Rules in surds is that any number of the power of 0

Main Question or Discussion Point

One of the rules in surds is that any number of the power of 0 must equal 1
And as Infinity is nothing more than a never-ending number then does it stand to reason that Infinity to the power of 0 must equal 1 but my question is what is 1
Is 1 (in this situation) an abstract concept or does 1 equal one lot of Infinity?

many thanks
Ryan

CompuChip
Homework Helper

Infinity is not a number.
0 is a number, 1 is a number, infinity is a concept.

Infinite

In-finite

Infinite: is not finite

Infinite: cannot be placed within bounds

Infinite: is not bounded

Infinite: not an object, just a concept

http://en.wikipedia.org/wiki/Infinity

Under "Real Analysis"
"Infinity is often used not only to define a limit but as a value in the affinely extended real number system. "

Under "Complex Analysis"
"In this context it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of infinity at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations."

I mean, infinity is as real as the infinitesimal right?

EDIT: Also I should mention cardinality... So some infinities are larger than others!
EDIT2: lol at infinity(wolfspirit)

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I would also like to point out, in math you can typically define things as you please. For example take the word “normal”. “Normal” has at least 10 definitions I know, and I’m sure many more.

But, there is a problem with how you define infinity. Let's call your version of infinity infinityws. If you want to call infinityws a never ending number, you would have to spell out what “never ending” means; since it’s not a well defined concept already (at least in this context). The problem with this is you probably would want to define “never ending” by referring to either some property a number can’t have, and thus commit a http://en.wikipedia.org/wiki/Category_error" [Broken], or by referring to infinityws which would make your definition equivalent to infinityws is a number that is infinityws.

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It really isn't difficult at all:

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It really isn't difficult at all:
This comic isn't quite accurate. Though it is creepy.

http://en.wikipedia.org/wiki/Infinity

Playing devil's advocate is great, but given the OP's wording, he isn't quite ready for working with extended real numbers, and they are just going to cause unnecessary confusion.

Infinity is not a number.
This is all the OP really needs to understand.

Seems pretty accurate to me. :O

The infinity of the real numbers could fill up the requirements for any form of infinity. >_>

Seems pretty accurate to me. :O

The infinity of the real numbers could fill up the requirements for any form of infinity. >_>
There are no "real numbers within an integer".

If the comic means "in between", keep in mind that there are an infinite number of rationals between the integers too. But the rationals are countable.

Cardinality is defined independently of field ordering, so you can't talk about how many numbers there are "in between" others to make any judgment on cardinality.

It meant in between

The comic expressed the difference between countably infinite, and incountably infinite

CRGreathouse
Homework Helper

The comic expressed the difference between countably infinite, and incountably infinite
Not really; it seemed to distinguish between "not dense in the reals" and "dense in the reals". OF course this does not correspond to cardinality...

jambaugh
Gold Member

Note: 0^0 Is undefined. So you have to be more careful with the scope of the "rule"
The rule is that any non-zero real number to the 0th power is 1. This excludes 0 and infinity. To deal with those cases you must "sneak up on them" with limits from Calculus.

One can consider how x^y behaves as x approaches 0 or infinity and as y approaches 0.
(Imagine plotting in 3 dimensions the surface z= x^y.)
One then finds that for different paths in the x-y plane this number (z) will approach different values in these limits so we can't give one fixed rule for the actual value.

CompuChip