# non-base 10 number systems **URL:** \ **Category:** Factual Questions\ **Created:** [March 20, 2002, 2:26pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208 "2002-03-20T14:26:33Z")\ **Posts on this page:** 15\ **Page:** 2
**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\ **Post date:** [March 20, 2002, 4:45pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/21 "2002-03-20T16:45:07Z")
I made my last post before seeing what **David Simmons** and **friedo** wrote and they seemed to have answered my question/problem. Thanks… ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [March 20, 2002, 4:46pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/22 "2002-03-20T16:46:46Z")
> [@](#): > > \*Originally posted by Whack-a-Mole \* > \*\* > > Not at all. I think I am still missing something here so I was using examples of groupings of tens and groupings of 12 to suggest which is more intuitive and \*not to give a lesson on kindergarten math. > > \*\* Sorry I insulted you. I withdraw. You can think, or not, about number systems any way you choose. ---
**Author:** ![yabob](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yabob/32/2821_2.png) [@yabob](https://boards.straightdope.com/u/yabob)\ **Post date:** [March 20, 2002, 4:57pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/23 "2002-03-20T16:57:26Z")
We may have been slightly better off if we had evolved with an extra finger on each hand. There are small advantages in everyday use to having a base with a lot of factors, such as the decimal representation of common fractions. 10 has factors of 2 and 5. Hence 1/3 comes out as a repeating decimal - 0.33333 … And because there’s only one factor of 2, 1/4 comes out to two places. If we were using base 12, the commonly used fractions 1/2, 1/3 and 1/4 would all come out as single place decimals - 0.6, 0.4, 0.3. True, 1/5 would be a repeating decimal (0.222 …), but that’s probably an adequate trade. Similar advantages result from ability to factor your base in areas like the sorts of coinage we would use - we would be able to have more coins of convenient sizes which were even multiples of the larger denomination. ---
**Author:** ![knock\_knock](https://avatars.discourse-cdn.com/v4/letter/k/22d042/32.png) [@knock\_knock](https://boards.straightdope.com/u/knock_knock)\ **Post date:** [March 20, 2002, 5:09pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/24 "2002-03-20T17:09:09Z")
I agree with **David Simmons** and **friedo**. The only number base that makes any sense is base 10. And if we ever heard from aliens (no matter how many fingers/tentacles they have), they would all tell us that they use base 10 as well. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [March 20, 2002, 5:19pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/25 "2002-03-20T17:19:22Z")
> [@](#): > > \*Originally posted by Whack-a-Mole \* > \*\*If I’m missing something here I think it may be on the base definition of a base numbering system and exactly how they are supposed to work. \*\* Are you aware that 10^4 (base ten) is 10,000, and 10^4 (base 12) is 10,000? The numbers are different–10^4 (base 12) is 20,736 (base 10)–but manipulating them is just as easy. In base 12, 10,000 divided by 10 is 1,000. ---
**Author:** ![Beetle99](https://avatars.discourse-cdn.com/v4/letter/b/35a633/32.png) [@Beetle99](https://boards.straightdope.com/u/Beetle99)\ **Post date:** [March 20, 2002, 5:43pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/26 "2002-03-20T17:43:38Z")
Thanks everyone! Knock Knock…i think you may have missed what frieda and Dave said. I believe they were saying it doesnt matter whether a number system is base 10 or 8 or 12. It just is an assistance with arithmetic. Quantities are still quantities…and prime numbers and things all remain the same… And I guess PI is still 3.14xxxxxx…right? ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [March 20, 2002, 5:47pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/27 "2002-03-20T17:47:06Z")
> [@](#): > > \*Originally posted by Manatee \* > \*\* > > I thought that the “teen” part of 13-19 originated with “ten” (so “three and ten,” “four and ten,” and so on). If so, how is that a unique name? (Eleven and Twelve, though, screw up my answer and certainly support yours).\*\* Not at all. Eleven and twelve support it, because they come from Germanic roots that add to ten. Or, from [dictionary.com](http://dictionary.com) ([http://www.dictionary.com/search?q=eleven](http://www.dictionary.com/search?q=eleven)): “Thus, eleven is literally “one-left” (over, that is, past ten), and twelve is “two-left” (over past ten).” ---
**Author:** ![Padeye](https://avatars.discourse-cdn.com/v4/letter/p/a9a28c/32.png) [@Padeye](https://boards.straightdope.com/u/Padeye)\ **Post date:** [March 20, 2002, 6:09pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/28 "2002-03-20T18:09:17Z")
> [@](#): > > \*Originally posted by David Simmons \* > \*\*To convert a binary number to octal: Say 1011101, start from the right and divide it into groups of three, adding zeros to the left as needed. 001 011 101, then write the digit corresponding to each group of three: 001 = 1, 011 = 3, 101 = 5 and _voyla! (just a little joke you French speaking dopers)_, the octal number is 135. \*\* That has actually been put into use. When I worked on F-14s in the navy the computer used 24 bit memory registers which were displayed in 8 digits of octal. With a little practice we could easily memorize critical status bits and do the octal to binary conversion in our heads. This made for quick troubleshooting, a good thing when standing in the nose wheel well of a plane about to launch talking to the crew on a headset. FYI _voila_ ---
**Author:** ![Jman](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jman/32/4416_2.png) [@Jman](https://boards.straightdope.com/u/Jman)\ **Post date:** [March 20, 2002, 6:17pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/29 "2002-03-20T18:17:53Z")
Some of you are misunderstanding what a number base is. I’ll try to make it make sense: Base - 10. This means that we have 10 distinct digits for counting. When we run out, we move one place to the left (the Tens place) and start counting again. In our base 10, the symbols are 0,1,2,3,4,5,6,7,8,9. When you run out of numbers (ie 1 more than 9,) you put a 1 in the next place and start over. In a different base, say base 16 (hexadecimal), you have 16 different symbols to use for counting. In our numeric system, we use the first 6 digits of the alphabet to supplement our numbers. So, the symbols are: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. When you get to F, you move over one place and start over again. (ie 1 more than F = 10). In this same way, 10\*10=100 However, there are more parts per place than in base 10, so 10 in base 16 is equivalent to 16 in base 10. So, if I see the number 6D in base 16, and I want to change it to base 10 to get an understanding of it (since we’re accustomed to base 10 numbers), I simply say “there are 6 groups of 16, and D groups of 1.” Since D=13, we have 6\*16+13 = 109 in base 10. So, 6D in base 16 is the same number as 109 in base 10. However, the math is the same…when you get to the end of a place (run out of numbers), you increase the next place by one and start over, just like in base 10. If we had 16 fingers, and base 16 was natural to you, doing 4F_15 in hex would be as easy as doing 79_21 in decimal (they’re the same calculation.) Hope this cleared things up. ---
**Author:** ![Balance](https://avatars.discourse-cdn.com/v4/letter/b/ccd318/32.png) [@Balance](https://boards.straightdope.com/u/Balance)\ **Post date:** [March 20, 2002, 6:25pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/30 "2002-03-20T18:25:03Z")
> [@](#): > > \*Originally posted by jsleek \* > I think the Babylonians had a base 60 system. I suppose it derived from 360 days in the year. That is, the sun moves through the stars one degree a day. Thus, our 360-degree circles and so forth. I don’t have anything of real substance to add. I just wanted to comment that my 3rd-grade math teacher gave us a test composed entirely of base-60 arithmetic, complete with a chart of symbols for the digits. ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [March 20, 2002, 6:49pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/31 "2002-03-20T18:49:33Z")
> [@](#): > > \*Originally posted by yabob \* > \*\*We may have been slightly better off if we had evolved with an extra finger on each hand. > > There are small advantages in everyday use to having a base with a lot of factors, such as the decimal representation of common fractions. 10 has factors of 2 and 5. Hence 1/3 comes out as a repeating decimal - 0.33333 … And because there’s only one factor of 2, 1/4 comes out to two places. > > If we were using base 12, the commonly used fractions 1/2, 1/3 and 1/4 would all come out as single place decimals - 0.6, 0.4, 0.3. True, 1/5 would be a repeating decimal (0.222 …), but that’s probably an adequate trade. > > Similar advantages result from ability to factor your base in areas like the sorts of coinage we would use - we would be able to have more coins of convenient sizes which were even multiples of the larger denomination. \*\* The real advantage to grouping the counting of items by groups of units of some base number is in arithmetic. How, for example would you ever multiply XIII\*VII? Using number systems to some base it is easy. You simply construct a table of products of two numbers up to the numbers that are one less than the base by repetitive addition, or even simple counting if necessary. This table combined with a number system using a base number and positional notation facilitates the multiplication of any two numbers of any size at all. Likewise with other arithmetic operations. ---
**Author:** ![panamajack](https://avatars.discourse-cdn.com/v4/letter/p/47e85d/32.png) [@panamajack](https://boards.straightdope.com/u/panamajack)\ **Post date:** [March 20, 2002, 7:04pm UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/32 "2002-03-20T19:04:01Z")
> [@](#): > > \*Originally posted by Beetle99 \* > > And I guess PI is still 3.14xxxxxx…right? Well no … numbers after the radix point[sup]\*[/sup] are fractions, with the denominator as a negative power, just as to the left of the radix point. So [sym]p[/sym] has a different notation for each base, and usually some sort of algorithm is used to calculate it. (This is the same method as in base 10, so there’s still no difference.) The way these digits are expressed is well-illustrated at this site, which also lets you have some fun : [Search for your name in the digits of PI!](http://www.angelfire.com/mt/ofolives/pisearch.html) What this guy has done is convert [sym]p[/sym] to base 27, using the alphabet for the digits. Long names aren’t likely to be found, since he doesn’t have a lot of digits. But the search is very helpful in explaining how fractions work in other bases. [sup]\*[/sup] Radix is used as a synonym for base, so the radix point is the division between fractions and integers. When working with a particular base, you can use the name of the base for the radix point, hence ‘decimal point’, ‘binary point’, ‘sexagesimal point’, etc. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [March 21, 2002, 1:29am UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/33 "2002-03-21T01:29:59Z")
It’s very easy to find ten to any power in base ten, but it’s difficult to find powers of a dozen in base ten. Likewise, it’s very easy to find powers of a dozen in base dozen, but it’s very difficult to find powers of ten in base dozen. Note: I’m using the words here, instead of writing the symbols “10” and “12”, because those symbols mean different things in different bases. A dozen is represented by the symbol “12” in base ten, but that same number is represented by the symbol “10” in base dozen. If I want to take a dozen to the third power, in base dozen, all I have to do is move the dozimal place three to the right: A dozen to the third power, in base dozen, is written “1000”. Now, it might be hard to figure out what the symbol for that number is in base ten, but if we were used to using base dozen, we wouldn’t care about that, because we wouldn’t be trying to convert to base ten. ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [March 21, 2002, 1:34am UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/34 "2002-03-21T01:34:55Z")
I don’t se how its possible to have an irrational number base system (such as a multiple of e or pi). How would base -10 or base 10i work out? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 21, 2002, 1:36am UTC](https://boards.straightdope.com/t/non-base-10-number-systems/99208/35 "2002-03-21T01:36:47Z")
> [@](#): > > \*Originally posted by Enola Straight \* > \*\*I don’t se how its possible to have an irrational number base > system (such as a multiple of e or pi). > > How would base -10 or base 10i work out? \*\* Pretty much exactly the same as any other base, although if the base is irrational, no rational number will have a representation of finite length. Since i[sup]2[/sup] = -1, you just have to use a lot of zeroes. [Previous page](https://boards.straightdope.com/t/non-base-10-number-systems/99208.md?page=1)