# A Basic Question\[about math\]--edited title **URL:** \ **Category:** Factual Questions\ **Created:** [April 22, 2010, 5:08pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005 "2010-04-22T17:08:41Z")\ **Posts on this page:** 18\ **Page:** 3
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [April 23, 2010, 1:38pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/41 "2010-04-23T13:38:52Z")
> [@Keeve](#): > > I see that some mod has changed the title of the thread from A Basic Question to A Basic Question [about math]. > > My guess is that he didn’t get the joke, that “basic” was referring to the thread being about bases. Joke or not, the title was unclear. ---
**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\ **Post date:** [April 23, 2010, 1:50pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/42 "2010-04-23T13:50:50Z")
> [@Frylock](#): > > Sorry I wasn’t clear: I’m interested in the claim that you _can’t_ do this in base 10. Is that proven, or is it just that we don’t know a way? Ah. I think it’s the case that we merely don’t know a way yet to do it in base 10, not that it’s been proven impossible. (And that’s actually what I read **Chronos** to be referring to.) ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [April 23, 2010, 3:11pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/43 "2010-04-23T15:11:06Z")
> [@](#): > > Oh, and I also once encountered a problem in a program I was writing where it was most natural to work in base 11, but that’s kind of a fluke. So you’re the guy who had to set up Nigel Tufnel’s studio. ---
**Author:** ![K364](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/k364/32/5_2.png) [@K364](https://boards.straightdope.com/u/K364)\ **Post date:** [April 23, 2010, 4:32pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/44 "2010-04-23T16:32:08Z")
Not sure if it has been mentioned, but base 12 would link up with time and calender. For example, 100 months is 10 years. 200 hours is 10 days. (all base 12 numbers, hope this is correct!) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 23, 2010, 6:14pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/45 "2010-04-23T18:14:02Z")
> [@Capt.Ridley\_s\_Shooting\_Party](#): > > It’s more “efficient” in the following (possibly useless) sense: given an abacus with a fixed set of beads, with what base can you count the highest? > > Generally, the answer is given by the function f(n) = n^(K/n) where K is the number of beads on your abacus. Differentiating, then setting equal to zero, we find the maximum at e. Ooh… that’s interesting; I didn’t think of that. That perfectly explains the efficiency claim; I retract my skepticism. Of course, abacuses are discrete, so putting this another way, making sense of the continuous aspect: for any event which may or may not occur, we can define the “information it is responsible for” as the information contained in semilearning about the event (learning of the event if it does occur, and learning nothing otherwise). That is, the information an event is responsible for is defined for any particular case as the information contained in learning that the event occurred, in those cases where it does occur, and as 0 in those cases where it does not occur. For example, whenever one learns the value of a random integer between 0 and 999, one learns three decimal digits of information, but different events are responsible for different parts of that information; if I learn that the random number is 538, then “The first digit is 5”, “The middle digit is 3”, and “The last digit is 8” are each responsible for 1 decimal digit of the information I learnt, while “The number is even” is responsible for 1 bit of the information I learnt, and “The last digit is 2” is responsible for none of the information I learnt. Alright, so this defines information-responsibility. Then, for any event, we can ask, on probabilistic average, how much information is it responsible for? For example, half the time, “The number is even” is responsible for 1 bit of information and half the time it is responsible for 0 information, so, on average, this is responsible for only half a bit of information. (“The number is odd” is also responsible for only half a bit of information on average, and the information contained in learning a number’s parity is the information contained in semilearning both of these, and thus their sum, 1 full bit on average (and, indeed, all the time), as expected). On the other hand, while “The number’s last digit is 4” is sometimes responsible for a full decimal digit, which is much more than a bit, on average, it is only responsible for a tenth of a decimal digit, which is less than half a bit. So an event with probability 1/2 is responsible for more information on average than an event with probability 1/10, even though learning the latter is learning much more information than learning the former. In this sense, base-2 digits are more efficient than base-10 digits. Of course, the average magnitude of the information an event is responsible for is the event’s probability \* the event’s information-magnitude [thus, p \* log(p) for an event with probability p, per the previous discussion]. This is maximized when the probability is 1/e and the information-magnitude is 1 nat. Accordingly, nats (i.e., base-e “digits”) are the most efficient information-magnitudes of all, in terms of expected information-responsibility. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 23, 2010, 6:30pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/46 "2010-04-23T18:30:57Z")
> [@Indistinguishable](#): > > [thus, p \* log(p) for an event with probability p, per the previous discussion] (To clarify, when I wrote log(p), I meant the function which sends a probability p to the corresponding magnitude of information. If the base of the logarithm is q, then the information is measured in units of the information magnitude corresponding to a probability of q. So base 1/2 logarithms measure a probability’s information magnitude in bits, and base 1/10 logarithms measure it in decimal digits, while base 2 logarithms oppositely measure it in negative bits, and so on. So perhaps I should’ve just written -log(p) instead.) ---
**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\ **Post date:** [April 23, 2010, 6:37pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/47 "2010-04-23T18:37:51Z")
> [@Keeve](#): > > I see that some mod has changed the title of the thread from A Basic Question to A Basic Question [about math]. > > My guess is that he didn’t get the joke, that “basic” was referring to the thread being about bases. Did you actually understand what the thread would be about, without opening the thread? I couldn’t tell if it was about different number systems, or about chemicals with large Ph, or about a computer language, or about military installations, or something else. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 23, 2010, 6:55pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/48 "2010-04-23T18:55:48Z")
> [@Indistinguishable](#): > > (To clarify…) I should also clarify that sometimes I write “digit” and such things to mean a quantity of information (e.g., a “digit” as the information in learning which of 10 many possibilities holds) and sometimes I write it to mean a particular event (e.g., taking each of 10 many possibilities as a particular digit; each particular “digit” in this latter sense is responsible for only 1/10 of a “digit” of information, per the former sense). Hopefully, the context should generally make clear which I mean. ---
**Author:** ![mnemosyne](https://avatars.discourse-cdn.com/v4/letter/m/c4cdca/32.png) [@mnemosyne](https://boards.straightdope.com/u/mnemosyne)\ **Post date:** [April 23, 2010, 8:50pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/49 "2010-04-23T20:50:22Z")
> [@Napier](#): > > That is debated. Moles aren’t units like the other ones. The Wikipedia article covers this a little: > “Since its adoption into the International System of Units, there have been a number of criticisms of the concept of the mole being a unit like the metre or the second.[6] These criticisms may be briefly summarised as: > amount of substance is not a true physical quantity (or dimension), and is redundant to mass, so should not have its own base unit; > the mole is simply a shorthand way of referring to a large number.” I did not know that. Though as a (former) chemist, I’ve only ever encountered it in terms of “per mole” and in calculations done on a molar basis in which unit analysis would require the number to be in mol[sup]-1[/sup], and so in that context, for practical applications, I still maintain that it isn’t exactly dimensionless. As of now, it is a unit in the SI system, and that’s good enough for me, although I can see the reasoning behind the criticism. So my nitpick has its own nit. Such is life! 😉 ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [April 23, 2010, 9:22pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/50 "2010-04-23T21:22:43Z")
> [@Frylock](#): > > Sorry I wasn’t clear: I’m interested in the claim that you _can’t_ do this in base 10. Is that proven, or is it just that we don’t know a way? We don’t know a way. It wouldn’t at all surprise me if some day, someone finds a way. ---
**Author:** ![HoboStew](https://avatars.discourse-cdn.com/v4/letter/h/f17d59/32.png) [@HoboStew](https://boards.straightdope.com/u/HoboStew)\ **Post date:** [April 23, 2010, 9:49pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/51 "2010-04-23T21:49:08Z")
One nice thing about base 10 is that it scales well. 10^8 = 100000000. I have no idea what 12^8 is, or 8^8, etc. ---
**Author:** ![Bosstone](https://avatars.discourse-cdn.com/v4/letter/b/22d042/32.png) [@Bosstone](https://boards.straightdope.com/u/Bosstone)\ **Post date:** [April 23, 2010, 10:08pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/52 "2010-04-23T22:08:29Z")
never mind ---
**Author:** ![Gary\_Robson](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gary_robson/32/3448_2.png) [@Gary\_Robson](https://boards.straightdope.com/u/Gary_Robson)\ **Post date:** [April 23, 2010, 10:52pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/53 "2010-04-23T22:52:14Z")
> [@HoboStew](#): > > One nice thing about base 10 is that it scales well. 10^8 = 100000000. I have no idea what 12^8 is, or 8^8, etc. All bases scale well if you’re not converting the numbers to some other base. 8^8 _in base 8_ would be written as 10^10, or 100,000,000. Similarly, 12^8 _in base 12_ would be written as 10^8, which would be 100,000,000 (or maybe 1,0000,0000 😉 ) ---
**Author:** ![Apex\_Rogers](https://avatars.discourse-cdn.com/v4/letter/a/e480ec/32.png) [@Apex\_Rogers](https://boards.straightdope.com/u/Apex_Rogers)\ **Post date:** [April 23, 2010, 10:54pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/54 "2010-04-23T22:54:37Z")
> [@HoboStew](#): > > One nice thing about base 10 is that it scales well. 10^8 = 100000000. I have no idea what 12^8 is, or 8^8, etc. This is inherent to the base system. 12^8 in base 12 is 100000000, and 8^8 in base 8 is 100000000 as well. You can see this by breaking the number down into digits and multiplying them by the power that they represent (ie 0 ones, 0 tens, 0 hundreds, … 1 hundred-million): 10^8 = 1_10^8 + 0_10^7 + 0_10^6 + … + 0_10^0 12^8 = 1_12^8 + 0_12^7 + 0_10^6 + … + 0_12^0 ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 23, 2010, 11:49pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/55 "2010-04-23T23:49:11Z")
I’m gonna take a wild guess that **HoboStew** was joking, guys… ---
**Author:** ![HoboStew](https://avatars.discourse-cdn.com/v4/letter/h/f17d59/32.png) [@HoboStew](https://boards.straightdope.com/u/HoboStew)\ **Post date:** [April 24, 2010, 12:13am UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/56 "2010-04-24T00:13:06Z")
> [@Indistinguishable](#): > > I’m gonna take a wild guess that **HoboStew** was joking, guys… Actually no. I wasn’t. I’m just retarded. Move along! :smack: ---
**Author:** ![Kimmy\_Gibbler](https://avatars.discourse-cdn.com/v4/letter/k/bbe5ce/32.png) [@Kimmy\_Gibbler](https://boards.straightdope.com/u/Kimmy_Gibbler)\ **Post date:** [April 24, 2010, 3:08am UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/57 "2010-04-24T03:08:16Z")
Fifty-six posts in and nobody has suggested [quater-imaginary](http://en.wikipedia.org/wiki/Quater-imaginary_base) yet!? The dork quotient really has fallen on this board. ---
**Author:** ![Keeve](https://avatars.discourse-cdn.com/v4/letter/k/f07891/32.png) [@Keeve](https://boards.straightdope.com/u/Keeve)\ **Post date:** [April 25, 2010, 9:44pm UTC](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005/58 "2010-04-25T21:44:19Z")
> [@Kimmy\_Gibbler](#): > > Fifty-six posts in and nobody has suggested [quater-imaginary](http://en.wikipedia.org/wiki/Quater-imaginary_base) yet!? The dork quotient really has fallen on this board. That is one whacked-out system! 100 is exactly halfway between 101 and 203? But it works! I have got to read that article a few more times!!! [Previous page](https://boards.straightdope.com/t/a-basic-question-about-math-edited-title/537005.md?page=2)