Irrational digits countably infinite?

In summary, the conversation discusses the concept of countable infinity in relation to the set of digits in an irrational number. It is concluded that the set of digits in an irrational number is countably infinite, as it can be put in a 1 to 1 relation with the set of all natural numbers. It is also mentioned that the set of rational numbers is countable, but the set of irrational numbers is not. There is a theorem stating that any set of positive numbers with a cardinality greater than a countable set will have a sum that exceeds any finite number. Additionally, there is an explicit isomorphism between the sequence of digits and the natural numbers. The conversation ends with a discussion about the infinite number of irrationals
  • #1
Crosson
1,259
4
Is the set of digits of an irrational number countably infinite?

I suspect the answer has to do with long division.
 
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  • #2
do you mean to ask how many decimal entries does an irrational number have when written as an infinited ecimal?

isn't there one entry for each (negative) integral power of 10? (not counting the integral part of the number).
 
  • #3
I interpret this as asking about the cardinality of the set of digits in the decimal expansion of an irrational number. One difficulty with that is that, strictly speaking a "set" does not have multiple instances of the same thing: the "set of digits" of any number is a subset of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and so is finite!

But what you MEAN, I feel sure, is "counting" the digits- that is labeling the first digit as d1, the next d2, how many digits are there? The answer is simply that doing that IS counting them. The fact that you CAN do that means that the set is countably infinite. A set is countably infinite if it can be put in a 1 to 1 relation with the set of all natural numbers- "listing" a set, so that there is a "first", a "second", etc. is obviously doing that. In fact, considering terminating decimals as ending with an infinite string of 0s (0.5 is 0.500000...) then the decimal expansions of ALL numbers are countably infinite.
 
  • #4
I think it is a theorem that any set of positive numbers all greater than zero, if added together will exceed any finite sum, if the cardinality of the set exceeds a countable set.

Thus if we went from decimal place to decimal place and somehow exceeded a countable set of non-zero terms, the sum would exceed any finite number.
 
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  • #5
I think that mathwonks proof is the most elegant...he found an explicit isomorphism between the sequence (not set) of digits and the natural numbers. Thanks.
 
  • #6
rationals are countable, but irrationals are not. Between any two numbers there are infinite irrationals, and you can't know its exact value.
 
  • #7
Well, there are also infinitely many rationals between any two real numbers. Just countably infinitely many~
 

What does it mean for "Irrational digits countably infinite"?

"Irrational digits countably infinite" means that there are an infinite number of irrational numbers between any two rational numbers. This means that the set of irrational numbers, while still uncountable, is not as large as the set of real numbers.

Why are irrational digits countably infinite?

Irrational digits are countably infinite because they can be put into a one-to-one correspondence with the natural numbers. This means that there is a way to label each irrational number with a unique natural number, even though there are an infinite number of them.

How are irrational digits counted?

Irrational digits are not typically counted in a traditional sense because they are uncountable. However, they can be represented by using decimal expansions or other numerical representations.

What is an example of an irrational digit?

One commonly known example of an irrational digit is pi (π), which is approximately equal to 3.14159. It is an irrational number because it cannot be expressed as a ratio of two integers and has an infinite number of non-repeating digits in its decimal expansion.

Why is the concept of irrational digits important in math and science?

The concept of irrational digits is important in math and science because it helps us understand the nature of real numbers and the concept of infinity. It also has practical applications in fields such as computer science, where rational and irrational numbers are used in calculations and algorithms.

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