# Irrational Numbers In Non-Base 10 Systems **URL:** \ **Category:** Factual Questions\ **Created:** [July 13, 2005, 3:02pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341 "2005-07-13T15:02:30Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![ZomZom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zomzom/32/5620_2.png) [@ZomZom](https://boards.straightdope.com/u/ZomZom)\ **Post date:** [July 13, 2005, 3:02pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/1 "2005-07-13T15:02:30Z")
Are irrational numbers simply an artifact of the particular numbering system being used or are they common accross numbering systems. For example, is 1/3 (.33 repeating) also irrational in all other numbering systems? ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [July 13, 2005, 3:07pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/2 "2005-07-13T15:07:33Z")
If the base is divisible by 3, then the expansion terminates. For example, 1/3 is 0.4 base 12. If the base is not divisible by 3, then the expansion repeats indefinitely. For example, 1/3 is 0.0101010101010101010101… base 2. (Note that fractions like 1/3 are called rational numbers. Irrational numbers are those which cannot be expressed as a fraction, and irrational numbers have non-terminating, non-repeating expansions in any integer base.) ---
**Author:** ![bordelond](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bordelond/32/150_2.png) [@bordelond](https://boards.straightdope.com/u/bordelond)\ **Post date:** [July 13, 2005, 3:07pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/3 "2005-07-13T15:07:43Z")
> [@mangoldm](#): > > Are irrational numbers simply an artifact of the particular numbering system being used or are they common accross numbering systems. For example, is 1/3 (.33 repeating) also irrational in all other numbering systems? 1/3 would be rendered as 1/10 in base-3. The decimal form would be .1 . Some irrational numbers aren’t tamed quite so easily. My understanding is that pi is irrational in _any_ integer-based counting system. ---
**Author:** ![Plynck](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Plynck](https://boards.straightdope.com/u/Plynck)\ **Post date:** [July 13, 2005, 3:08pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/4 "2005-07-13T15:08:35Z")
> [@mangoldm](#): > > Are irrational numbers simply an artifact of the particular numbering system being used or are they common accross numbering systems. For example, is 1/3 (.33 repeating) also irrational in all other numbering systems? Can’t answer your question, but a quick correction: 1/3 is by definition rational (as in “can be expressed as a ratio”). Any number that repeats can be factored to a ration. Pi is an example of an irrational number. I leave the complicated stuff for better minds than mine. plynck ---
**Author:** ![Bricker](https://avatars.discourse-cdn.com/v4/letter/b/977dab/32.png) [@Bricker](https://boards.straightdope.com/u/Bricker)\ **Post date:** [July 13, 2005, 3:10pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/5 "2005-07-13T15:10:09Z")
Again, 0.3333 (repeating) is NOT irrational. A rational number is any number that can expressed in the form _p/q_, _p_ and _q_ integers, _q_ not 0. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 13, 2005, 3:10pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/6 "2005-07-13T15:10:35Z")
1/3 is a rational number. Recall that a number r is rational if there are integers p and q such that r = p/q. 1/3 is rational (p = 1, q = 3). Irrational numbers are things like sqrt(2), [symbol]p[/symbol], etc. These are irrational in any integer base and have non-terminating, non-repeating representations. ---
**Author:** ![Plynck](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Plynck](https://boards.straightdope.com/u/Plynck)\ **Post date:** [July 13, 2005, 3:11pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/7 "2005-07-13T15:11:44Z")
> [@Plynck](#): > > … Any number that repeats can be factored to a ration… :smack: \*\*\*…ratio… \*\* \* ---
**Author:** ![ZomZom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zomzom/32/5620_2.png) [@ZomZom](https://boards.straightdope.com/u/ZomZom)\ **Post date:** [July 13, 2005, 3:12pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/8 "2005-07-13T15:12:12Z")
Ah, I get it – ratio, rational, and the light bulb dings. I learned two things from one question – certainly getting my money’s worth this morning. ---
**Author:** ![Schnitte](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/schnitte/32/9033_2.png) [@Schnitte](https://boards.straightdope.com/u/Schnitte)\ **Post date:** [July 13, 2005, 3:14pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/9 "2005-07-13T15:14:42Z")
1/3 is not an irrational number because it can be expressed as a fraction, albeit not a decimal one. Pi is the standard example of an irrational number. But regarding your question: Imagine a base-3 numbering system, where 1 (Decimal) = A, 2 = B, and zero = 0. 3 would be A0, 4 would be AA, 5 would be AB (1_3 + 2_1), 6 would be B0 (2_3 + 0_1), 7 would be BA (2_3 + 1_1), 8 would be BB (2_3 + 2_1), and 9 finally would be A00 (1\*3²). More general, every integer can be expressed as x\*3[sup]1[/sup] + y _3² + z_3³, ad so on, ad infinitum. This can be done analogously for numbers smaller than one. You could express it as multiples of 3[sup]-1[/sup], 3[sup]-2[/sup] and so on. So in our system, 1/3 ought to be 0.A. ---
**Author:** ![Amazon\_Floozy\_Goddess](https://avatars.discourse-cdn.com/v4/letter/a/779978/32.png) [@Amazon\_Floozy\_Goddess](https://boards.straightdope.com/u/Amazon_Floozy_Goddess)\ **Post date:** [July 13, 2005, 3:18pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/10 "2005-07-13T15:18:52Z")
Just want to say how I envy you math people. I checked this out out of curiosity and I may as well be trying to read hieroglyphics. ☹ ---
**Author:** ![aahala](https://avatars.discourse-cdn.com/v4/letter/a/a88e4f/32.png) [@aahala](https://boards.straightdope.com/u/aahala)\ **Post date:** [July 13, 2005, 3:36pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/11 "2005-07-13T15:36:47Z")
Irrational numbers are the same across all bases. I have a truly marvelous demonstration of this proposition which this post is too small to contain.😃 ---
**Author:** ![Hypnagogic\_Jerk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hypnagogic_jerk/32/4252_2.png) [@Hypnagogic\_Jerk](https://boards.straightdope.com/u/Hypnagogic_Jerk)\ **Post date:** [July 13, 2005, 3:43pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/12 "2005-07-13T15:43:06Z")
It should be mentioned that, as was said in a previous [thread=322665]GQ thread on irrational bases[/thread], rationality and irrationality are properties of a **number** while termination is a property of a number’s **representation**. 1/3 is a rational number since it can be expressed as a ratio of integers. In base 10, its representation is nonterminating (but periodic), while in base 3, it is terminating. We could presumably find bases (irrational bases, I guess) in which the representation of 3 would be neither terminating nor periodic, but the number itself would still be rational. ---
**Author:** ![Padeye](https://avatars.discourse-cdn.com/v4/letter/p/a9a28c/32.png) [@Padeye](https://boards.straightdope.com/u/Padeye)\ **Post date:** [July 13, 2005, 4:48pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/13 "2005-07-13T16:48:03Z")
This is the other side of the “could there be a base pi number system” so that pi would no longer be irrational. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [July 13, 2005, 5:09pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/14 "2005-07-13T17:09:14Z")
> [@Padeye](#): > > This is the other side of the “could there be a base pi number system” so that pi would no longer be irrational. In a base pi system, while pi would be just 10, all integers greater than 3 would have non-terminating representations, e.g. 4 would be something like 10.22012202… (Arithmetic would become extremely hard in such a system). ---
**Author:** ![Section\_408](https://avatars.discourse-cdn.com/v4/letter/s/f08c70/32.png) [@Section\_408](https://boards.straightdope.com/u/Section_408)\ **Post date:** [July 13, 2005, 5:20pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/15 "2005-07-13T17:20:44Z")
> [@aahala](#): > > I have a truly marvelous demonstration of this proposition which this post is too small to contain.😃 Are we going to have to wait 300+ years to see it? ---
**Author:** ![iamthewalrus\_3](https://avatars.discourse-cdn.com/v4/letter/i/258eb7/32.png) [@iamthewalrus\_3](https://boards.straightdope.com/u/iamthewalrus_3)\ **Post date:** [July 13, 2005, 5:23pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/16 "2005-07-13T17:23:32Z")
Base pi would make for very difficult arithmetic, but some other bases would make arithmetic simpler. Base-60 is rather convenient for doing hand calculations in because 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12 and 1/15 all have terminating decimal notations. Adding 1/3 + 1/5 is a hassle in base-10, but is really easy in base-60. Also, for an irrational base numbering system, base-e is pretty useful for lots of things. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [July 13, 2005, 6:04pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/17 "2005-07-13T18:04:20Z")
> [@iamthewalrus(:3=](#): > > Base-60 is rather convenient for doing hand calculations in because 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12 and 1/15 all have terminating decimal notations. Adding 1/3 + 1/5 is a hassle in base-10, but is really easy in base-60. When I add 1/3 and 1/5 by hand, I get 8/15. ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [July 13, 2005, 6:06pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/18 "2005-07-13T18:06:48Z")
> [@iamthewalrus(:3=](#): > > Base-60 is rather convenient for doing hand calculations in because 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12 and 1/15 all have terminating decimal notations. Adding 1/3 + 1/5 is a hassle in base-10, but is really easy in base-60. Maybe, but those multiplication tables in base 60 would be a bear! Every kid would have to rote memorize what (base 10) 37 times 49 is, etc. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 13, 2005, 6:13pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/19 "2005-07-13T18:13:29Z")
> [@Padeye](#): > > This is the other side of the “could there be a base pi number system” so that pi would no longer be irrational. [symbol]p[/symbol] is irrational in any representation system. See post #12 for details. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [July 13, 2005, 7:02pm UTC](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341/20 "2005-07-13T19:02:18Z")
> [@Orbifold](#): > > When I add 1/3 and 1/5 by hand, I get 8/15. As a general rule, if p and q are different prime numbers, then: 1/p + 1/q = (p+q)/(pq) [Next page](https://boards.straightdope.com/t/irrational-numbers-in-non-base-10-systems/312341.md?page=2)