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The number $\frac{22}{7}$ is irrational in our base-$10$ system, but in, say, base-$14$, it is rational (it comes out to $3.2$ in that system).

It's easy for fractions that are irrational as decimals, as you can just represent them in a base that's double the denominator of the fraction. However, what if I have a number like $\pi$, or $\log(2)$?

For those numbers, it could easily be represented as a rational number if it is in base-($\pi\cdot 2$) or base-($\log(2)\cdot 2$), but is it possible to represent them in any rational-based number system?

Winther
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mdlp0716
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    No. An integer is an integer when written in any (integer) base and so is any rational number. A base is just about representing a number with digits - it does not change what the number is. – Winther Jan 02 '17 at 20:44
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    "Rational" means "can be written as a ratio of integers", so $22/7$ is rational. You may be confusing this with the fact that it is sometimes used as a (very) rough approximation to $\pi$, which is irrational. – Austin Mohr Jan 02 '17 at 20:44
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    π is irrational -- 3.1415926.... whereas, 22/7 is rational, it is 3.142857142857.. repeating, in a predictable manner – Saketh Malyala Jan 02 '17 at 20:45
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    Are you asking about terminating and non-terminating expansions of fractions to different bases? – Mark Bennet Jan 02 '17 at 20:48
  • Wait, I thought irrational numbers were simply numbers that are non-repeating and went on forever? That's the definition I remember learning a while ago, but is that a flawed definition for this kind of application, or just a wrong definition period? – mdlp0716 Jan 02 '17 at 21:09
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    Ok I looked it up, you guys are right, I had a flawed definition of irrational numbers. I guess you learn something new every day! – mdlp0716 Jan 02 '17 at 21:11
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    It is true that irrational numbers are the ones that have infinite non-repeating decimal expansions. That is not the definition, but it is correct. $\frac {22}7$ has a repeating decimal expansion and therefore is not irrational. – Ross Millikan Jan 02 '17 at 21:21
  • Related: http://math.stackexchange.com/questions/2073041/what-is-so-wrong-with-thinking-of-real-numbers-as-infinite-decimals/2073186#2073186 – mweiss Jan 02 '17 at 21:27
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    @RossMillikan I guess my calculator didn't show enough digits for me to realize that it did in fact repeat. – mdlp0716 Jan 02 '17 at 21:28
  • @mdlp0716 Please have a look at the link in my comment directly above your last one -- I think it is very relevant for your question. – mweiss Jan 02 '17 at 21:30
  • @mweiss I looked at it and it definitely seems helpful. – mdlp0716 Jan 02 '17 at 21:33
  • A rational (quotient of integer) always has a terminating representation in some number base system. – Henricus V. Jan 02 '17 at 21:40
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    The repeats can be quite long. Once you reduce the fraction to lowest terms and remove the factors $2$ and $5$ from the denominator, it will repeat within one less decimals than the denominator. $7$ achieves this limit with a repeat that is $6$ long. $17$ has a repeat of $16$ decimals and $109$ has one that is $108$ decimals long, but they will always repeat. – Ross Millikan Jan 02 '17 at 21:50
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    @RossMillikan very informative, I knew the repeats could be log but I never knew it was so simple to figure out exactly how long the repeat would be! – mdlp0716 Jan 02 '17 at 21:55

3 Answers3

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Whether a number is rational or not is independent of the base in which the number may be expressed.

On the other hand the fraction $\frac ab: a,b\in \mathbb Z, b\gt 0$ may terminate or eventually recur when expressed as a "decimal" (Hardy could find no better word - Hardy and Wright, Introduction to the Theory of Numbers). Choosing the base $b$ automatically ensures that the expression terminates. This doesn't work if $b=1$, but then you have an integer anyway.

Mark Bennet
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Always refer to definitions, a number $x$ is called irrational iff $\forall p,q \in \Bbb Z : x\neq\frac pq$, that is when you can't express it as ratio of two integers, not based on how it looks using a different number system. EDIT: This means that you can never find two integers to precisely equal $\pi$ for example, $\frac 31$, $\frac{22}{7}$, $\frac{333}{106}$, $\frac{355}{113}$, $\frac{103993}{33102}$ $\dots$ won't equal $\pi$, they're all finite decimals, the real irrational $\pi$, has unending decimals.

fuzzypixelz
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  • Ok, I have no idea what "iff ∀p,q∈Z:x≠pq" means. Explain it to me like you are explaining it to someone halfway through high school pre-calc – mdlp0716 Jan 02 '17 at 21:07
  • @mdlp0716 "iff" means "if and only if," "$\forall$" means "for all, "$\in$" means "in", and ":" here means "it is the case that." So the expression translates to: "$x$ is called irrational if and only if, for all integers $p$ and $q$, $x\not={p\over q}$". That is, being irrational means not being a ratio of integers. – Noah Schweber Jan 02 '17 at 21:26
  • The answer is OK up to the words "finite decimals", where it becomes very misleading. The number $\frac{22}{7}$ has an unending decimal representation in base ten. The part just before the edit is more accurate: – David K Mar 11 '19 at 12:47
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The number $22/7$ is not irrational, regardless of the number system one uses to express it. On the other hand $\pi$ is irrational, regardless of the number system one uses to express it. They are different numbers: $22/7$ is a good rational approximation to $\pi$, but they are not equal.

It is unfortunate that students are taught to think of "nonrepetition" as the essential quality that distinguishes rational numbers from irrational numbers. It is true that if a number is rational, then its decimal representation will eventually either terminate or repeat, but it might take a very long time, and just looking at a string of digits is not enough evidence to conclude whether or not the string represents the beginning of a repeating decimal or a non-repeating decimal. The real distinction between rational and irrational numbers lies in whether it is possible to express the number as a ratio of integers. If it is possible, then the number is rational; if it is not possible, then the number is irrational. The fact that rational numbers correspond to decimal representations that terminate or repeat is an important and interesting consequence of the definition, but it is not really the essence of the distinction.

For more on this, see my answer and the discussion in the comments beneath it at https://math.stackexchange.com/a/2073186/124095.

Community
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mweiss
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