# Is the base of a number counting system irrelevant to mathematics? **URL:** \ **Category:** Factual Questions\ **Created:** [July 10, 2006, 1:43am UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806 "2006-07-10T01:43:03Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![BJMoose](https://avatars.discourse-cdn.com/v4/letter/b/5fc32e/32.png) [@BJMoose](https://boards.straightdope.com/u/BJMoose)\ **Post date:** [July 10, 2006, 5:07pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/21 "2006-07-10T17:07:24Z")
> [@BJMoose](#): > > There is one area where the number base has a practical relevance: the conversion of fractions to “decimal” numbers. In base 10 the only fractions that can be converted to a precise decimal number are those whose denominators are whole number multiples of 2 and/or five - the factors of 10. Thus in base 10 the fraction 1/3 converts to the imprecise 0.333… . But in base 3, 1/3 is 0.1, the first positon to the right of the decimal, er, trimal point representing thirds (in the same base, 1/2 is 0.111… [one third plus one ninth plus one twenty-seventy &c. &c.). > > I wouldn’t dare try to restate 1/3 in base pi. . . . D’oh! Would you believe, twenty-seventh (for “twenty-seventy” above). ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 10, 2006, 5:20pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/22 "2006-07-10T17:20:25Z")
Here are the rules for digits in base n:[ul] [li]If |n| \> 1, the symbols are the integers in the range [0, |n|) and everything works exactly like you’d expect.[/li][li]If 1 \> |n| \> 0, the symbols are the integers in the range [0, |1/n|). Number representations may extend infinitely to the left, but will terminate to the right. For instance, (1/3)[sub]10[/sub] = …333.[sub]1/10[/sub].[/li][li]If |n| = 1 or |n| = 0, it’s just not going to work.[/li][li]If n \< 0, you don’t need a sign marker (a negative sign, a bit, etc.) because that information is already contained in the digits.[/li][/ul]As **Exapno Mapcase** mentions, you can use any symbols you like, but since they have to behave exactly like integers with respect to addition and ordering, what’s the benefit of using something odd? Wikipedia has [articles on non-standard positional numeral systems](http://en.wikipedia.org/wiki/Category:Non-standard_positional_numeral_systems), some of which are not base n for any n. ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [July 10, 2006, 5:32pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/23 "2006-07-10T17:32:07Z")
> [@jayjay](#): > > As **TJDude** said above, you’d need 3 symbols: 0, 1, and pi. Pi is only 3.1415… in base 10. In base pi, pi would simply be pi. Just to make it clear, you need three symbols, for example 0, 1 and **2**. Pi would be 10. ---
**Author:** ![sinjin](https://avatars.discourse-cdn.com/v4/letter/s/ebca7d/32.png) [@sinjin](https://boards.straightdope.com/u/sinjin)\ **Post date:** [July 10, 2006, 5:59pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/24 "2006-07-10T17:59:10Z")
> [@naita](#): > > Just to make it clear, you need three symbols, for example 0, 1 and **2**. > Pi would be 10. Right… I’ll come in again. You need **four** symbols, 0, 1, 2, and 3. I will put them into bold-face type to make them really pop. **0** , **1** , **2** , **3**. Ok, so what do these symbols represent? Well, **0** = 0 **1** = Pi/4 **2** = Pi/2 **3** = 3Pi/4 **10** = Pi **11** = 5Pi/4 et cetera. At least, this is how I would guess it would work. If I remember I’ll come back tonight and see how this thread is going. sinjin For the record, I have most of a PhD in mathematics and at no point in my mathematical career has this ever been an issue. I’ll try to decide just what it means to have irrational bases and if I can write it up nicely without using .pdf files. 🙂 ---
**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\ **Post date:** [July 10, 2006, 6:08pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/25 "2006-07-10T18:08:27Z")
> [@sinjin](#): > > You need **four** symbols, 0, 1, 2, and 3. Correct. (Cite: **ultrafilter’s** post.) > [@sinjin](#): > > Ok, so what do these symbols represent? Well, > > **0** = 0 > **1** = Pi/4 > [. . .] No, those symbols represent exactly what they would in any base: the integers zero, one, two, and three. ---
**Author:** ![Lemur866](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lemur866/32/434_2.png) [@Lemur866](https://boards.straightdope.com/u/Lemur866)\ **Post date:** [July 10, 2006, 6:17pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/26 "2006-07-10T18:17:23Z")
But how can that work? If 1=1, 2=2, 3=3 and 10=pi, how can you do arithmetic? What would 2+2 equal? It can’t equal 10. What does 11 equal? Pi+1? But 100 has to eqal pi^2, right? What does 20 equal, 2pi? If you have a non-integer base, then I can’t see how any integer in that counting system could be a decimal integer. ---
**Author:** ![CookingWithGas](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cookingwithgas/32/485_2.png) [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)\ **Post date:** [July 10, 2006, 6:22pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/27 "2006-07-10T18:22:25Z")
> [@Bytegeist](#): > > No, those symbols represent exactly what they would in any base: the integers zero, one, two, and three. I’m still not getting this. Wouldn’t 3.3 represent 3 + 3/[symbol]p[/symbol], which is about 3.955[sub]10[/sub]? But that is greater than 10[sub][symbol]p[/symbol][/sub]. So you can define a number base such that 3.3 \> 10? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 10, 2006, 6:28pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/28 "2006-07-10T18:28:13Z")
> [@CookingWithGas](#): > > I’m still not getting this. > > Wouldn’t 3.3 represent 3 + 3/[symbol]p[/symbol], which is about 3.955[sub]10[/sub]? But that is greater than 10[sub][symbol]p[/symbol][/sub]. So you can define a number base such that 3.3 \> 10? In base n, a[sub]k[/sub]…a[sub]0[/sub].a[sub]-1[/sub]…a[sub]-m[/sub] represents sum( a[sub]i[/sub]n[sup]i[/sup], -m \< i \< k ). The second three only represents 3/[symbol]p[/symbol] because of its position. ---
**Author:** ![Lemur866](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lemur866/32/434_2.png) [@Lemur866](https://boards.straightdope.com/u/Lemur866)\ **Post date:** [July 10, 2006, 6:39pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/29 "2006-07-10T18:39:27Z")
Back up a minute there. In base pi, what does 2+2 or 3+3 equal? How would you write that? 1=1 2=2 3=3 10=pi 11=pi+1 12=pi+2 13=pi+3 20=2pi 21=2pi+1 22=2pi+2 23=2pi+3 30=3pi 31=3pi+1 32=3pi+2 33=3pi+3 100=pi^2. It seems to me that 2 and 3 are pretty useless characters. How do you write 4pi? How do you write pi/2? I mean, in base pi, shouldn’t our standard base 10 integers be irrational numbers? ---
**Author:** ![jayjay](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jayjay/32/6765_2.png) [@jayjay](https://boards.straightdope.com/u/jayjay)\ **Post date:** [July 10, 2006, 6:44pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/30 "2006-07-10T18:44:50Z")
> [@gazpacho](#): > > What? Where is TJDude saying that. There is something I am missing here. Why would there be a symbol for pi? In base 10 there is no symbol for 10. It is made with two symbols? You’re right…I thought I understood his post but I had it garbled. Never mind… ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 10, 2006, 6:46pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/31 "2006-07-10T18:46:06Z")
> [@Lemur866](#): > > I mean, in base pi, shouldn’t our standard base 10 integers be irrational numbers? No, but they do have non-terminating and non-repeating representations. ---
**Author:** ![Voyager](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/voyager/32/133_2.png) [@Voyager](https://boards.straightdope.com/u/Voyager)\ **Post date:** [July 10, 2006, 6:48pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/32 "2006-07-10T18:48:08Z")
perhaps the source of confusion here is that we are conflating the integers 1, 2, and 3 with the symbols 1,2, and 3 in **sinjin** ’s numbering system. Perhaps if we used p, q, r and s it might be clearer. In a system of base pi, you would not be able to exactly represent a standard integer. We can keep 0 Actually, I’d think the number of symbols between 0 and “10” (pi) would be arbitrary. You’d have just as good a system if you used p = pi / 8 q = pi / 4 etc. In general, if you choose n symbols between 0 and pi, the ith would be i pi / n. The one glitch I can see is how you prove things using real integers when real integers are not well defined in the system. You would pretty much need to define integers anyhow independently. On preview, **Lemur866** brings up a good point. You need integers anyhow to express what pi+pi is in a positional numbering system. The problem is that 1 is fundamental as the multiplicative identity, so I’m not sure how you’d express this in a system where 1 is not directly and exactly expressible. Or could we just assign a symbol to it, like we do for, say, pi and e, which are not expressible in our standard number systems. ---
**Author:** ![Voyager](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/voyager/32/133_2.png) [@Voyager](https://boards.straightdope.com/u/Voyager)\ **Post date:** [July 10, 2006, 6:53pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/33 "2006-07-10T18:53:38Z")
> [@Bytegeist](#): > > Well, base **zero** might present a few difficulties. > > And, base one isn’t much better. Base 0 is not well defined. Base 1, on the other hand is. I’ve written some parody papers on base 1 arithmetic, and you can prove, using information theory, that base 1 machines need no power. (They also produce no information. ) I call the basic unit of base 1 a nit, so a base 1 engineer is a nit picker, and a genius at base 1 arithmetic is a nitwit. BTW, I missed **Exapno** ’s post on the arbitrariness of symbols when I did mine, but at least I applied it to the base pi problem. ---
**Author:** ![CookingWithGas](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cookingwithgas/32/485_2.png) [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)\ **Post date:** [July 10, 2006, 7:00pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/34 "2006-07-10T19:00:01Z")
> [@ultrafilter](#): > > In base n, a[sub]k[/sub]…a[sub]0[/sub].a[sub]-1[/sub]…a[sub]-m[/sub] represents sum( a[sub]i[/sub]n[sup]i[/sup], -m \< i \< k ). The second three only represents 3/[symbol]p[/symbol] because of its position. We agree completely on that but it doesn’t address my question. I am responding to the post that says the digits in base pi are the decimal digits 0-3. Then by definition (as you have detailed) 3.3 \> 10. This seems as nonsensical as saying 9.9 \> 10 if you’re in base 10. ---
**Author:** ![Clothahump](https://avatars.discourse-cdn.com/v4/letter/c/51bf81/32.png) [@Clothahump](https://boards.straightdope.com/u/Clothahump)\ **Post date:** [July 10, 2006, 7:01pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/35 "2006-07-10T19:01:03Z")
> [@Tripler](#): > > I wish I had [3.14159265 fingers](http://www.fat-pie.com/salad.htm). Dammit, **Tripler**!! ::Searching for something to gouge my mind’s eye out. Perhaps a rusty spoon?:: ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 10, 2006, 7:06pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/36 "2006-07-10T19:06:26Z")
> [@CookingWithGas](#): > > We agree completely on that but it doesn’t address my question. I am responding to the post that says the digits in base pi are the decimal digits 0-3. Then by definition (as you have detailed) 3.3 \> 10. This seems as nonsensical as saying 9.9 \> 10 if you’re in base 10. With all the other things that behave oddly in base [symbol]p[/symbol], I wouldn’t be surprised if our intuition about magnitudes is off. It doesn’t matter, though–as long as there’s a one-to-one correspondence between representations and real numbers, every other property is up for grabs. > [@Voyager](#): > > Actually, I’d think the number of symbols between 0 and “10” (pi) would be arbitrary. You’d have just as good a system if you used > > p = pi / 8 > q = pi / 4 > > etc. In general, if you choose n symbols between 0 and pi, the ith would be i pi / n. True, but unless you do it as I have, it’s not base [symbol]p[/symbol]. ---
**Author:** ![Voyager](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/voyager/32/133_2.png) [@Voyager](https://boards.straightdope.com/u/Voyager)\ **Post date:** [July 10, 2006, 7:25pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/37 "2006-07-10T19:25:24Z")
> [@ultrafilter](#): > > With all the other things that behave oddly in base [symbol]p[/symbol], I wouldn’t be surprised if our intuition about magnitudes is off. It doesn’t matter, though–as long as there’s a one-to-one correspondence between representations and real numbers, every other property is up for grabs. > > True, but unless you do it as I have, it’s not base [symbol]p[/symbol]. The problem is that the coefficients are not directly representable in base p, where p is irrational! But since you need an integer number of symbols, traditional integers factor into things anyhow. So, you can’t represent integers exactly in the base selected, but they are used as if represented exactly in arithmetic in this base. This shows, if nothing else, you can’t reason as simply about irrational bases as you can about integer bases. ---
**Author:** ![Lemur866](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lemur866/32/434_2.png) [@Lemur866](https://boards.straightdope.com/u/Lemur866)\ **Post date:** [July 10, 2006, 7:30pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/38 "2006-07-10T19:30:53Z")
OK, I think I’ve had an intuition. In base pi, the symbols 1, 2, and 3 are equivalent to _e_ and pi and phi. They are interesting numbers, but you can’t do much with them, except 1. So you can write 2+2, but there isn’t a simple number equivalent to our base 10 4, just like there’s no simple way to write pi + pi or pi\*\*e\* in base 10. You just have to leave those pi’s in. So… 0=0 1=pi^0, or 1 10=pi^1, or pi 100=pi^2 1000=pi^3 etc. The only reason 1 doesn’t equal pi is that it’s so much easier if 1=1, that way you can still do arithmetic. And you’ve got some special characters 2 and 3 but using them could cause problems, so 3.3 is analgous to writing _e.e_ in base 10. One more thing. What’s the difference between a nonterminating, nonrepeating number and an irrational number? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 10, 2006, 7:30pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/39 "2006-07-10T19:30:54Z")
> [@Voyager](#): > > The problem is that the coefficients are not directly representable in base p, where p is irrational! The coefficients aren’t written in base pi. They’re written as whatever set of symbols you care to use. > [@](#): > > So, you can’t represent integers exactly in the base selected Sure you can. It just requires an infinite number of digits. Think like a mathematician, not a computer scientist. ---
**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\ **Post date:** [July 10, 2006, 7:36pm UTC](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806/40 "2006-07-10T19:36:11Z")
> [@CookingWithGas](#): > > We agree completely on that but it doesn’t address my question. I am responding to the post that says the digits in base pi are the decimal digits 0-3. Then by definition [. . .] 3.3 \> 10. This seems as nonsensical as saying 9.9 \> 10 if you’re in base 10. If it’s any consolation, 10[sub]pi[/sub] can be rewritten as 3.01102111[sub]pi[/sub] . . ., which makes it look nice and small again compared to 3.3[sub]pi[/sub], like it’s “supposed” to. But in general, all numbers except zero are going to have multiple representations in base pi, and that’s going to obscure their relative ordering — at least to the human eye. [Previous page](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806.md?page=1) [Next page](https://boards.straightdope.com/t/is-the-base-of-a-number-counting-system-irrelevant-to-mathematics/363806.md?page=3)