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

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Discussion Overview

The discussion revolves around the concept of infinity, particularly in relation to the rules of surds and the implications of raising infinity to the power of zero. Participants explore the nature of infinity, its mathematical definitions, and its conceptual status compared to numbers like zero and one.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that any number to the power of zero equals one and questions whether infinity to the power of zero should also equal one, leading to a discussion about the nature of one in this context.
  • Another participant emphasizes that infinity is not a number, contrasting it with defined numbers like zero and one.
  • Some participants elaborate on the concept of infinity, describing it as unbounded and not an object, but rather a concept.
  • A participant references mathematical contexts where infinity is treated as a value, such as in real and complex analysis, suggesting that infinity can be considered in certain mathematical frameworks.
  • There is a discussion about the need for precise definitions when discussing infinity, with one participant highlighting the ambiguity in defining "never ending" in relation to infinity.
  • Another participant points out that the original poster may not be ready to engage with more complex concepts like extended real numbers, reiterating that infinity is not a number.
  • Some participants discuss the differences between countably infinite and uncountably infinite sets, referencing a comic that illustrates these concepts.
  • A later reply notes that the expression 0^0 is undefined and emphasizes the need for caution when applying rules about powers to cases involving zero and infinity.

Areas of Agreement / Disagreement

Participants generally disagree on the status of infinity, with some viewing it as a concept rather than a number, while others explore its mathematical implications. The discussion remains unresolved regarding the application of the power rules to infinity and the definitions surrounding it.

Contextual Notes

Participants express varying definitions and interpretations of infinity, highlighting the complexity and ambiguity in its mathematical treatment. The discussion also touches on the limitations of applying standard rules of exponents to cases involving zero and infinity.

wolfspirit
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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
 
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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

To play devils advocate:

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" , 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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G037H3 said:
It really isn't difficult at all:

This comic isn't quite accurate. Though it is creepy.

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

To play devils advocate:

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.
 
  • #10


It meant in between

The comic expressed the difference between countably infinite, and incountably infinite
 
  • #11


G037H3 said:
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...
 
  • #12


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.
 
  • #13


LOL, funny how such a simple question always ends up in (the same) long topics :)
 

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