Irrational digits countably infinite?

[Crosson]

Is the set of digits of an irrational number countably infinite?

I suspect the answer has to do with long division.

 

[mathwonk]

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).

 

[HallsofIvy]

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.

 

[robert Ihnot]

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.

 

[Crosson]

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.

 

[MiGUi]

rationals are countable, but irrationals are not. Between any two numbers there are infinite irrationals, and you can't know its exact value.

 

[Data]

Well, there are also infinitely many rationals between any two real numbers. Just countably infinitely many~