1
$\begingroup$

For example, 0.10 (decimal) is a rational number in base 10 but is irrational (a repeating fraction) in binary (base 2). Is there a number base where PI (3.14159... decimal) is a rational number. That is, accurately expressed with a fixed number of digits.

I am not a math guy, so I may have expressed this question inelegantly.

$\endgroup$
2
  • 1
    $\begingroup$ I’m voting to close this question because it is not about Computational Science and is about general Mathematics. This already existing question on Mathematics SE seems to answer; hence, no migration suggested. If this does not answer it, feel free to post on Math SE explaining what is not covered by the aforementioned post -> or edit this question here to particularly target what is on-topic at Computational Science SE. $\endgroup$
    – Anton Menshov
    Apr 6 at 19:26
  • 1
    $\begingroup$ The statement about 1/10 being irrational in base-2 is wrong. It is a number that does not have a finite number of digits after the decimal (binary) point, but it is still a rational number: a fraction of two integers. $\endgroup$ Apr 6 at 21:05

1 Answer 1

1
$\begingroup$

No, not for any integer base. Irrational doesn't only mean it has infinitely repeating digits (whatever the base may be), it also means that that number cannot be expressed as a ratio of two integers excluding when 1 is in the denominator; hence the term rational and irrational.

For example, 0.33333... is infinitely repeating but still rational because it can be expressed as 1/3, a ratio, and thus rational.

Therefore, 0.10 is still rational even in base 2 and in any integer base, since 0.10 can be written as the following ratios: 1/10 = 2/20 = 35/350 = etc.. While it will have an infinite expansion in base 2, this alone does not make it irrational. Simply put, the value and its mathematical properties don't depend on the base you express it in, it is independent from that.

PI is not rational in any integer base simply because the value of a number doesn't change when you switch their base. It's irrationality has nothing to do with the base you chose to express it in, it is simply its nature to not be reduceable to a ratio of integers, quite similar to how prime numbers' primality doesn't change by merely switching the base you represent them in.

But the gist of your question seems more interested in if there is any numeric base to represent PI in such a way that its expansion is finite in that base. The answer to that would be any base that is a multiple of PI itself. But that would be an impractical system since you are just redefining the magnitude of what you consider to be integers.

$\endgroup$
1
  • 1
    $\begingroup$ OK, I get it. Thanks for explaining the definitions. $\endgroup$ Apr 8 at 14:03

Not the answer you're looking for? Browse other questions tagged or ask your own question.