Undercurrent

The short answer is, it depends on if we choose to define it. In some applications, a specific infinity element is added to the complex plane and in that system, z/0 = infinity and z/infinity = 0. In other applications we talk about complex numbers with one finite component and one infinte component with some meaning. In a bunch of contexts, we simply decline to define what 1/0 equals and treat it as an anomoly if it comes up.

The "limits of mathematics" are revealed in the halting problem and Godel's theorem. The lack of a sensible definition for division by zero is more of an intersting anomoly than anything else. After all, we can do decent math without defining division at all.

m.

DNAunion

DNAunion Well, I and a few other people disagree.


DNAunion: Those were the math books I had on my bookshelf. Just for some icing on the cake, here’s everything my dictionary says about this. Note that it defines an irrational number in both ways (a number whose decimal representation is a nonterminating, nonrepeating decimal; and as a number that can’t be written in the form n/d where both n and d are integers).



DNAunion: How about a web link or two?


DNAunion: Let me try to head off some potential strawmen that people might try (not necessarily the person this post is a direct response to).

I have NOT claimed that the “non-repeating, non-terminating decimal representation” definition is the only correct one for an irrational number, nor have I claimed that the “cannot be expressed as a fraction of integers” in incorrect. I have in fact confirmed that the “cannot be written as a fraction of integers” definition is a correct definition of an irrational number. My position is that both are correct. Therefore, people can show one, five, ten, twenty, or a hundred sources defining an irrational number as a number that cannot be written as a fraction consisting of two integers and it won’t counter me in the least. To counter me, one must show that the “definition” I originally used, based on non-repeating, non-terminating decimal representations, is incorrect. And that will be hard to do, since I have just shown valid mainstream mathematics sources that state it is a valid definition of an irrational number.

[ December 02, 2002: Message edited by: DNAunion ]</p>

DNAunion

DNAunion: I think several people missed the point I was making about the 30-digit decimal number thought experiment. Let me back up.

Principia objected to “my definition” of an irrational number because, quote:

(1) “non-terminating [by itself] is never a sufficient criteria”

and

(2) “non-repeating is impossible to apply, unless one wishes to calculate every single digit.”

Objection one does not counter anything I said since I made it clear that both non-terminating AND non-repeating were required (in other words, non-terminating is a necessary, but not sufficient condition, for a decimal representation of a number to flag it as being irrational: just as I stated).

Objection two doesn’t work because of the example I gave. I indirectly handed Prinicipia a 30-digit decimal number that ended in ellipses and asked him if he could tell me whether or not it could be expressed as a fraction of two integers. He couldn’t. Exactly! He couldn’t make that determination either unless someone “calculate[d] every single digit”. So his objection to "my definition" works against "his own definition" as well.


Okay, so let me put it this way. I have a decimal-represented number in mind that is non-repeating and non-terminating. Can someone tell me if it can be written as a fraction with both the numerator and the denominator being integers? No, because no one except for me knows the number. So I can tell you absolutely that the number is an irrational number, but those who use the “cannot be written as n/d where both n and d are integers” definition can’t.

That is, unless that person accepts that knowing that a number has a decimal representation that is non-repeating and non-terminating is equivalent to knowing that the number is irrational, or that it is equivalent to knowing that the number cannot be written in n/d fashion with both n and d begin integers. Thus, they would have to accept that the two definitions of interest are equally valid.

Principia

The emphasis is exactly the point, DNA. Do you (or in fact, can you) really "know" the number in its decimal representation? Let's try your thought experiment again. What is the 100 millionth digit? What is the 10^100 th digit? Can you give me any sequence of N digits that I request? And after that, have you given us any information that determines conclusively whether or not the number is irrational? No.

You, in fact, have a digit-generating algorithm in your head to make up sequences that you think are consistent with 'non-repeating.' But you can never prove to anyone by spouting out finite sequences of digits unless you disclose the closed-form representation of your algorithm, which is tantamount to permitting an analysis based on the n/d formulation. In other words, the definitions are not equivalent. 'Yours' is utterly useless in determining a priori whether a number is irrational. Or do you deny that your question is about the reverse inference? Let's take a look:
Note that I (or any of the other math texts you cited) would completely agree that an irrational number is consequentially 1) not representable as the ratio of two rational numbers and 2) non-repeating and non-terminating in its decimal representation. Hell, I even proved to you that your 0.1 example in binary was premature and flawed. But that was not my point. In all the cases you cited, people are making a forward inference (or equivalently the negation of the forward inference): Irrational number -&gt; 1) and 2) or not[1) and 2)] -&gt; not irrational number. My argument is that only 1) permits a reverse inference (i.e. 1) -&gt; irrational number), which is what you are after. 2) is unknowable by any pratical means, because it only begs a 'why is it non-repeating?' question -- after which, you're stuck spouting finite sequences which are inconclusive. In other words, the only way to know 2) a priori is to reveal some additional property about the number other than that it is "non-terminating and non-repeating." Here, I'll make it simple for you. Prove to us that: is non-repeating in order to show to us that it is irrational without invoking any circular logic or additional information (e.g. that you know the number 'is' irrational).

BTW, we all know that you are trying to have people lose track of your original (laughable) question, which has already been answered to great detail (and your great embarrassment, if I might add):
Any other questions along this line?

[ December 03, 2002: Message edited by: Principia ]</p>

DNAunion

DNAunion: Uhm, no you didn’t.

UglyManOnCampus reminded me that another definition of an irrational number is, a number that can’t be written in n/d form with both n and d being integers (and d != 0). He - UglyManOnCampus - is the one who exposed my oversight, not you. That is, as soon as UglyManOnCampus pointed this out, I and everyone else knew immediately that 1/10 in binary is still rational. All you did was “reprove” something everyone already knew…big deal.

DNAunion: No, only a stupid person would think that (so I hope you are speaking only for yourself, and not for the others that you include in your statement).

I already freely admitted that my original question was flawed, in my first post after the thread starter. You know, the first post I made after UglyManOnCampus pointed out my oversight.

That topic is over – it was over 15 posts ago. Time to move on Principia.

DNAunion: Anyone who kept giving great detail why my original question was flawed were adding nothing to the discussion - UglyManOnCampus was the one who killed the horse – anyone who did any more was just beating a dead horse for their own pleasure.

DNAunion: Uhm, I am not embarrassed to have overlooked something, especially considering that I explicitly stated in my thread-starting post that I didn’t give the matter much thought before posting.

[ December 03, 2002: Message edited by: DNAunion ]</p>

DNAunion

DNAunion: In case you somehow missed it, Principia, the NEW topic I have been discussing in this thread (since the ORIGINAL topic was dropped over 15 posts ago) is Wade-w's assertion that "my original definition" of an irrational number is incorrect.

Sorry, but "my original defintion" is valid: an irrational number is a real number whose decimal representation is both non-terminating and non-repeating. That another defintion exists, or is preferred, is not the issue. To show me to be wrong on this issue one needs to show that "my original definition" of an irrational number is incorrect - the claim that Wade-w made and that I rejected.

You have now confirmed that "my original defintion" was a valid one - thereby showing Wade-w to be wrong.

But, instead of simply acknowledging the validity of "my original defintion", you also start playing your typical in-your-face taunting games. You know, your primary method of "argumentation" (at least against me).

[ December 03, 2002: Message edited by: DNAunion ]</p>

Claudia

DNAunion, your definition is a property equivalent to what is usually considered as the definition of irrationnal numbers. The fact that they are called irrationnal is a sign that non-rationnal is the first definition. Nevertheless, you could chose to use it as a definition, and call it a definition, why not. the problem is that for a definition it is useless, because you cannot show (for a given number) that this property/secondary definition is verified until you have first verified the first (and more usual) one.

Let us see your example. I have told you that you have a non countable infinity of numbers with the same first digits that you have shown. In fact, you have a countable infinity of rational numbers and an uncountable infinity of irrational numbers which share these same first digits. And for any additional digits you give (in finite number), it will still be true. So yes, if you add the information that the digits do not stop and do not repeat, we will know that it is an irrational number, but which one of the non countable infinity of possibilities? we do not know.

And if you say that it stop later or repeat, we still do not knof which of the countable infinity of rational numbers it is, unless you give us the exact algorithm which provides the digits.

So your definition is useless, unless you need no more that "any number of the infinite list of possible numbers, be these numbers rational or not." In which list you pick you will know. And?.

Principia

Simple. A definition is equivalently an axiomatic, logical if and only if statement. I have already argued that there is no logical way of deducing irrational numbers by using only the number's decimal representation (i.e. that 2) -&gt; irrational number is impractical and for all intents and purposes illogical, since one can never assert 'non-repeating' with absolute, logical certainty without invoking circular logic). The only valid if and only if statement is the following: a number is irrational if and only if it cannot be represented as a ratio of two rationals.
Nope. It is precisely the issue. Gee, Claudia gets what I am saying. Does anybody else besides DNA not get it? If so, then I have no problem leaving DNA in the dark on this one. I have been through his evasion tactic before. Remember, Rick? It had to do with your claim that the Center of mass of the Solar system oscillated because of planetary motion.

[ December 03, 2002: Message edited by: Principia ]</p>

Principia

Here is the demonstration of how DNA is wrong. I repeat my challenge to him, which he avoided in spite of all of his whining about me, from the previous page:
[ December 03, 2002: Message edited by: Principia ]</p>

Principia

And below I offer yet another one of an infinite number of useless, DNAunion-type 'definitions':

An idionumber is one which does not contain 1234 anywhere in its decimal representation.

I am thinking of an idionumber, and I tell you that it begins: 0.273240653682303929829016482931...

Now, you all say -- so what, prove to us that it is in fact an idionumber using only your definition... See, there is a reason why DNAunion-type 'definitions' aren't really valid definitions at all.

EDIT: And here's another DNAunion-type definition.

A humfer is by definition not a humfer. Is DNAunion a humfer???

EDIT: And here's the icing on the cake -- a terminology for the logical fallacy DNAunion is committing: <a href="http://www.everything2.com/index.pl?node_id=584928" target="_blank">a fallacy of definitions</a>. Specifically, he is advocating a definition that fails to elucidate the meaning of an irrational number, and he cannot avoid making the definition circular.

[ December 03, 2002: Message edited by: Principia ]</p>