.

http://www.mathforum.org/dr.math/faq/faq.0.to.0.power.html

But isn't it nice when everyone can be correct? :!)

 

[Leopold Infeld]

My calculator symbolically evaluates that limit to 1...

There's a more obvious counterexample:

<br /> \lim_{x \rightarrow 0} 0^x = 0<br />.




The whole thing about 0^0 depends precisely what ^ means. As an operation on real numbers, 0^0 is undefined. The reason is that (0, 0) is not in the domain of ^.

Okay, so you want the "philosophical" reason. A crucial property about real operations is that they're continuous within their domain.

However, ^ cannot be continuous at (0, 0) -- the classic examples demonstrating this fact are 0^x --> 0 and x^0 --> 1 as x --> 0.


However, there are other meanings to ^. For example, there's a definition of ^ that means repeated multiplication. The empty product is, by definition, 1, so anything to the zero-th power is equal to 1. In polynomials and power series, this definition is what ^ "really" means -- that's why one uses 0^0 = 1.

<br /> \lim_{x \rightarrow 0} 0^x = 0<br />.

Is this a possible limit?
I thought that the limit from the right isn't equal to that of the left for this case.
cuz 0^0.0000000000000001=0 cause the zillionth root of 0 is still o
but 0^-0.0000000000000001 is undefined if 0^0.0000000000000001=0 is correct.

Dont laugh guys if my arguments and terribly lame because i don't kno anything.

 

[master_coda]

<br /> \lim_{x \rightarrow 0} 0^x = 0<br />.

Is this a possible limit?
I thought that the limit from the right isn't equal to that of the left for this case.
cuz 0^0.0000000000000001=0 cause the zillionth root of 0 is still o
but 0^-0.0000000000000001 is undefined if 0^0.0000000000000001=0 is correct.

Dont laugh guys if my arguments and terribly lame because i don't kno anything.

That's correct. The limit doesn't exist, only the limit from the right. The limit from the left is just undefined.

 

[arildno]

Note that we may approach 0^{0} in the following manner, by letting x go to 0 from the positive side:

f(x)=(e^{-\frac{1}{x}})^{\alpha{x}}

But, evidently, we have f(x)=e^{-\alpha}
In this case therefore, we have \lim_{x\to0}f=e^{-\alpha}

 

[mathelord]

What the problem with wat i posed to thye forum concerning wat was done by my friend?I'd love to know so i can let the class be aware of the error

 

[arildno]

See post 51 by matt grime.

 

[mathelord]

but 0 time sany number whatever it is give 0 right

 

[Zurtex]

but 0 time sany number whatever it is give 0 right

What has a^b got to do with multiplying so many times when b is not a natrual number?

 

[arildno]

but 0 time sany number whatever it is give 0 right

There isn't any number log(0)

 

[Leopold Infeld]

log 0 is not a number

 

[leon1127]

i am sure this is right
set y= x^x
y=e^(x Ln[x])
lim[y,x,0]=lim[e^(x Ln[x]),x,0]
since it is continuous, it produces e^(lim[x Ln[x], x, 0]) = e^0 = 1

 

[Hurkyl]

That's (almost) correct; \lim_{x \rightarrow 0^+} x^x = 1.

 

[Zurtex]

That's (almost) correct; \lim_{x \rightarrow 0^+} x^x = 1.

Although perhaps their proof only covered that it's also true that \lim_{x \rightarrow 0^-} x^x = 1

 

[HallsofIvy]

Although perhaps their proof only covered that it's also true that \lim_{x \rightarrow 0^-} x^x = 1

What is true is that x^x is not defined for negative x so that doesn't even make sense.

 

[Zurtex]

What is true is that x^x is not defined for negative x so that doesn't even make sense.

\left( -\frac{1}{2} \right)^{-\frac{1}{2}} = -i \sqrt{2}



A couple years back the function x^x I loved studying, the properties of the function I really enjoy.

 

[Hurkyl]

You're talking about a different x^x function.

 

[BoTemp]

Note that we may approach 0^{0} in the following manner, by letting x go to 0 from the positive side:

f(x)=(e^{-\frac{1}{x}})^{\alpha{x}}

But, evidently, we have f(x)=e^{-\alpha}
In this case therefore, we have \lim_{x\to0}f=e^{-\alpha}

Interesting, but I'm not sure if f(x) = exp(-a) in the limit as x->0, because there's a 0/0 in the exponent.
I would say 0^0 is undefined, mainly because one can get different answers by attacking from different angles:

<br /> \lim_{x \rightarrow 0} 0^x = 0<br />

<br /> \lim_{x \rightarrow 0} x^x = 1<br />

The second I know how to prove with logs, the first I trust the arguments
given. Also, the proof of the second involves l'Hopitals rule, which only
gives limits, not exact values. For instance,

<br /> \lim_{x \rightarrow 2} (x-2)/(x-2)(x+3) = 1/5<br />

but if one were to define that as a function f(x), f(2) is undefined.
In conclusion, I would say treat 0^0 much as 0/0, basically by taking
limits when it comes up.

 

[arildno]

No, no you've misunderstood me (due to an omission I made).
I've said that the LIMIT of f is equal to e^{-\alpha}; I've not stated that f has been defined at x=0; that is; I should have written:
f(x)=e^{-\alpha},x\neq{0}
sorry about that one..

 

[shmoe]

Interesting, but I'm not sure if f(x) = exp(-a) in the limit as x->0, because there's a 0/0 in the exponent.

x/x=1 for all non-zero x. It's a limit, so we don't actually care what happens at x=0, just near it.

f(x)=(e^{-\frac{1}{x}})^{\alpha{x}}=e^{-\alpha}

for all x>0, so the limit in question is e^{-\alpha}

 

[arildno]

BoTemp was right in critizing me, shmoe; I hadn't made the proper restriction on x.

 

[shmoe]

How you stated it is fine to me, you were talking about a right hand limit. There's no reason for anyone to assume (or even care) if your function was defined at 0.

 

[Sabine]

i agree that 0^0= 1

a^x= 1/(a^(-x))

if a=x=0 then there is no way that 0^0= 0 because we'll have 0=infinite(1/0)

 

[Hurkyl]

if a=x=0 then there is no way that 0^0= 0 because we'll have 0=infinite(1/0)

No, there's no way that 0^0 = 0 because 0^0 is undefined.


If you like heuristic reasoning from identities, what about 0^x = 0?

 

[Rahmuss]

I don't think of 0 as being in the same number system as anything else really. It's more of a concept like infinity.

So you really can't do all of the same math with 0 as you can with other numbers. 0^0 makes no sense. Nor would log(0).

LOL... that's so funny... where I listed ['tex'] 0^0 [/'tex'] it put "infinity".

EDIT: And now it's back to 0^0. Hmmm...

 

[Hurkyl]

So you really can't do all of the same math with 0 as you can with other numbers.

You meant arithmetic. And that comes directly from the definitions -- division is defined for any nonzero denominator.

Incidentally, though, for each operation which is undefined at zero (such as 1/x), there's a corresponding operation which is undefined at one. (such as 1/(x-1)) So, in a very real sense, you can do exactly as much with zero as you can do with any other real number.

 

[cronxeh]

No, there's no way that 0^0 = 0 because 0^0 is undefined.


If you like heuristic reasoning from identities, what about 0^x = 0?

0^x = 0

defined: for all x > 0
undefined: for all x < 0, x = 0

 

[Rahmuss]

Wait! Did someone already do this?

0 \approx (1/\infty) Not quite; but approximately.

1/(1/\infty) \longrightarrow (1/1)/(1/\infty) \longrightarrow (\infty/1) * (1/1) \longrightarrow \infty*1 = \infty

Though 0 is a little less than 1/\infty. So what value is it really?

 

[jtox]

As far as math teachers are concerned, you may safely assume 0^0 = 1 (/End deliberate hand-waving mode). As a precaution, most documents or proofs that require its use (most that I've seen, anyway) will still explicitly state it as a useful interpretation before applying it.

 

[Zurtex]

Wait! Did someone already do this?

0 \approx (1/\infty) Not quite; but approximately.

1/(1/\infty) \longrightarrow (1/1)/(1/\infty) \longrightarrow (\infty/1) * (1/1) \longrightarrow \infty*1 = \infty

Though 0 is a little less than 1/\infty. So what value is it really?

You are wrong \frac{1}{\infty} does not make sense for real numbers as \infty is not an element of the real number set. As for sets in which it does exist, you need to learn their axioms and not assume that they are the same as the real numbers (otherwise they would be the real numbers).

 

[Sabine]

hurkyl 0^x=0 if x is different from 0

0^m*0^-m=0^0=1

or as i said before a^x=1\(a^(-x)) so if a=x=0 this means

0^0= 1\(0^(-0))
in this case 0^0 should b equal to one

x^0= 1 even for x=0

 

[matt grime]

0^m*0^-m=0^0=1

If you look very carefully you've just divided by zero: one of m or -m is negative (i'm assuming integral exponent)