Answerer

If N/0, where N is a complex number, do we call N/0 to be undefined again? Sorry for asking about other things, I'm just curious about the limits of math.

Principia

The power of mathematics is that one need not typically resort to 'tentative conclusions.'
There is no need to assume that taking a decimal representation to completion is possible. It is as possible as counting to infinity. But, the point should be clear by now. Suppose I tell the first 10 digits of a number I am thinking: 0.0101010101... Is it rational or irrational? Who knows?

BTW, it is theoretically possible to have a non-integer base system, and for that matter an irrational base system. But what is the practical use?

DNAunion

DNAunion: Principia, I was still editing my post when you replied (my fault for posting before fully proofreading). Can you look at the final version that I provide here and respond.

DNAunion: For example, can you tell me if that number can be written as n/d with both n and d being integers?

[ December 01, 2002: Message edited by: DNAunion ]</p>

Principia

In other words, if you multiply a complex number with an undefined number does it become defined? The obvious answer is no. Infinities and limits are powerful concepts in mathematics, but they can be extremely counterintuitive.

EDIT: I guess even in the case N = 0 it is still technically undefined...

[ December 01, 2002: Message edited by: Principia ]</p>

Principia

Definitively? No.

Demosthenes

They are undefined, although if you were using limits, 1/0 would be termed infinite and 0/0 still undefined. It's just elementary calculus.

beausoleil

You can't, of course, since it isn't a specific number. There's no clearly defined pattern so the ellipsis is used improperly, for one thing - any digit could follow the last 1 as far as I know, leading to 10 different numbers just by adding one digit. If you know how he was calculating it you might be able to determine whether it is irrational.

IrRATIOnal in this sense means, 'not being a ratio', by the way.

Klil H. Neori

1/0 would not be infinity if taken to a limit without some percautions. If you are talking about the complex compactification infinity, then, yes, you would be correct. But most people use the partial real signed infinity, in which case, the limit could be undefined, if you were taking the function 1/x to limit when x is taken to 0 - for any positivy number, I could choose another positive number y to make 1/y larger than it, and for every negative number, I could choose another negative number y to make 1/y smaller than it. So the limit is no more infinity than it is -infinity.

To conclude, one of the important parts of mathematics is taking such ill-defined things, such as "limit" or "infinity," and giving them exact definitions, from which results can logically follow. So, do define what you mean when you say infinity, or limit, or 1/0. 1/0, as a real, or complex, or any kind of number in any extension field, subfield or subring of the complex numbers is undefined. It's not a number, per se, and to add it in without cautionary notes and scope-limitations is always a risky endeveur.

Claudia

to expand on the "no" answer that you have obtained, there is a non countable infinity of real numbers which have these numbers as first digits...

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

Jesse

For any finite series of digits, there will always be an infinite number of rational numbers that starts out with those digits after the decimal place. In base 10 you can see this by dividing any integer by another number with the same number of digits, but all the digits are 9's...for example:

45/99 = 0.45454545...

123/999 = 0.123123123123...

458392027/999999999 = 0.458392027458392027458392027...

...and so on. You can do the equivalent in other bases too--for example, in base 3, 1201/2222 = 0.120112011201....

So, a finite number of digits will never be enough to tell if a number is rational or irrational.

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