Is there any numbers with infinite digits

In summary, the conversation discusses the concept of numbers with infinite digits. It is mentioned that there are an infinite number of integers, but no integers with an infinite number of digits. There are also different types of decimal numbers, including rational, irrational, and transcendental. The conversation concludes that there are no rational or irrational numbers with an infinite string of digits.
  • #1
amature83
5
0
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...)?

I appreciate your reply.

Thanks
ME
 
Physics news on Phys.org
  • #2
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
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.

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
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:
  • #5
mgb_phys & HallsofIvy: thanks a lot
 

FAQ: Is there any numbers with infinite digits

What is the concept of infinite digits in numbers?

Infinite digits in numbers refer to numbers that have an endless sequence of digits after the decimal point. These numbers cannot be fully represented or written down, as they go on forever.

Can a number have an infinite number of digits?

No, a number cannot have an infinite number of digits. While there are numbers with an infinite number of decimal places, they still have a finite number of digits. For example, pi has an infinite number of decimal places, but it is still a finite number with a defined value.

Are there any practical uses for numbers with infinite digits?

Yes, numbers with infinite digits have practical applications in fields such as mathematics, physics, and computer science. For example, they are used in calculating the area of a circle, modeling the behavior of chaotic systems, and in algorithms for data compression.

How do we represent numbers with infinite digits?

Numbers with infinite digits are usually represented using mathematical notation, such as the symbol for infinity (∞) or by using ellipses (…) to indicate an endless sequence of digits. They can also be approximated using rational numbers, such as fractions, to a certain degree of accuracy.

Is there a largest or smallest number with infinite digits?

No, there is no largest or smallest number with infinite digits. As long as we can continue adding digits after the decimal point, the number can always be made larger or smaller. This concept is known as infinity, which has no defined value or endpoint.

Back
Top