What is the value of 0^0?

We had this issue come up in calc while discussing exponential functions and couldn’t come up with a clear answer. Any one know anything useful.

0?

I’m not surprised no one came up with a clear answer considering this has been baffling the greatest m,athematical minds for decades.

The problem?

n^0 always =1. Don’t ask me why, but apparently having nothing of something in a multiplication makes it equal to 1 (something to do with parabolae, IIRC)

however

0^n always =0. Makes sense, nothing raised to the power 100 is still nothing.

That means, though, that

0^0 = both 0 AND 1 at the same time.

Maths is fucked up. :thud:

indeed it is, but I love it anyway. a guy on another board gave me this link www.faq.org/faqs/sci-math-faq/specialnumbers/0to0/

0^0 its a smiley, with a beak. You’re a genious.

I, personally, would have suggested putting this in the homework help board, but whatever…location isn’t a big deal.

Anyway…0^0 is one of those undefined values, like infinity^0 or 0^infinity.

These are the three undefined constants in math. I thidon’t even think they count as irrational or imaginary numbers, but I’m willing to be wrong on either statement, esp. the latter.

lim
x->0 a^x = 1, when a=any number except infinity or 0

lim a^x = infinity, when a=and number except 0
x->infinity

Those statements, ignoring the exceptions I gave, contradict. Hence the exceptions must be added; hence the three undefined constants.

Originally posted by Bing
[b]I, personally, would have suggested putting this in the homework help board, but whatever…location isn’t a big deal.

Anyway…0^0 is one of those undefined values, like infinity^0 or 0^infinity.

These are the three undefined constants in math. I thidon’t even think they count as irrational or imaginary numbers, but I’m willing to be wrong on either statement, esp. the latter.

lim
x->0 a^x = 1, when a=any number except infinity or 0

lim a^x = infinity, when a=and number except 0
x->infinity

Those statements, ignoring the exceptions I gave, contradict. Hence the exceptions must be added; hence the three undefined constants. [/b]

0^0 is not undefined. Infinity ^ 0 is not undefined. 0 ^ infinity is not undefined.

1. Infinity ^ 0, like any other number when raised to the 0th exponent, is equal to 1.

2. 0 ^ Infinity = 0 * 0 * 0 * 0 * 0 * … * 0 * 0 * 0 = 0.

And finally, 0 ^ 0 = 1.

Proof:

0 ^ 0 = ( 0 ^ 0 ) ( 1 ) (Anything is equal to itself multiplied by 1)

Since 0 ^ 0 has no actual terms associated with itself, the 0 ^ 0 from the right side can be eliminated (So can the 0 ^ 0 from the left, but we’ll leave that there)

Thus: 0 ^ 0 = <strike>( 0 ^ 0 )</strike> ( 1 )

Therefore: 0 ^ 0 = 1

Infinity still isn’t a number, so any numeric expression with infinity in it isn’t well defined… and anything divided by 0 is also undefined…

Allowing 0^0 to be 1 wouldn’t really cause any harm.

lim x->0 of x^x is 1

and there’s a bunch of other reasons why just defining it to be 1 would make sense…

Actually, google says it is 1…

some web pages say it depends on how it is being used…

Infinity isn’t a number, so both Bing and Xelo are wrong. Bing is confused by informalisms used in finding the limit of an expression, Xelo’s expressions don’t make sense because infinity is not a number.

And Xelo’s proof as to why 0^0 should be 1 only makes sense if that is how you are defining exponentiation.

Perhaps it’s just one of those things that is just defined as being 1, such as how 0! is equal to 1 for some reason.

Infinity is a number Vorp. It’s the conceptually largest possible number in the entire universe. And although it doesn’t have a numerical value, it can still be used in certain equations and have a numerical value due to mathematical laws.

Edit: And Flint, 0! is equal to 1 for the same reason I showed 0 ^ 0 = 1 in my above post.

The same as nothing multiplied with nothing. I have no idea what that becomes, but probably infinity.

Arcimedes used to do calculations with infinity in it. But he was mad, at least mad enough to sit and do maths while bathing. I mean, who does maths while bathing?

I asked my algebra professor. ANYTHING raised to the 0 power is 1. Including 0.

Xelo, infinity being a number has got to be the dumbest thing I’ve ever heard you say.

If infinity were a number, then infinity plus one would be a different number. Or is it now possible to increment a number and have the result be the original number?

Missed a word. It’s a conceptual number. But anyways, my point was that infinity can be used in certain equations and still have a defined answer. Infinity ^ 0 = 1. 0 ^ Infinity = 0.

Who honestly gives a flying fuck? What point is there to knowing what 0^0 equals? Someone give me one semi-practical application.

The practical use has something to do with derivatives, if I remember correctly. The point is that you simplify an equation, and wind up with things like infinity over zero or stuff like that. That’s where these things are undefined… because depending on what the original equation was, the infinity over zero you wind up with could mean different things about the original function.

Sorry, it’s been awhile since I’ve taken pure math classes.

Wouldn’t 0^0 power be undefined? The proof being apart of xelo’s original equation. Anything at all divided by 0 is undefined.

There is no division of 0 in an exponential equation. Only when the exponent becomes negative is the numerator/denomator switched.

Isn’t there some rule for exponents like the base can be any number but 0.