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Can I write the integer 2 with some zero before as : 002

Can I precede all integer by an infinity of zero : ..........002

Does it make sense to write an integer as an infinity of numerals ? For instance, is the infinite sequence of 1 (....................11111111111111111111111111111111111111) an integer?

For all these questions, can you give me references (book, article) at undergraduate level (better) or graduate level?


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Of course you can introduce that unusal definition, the question is "does it make sense". You can't simply say you "precede all integers by an infinity of zeroes", the concept of infinity is rather a limit than something you can work with, please use the search too to look for questions about infinity. – Listing Jan 24 '12 at 9:56
The decimal expansion of a number makes sense. For instance 0.99999999.... is equal to 1. I'm asking myself if this kind of notation is also used, not for the decimal part but "on the left" for the integer part. In the notation 0.999....., I suppose we employ a countably infinite set. I'm wrong with that? – Mathieu Vidal Jan 24 '12 at 10:08
@Mathieu, you have discovered the p-adics, which is far from wrong, although they are uncountable just like the reals. – Dan Brumleve Jan 24 '12 at 10:11
Thanks, I will read about it. – Mathieu Vidal Jan 24 '12 at 10:17

The decimal $...1111111$ is a 10-adic number. $n$-adics are similar to integers in some ways, and like real numbers in other ways. As an exercise, what is $...3333 \cdot 3$?

Also, in many computer programming languages, a leading zero is conventionally interpreted as indicating an octal base rather than decimal.

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The $p$ in $p$-adic stands for "prime," and 10 doesn't qualify. That's not to say you can't talk about the 10-adics, just that not everything that is true for the $p$-adics is true for the 10-adics. – Gerry Myerson Jan 24 '12 at 11:31
I think $n$-adic is fine. – Gerry Myerson Jan 24 '12 at 12:00
@Gerry, that's right. Wikipedia explains it by saying "The 10-adic numbers are generally not used in mathematics: since 10 is not prime, the 10-adics are not a field." However, it is a ring. Is the term "n-adic" correct for the general case? [Oops, I deleted and replaced this comment and it is appearing after Gerry's response to it.] – Dan Brumleve Jan 24 '12 at 12:01

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