# Is there any numbers with infinite digits

1. Nov 5, 2008

### amature83

Hello,

Is there any number with infinite digits (e.g. 34343329...) or just the decimal representation of a number could be infinite (e.g. 1/9 = .1111...)?

Thanks
ME

2. Nov 5, 2008

### mgb_phys

If you mean integers (whole numbers, with no decmal places) then yes there are an infinite number of them. You can see this because any number you give me I can always add another digit on the end.

There are a few types of decimals (real numbers) that also go on for ever.

Rational numbers are those that you can write as a fraction like 1/3

Irrational numbers are those that go on for ever but cannot be written as a shorter fraction. Numbers like the sqrt(2), you can write this as an equation but not a fraction.

Transcendental numbers are the 'most infinite' if you like. Not only do these numbers go on for ever but you cannot write numbers like pi or 'e' as any fraction or equation that isn't infinite.

3. Nov 5, 2008

### amature83

I think I didn't make the question clear. I am sorry about that..
I ask about the infinite string of digits, say infinite string of 1's: 11111... What is that number? I think it's not natural because the construction of N by successor (using Peano Arithm.) would never get us the "infinite" string. So, could we call it rational? My doubts about calling it rational is that -as far as I understand- in rationals the decimal expansion could be an infinite string strings but not the number itself.

ME

4. Nov 5, 2008

### HallsofIvy

No, there are no integers whose expression in base 10 requires an infinite number of digits. Specifically, 10n is an unbounded sequence. That means that, given any integer, a, there exist an integer N so that a< 10N. a has less than N decimal places.

A "rational number" is defined as a number of the form m/n where m and n are integers (and n is not 0) so, no, there are no rational numbers, or even irrational numbers, that have an infinite string of digits in front of the decimal point.

Last edited by a moderator: Nov 5, 2008
5. Nov 5, 2008

### amature83

mgb_phys & HallsofIvy: thanks a lot