I have often wondered if humans had simply had fewer fingers (or more) what would our math look like. What if it was based upon 9 or 8 or 13…
Anyone know of any research done on non-base 10 number systems or of any significant consequences?
I can’t answer your question. But, Isaac Asimov once speculated that if the thumbs had been ignored and the 8-based system used we could directly read binary number computer data printouts. That way we wouldn’t have to write a binary to decimal conversion program for output data.
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.
Of course, the whole of Computer Science is based on the base 2 (binary) numbering system, which is why (for e.g.) RAM comes in multiples of 2 (8Mb, 16Mb, … 128Mb, 256Mb, etc).
And time is a base 60 numbering system, at least for seconds and minutes…
I believe that the Babylonians had a numbering system based on the number 16, but that may just be my mind playing tricks on me…
Gp
There are lots of base numbering systems. Basically, pick any number you like and you could make it a base numbering system. The reason 10 is most common is it is easy to work with. However, there are lots of others some of which are quite common. See if you recognize any of these:
[ul]
[li] Binary (base 2)[/li]
[li] Octal (base 8)[/li]
[li] Groupings using 12 (base 12)[/li]
[li] Hexadecimal (base 16)[/li][/ul]
Probably the weirdest one I ever heard of was a base 60 numbering system used by the Sumerians. I have no idea how that worked…sounds like a nightmare to me.
So long as you’re using a positive integer for your base, there’s not much difference at all (at least in higher math). In fact, I can’t think of any significant differences with any base (irrational, negative, or complex), at least not outside of basic arithmetic. Remember, the base n representation is the representation of your number, not the number itself.
And there are vestigal elements of a base-20 system around. Note that in English we have unique names for the numbers up to 20, then switch to the more regular twenty-one, twenty-two, etc. That’s true of many (all?) of the Indo-European languages.
In French, the word for eighty is “four-twenties.”
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.
Often someone says 12 would be the best system, but I favor 16. I don’t often divide a wigit by 3, but I frequently halve and quarter my wigits.
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).
As a side track, didn’t most of medieval Europe use a base 12 system?
I’d rather have it the other way round. It’s easy to work with because it’s so common.
You may be right but it sure seems easier to do math in tens in your head than with most other base numbering systems and I don’t think that is entirely subjective.
Multiply by ten just move the decimal point to the right one space.
Divide by ten move the decimal place left one space.
Add by ten just change the number in the ‘ten’ location up one.
Subtract by ten just change the number in the ‘ten’ location down one.
Working with 100s, 1000s, etc. is equally simple.
I really don’t see how a base-12 or base-8 (or whatever) numbering system can be more intuitive. 1010=100 (or two zeros [one zero for each 10] following the one). 1212=144…what’s easier about that (or worse, 144144=20,736 vs. 100100=10,000)? For different applications different base numbering systems may be more useful but for day-to-day math base-10 seems to work nicely.
The long-hundred (120) has a Germanic origin (I keep seeing it in viking novels).
I’m pretty sure that this will answer all your number needs, including the reasons for using a sexagesimal system (look, I learnt a new word 
).
Note, I say “pretty sure” because I dropped off a quarter of the way into page 1 (Carol F. Justus appears to be very thorough).
Sorry, all of your rules work for any number system if your just substitute “the base” where you use '10." For example, in the binary system 1010 = 100, in the three based number system 1010 = 100 in the 254 based number system 1010 = 100 and so on and so on. In the octal system to multiply 12 by 10 simply add a 0 so 1210=120, octal.
There isn’t anything more “magical” or even “intuitive” about 10 (decimal)than any other base.
Ah but your comparing base-8 (and base-12) with base-10 number symbols. Couldn’t base-8 could be represented by:
1,2,3,4,5,6,7,10, and thus 10*10 in base-8 would equal our
base-10 64, but still seem like a natural value in a base-8 system. Represented this way, any base-N system could seem natrual.
Unless I’m way off base here.
Exactly the same claim can be made for any (integer, positive) base:
10[sub]hex[/sub]*10[sub]hex[/sub]=100[sub]hex[/sub]
10[sub]oct[/sub]*10[sub]oct[/sub]=100[sub]oct[/sub]
etc.
In this world (and especialy those of us who use the SI system) we happen to use ‘ten’ a lot, but only because it’s so easy.
I often work in hexadecimal notation, which is very usefull when fiddling with bits in computers, but only because the computer itself uses binary.
I don’t think it’s a very risky bet to say that if we had been born with eight fingers we’d be using octal notation - and thought that was the only ‘natural’ system.
I’m not 100% sure about this, but I think that if you were brought up using base 12 (or base 8, or whatever), you wouldn’t notice any problems.
With base 12, your numbering would be:
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, 10.
(where a & b are newly created numbers)
So 10*10 still gives you 100 (it’s just that in base 10, an archaic base you don’t use, that would refer to 144). Similarly, subtracting “10” would just change the number in the ‘ten’ location down one.
True but then what’s the point of any base numbering system? Math works the same in any system you care to name so why bother defining it at all with bases?
Perhaps it would have been better writing my examples as 10[sup]2[/sup] or 100[sup]2[/sup]. It would seem to me that the equivalent in a base-12 system would be 12[sup]2[/sup] or 144[sup]2[/sup]. I suppose in a base-12 numbering system you could still use scientific notation such as 6.2310[sup]9[/sup] so again I ask why bother with a base-12 (or whatever) at all? Certainly figuring out what 6.2312[sup]9[/sup] equals would be a nightmare without a calculator.
They used base ten digits to represent the base 60 groupings, apparently. So, they would count up to 60 using base 10, then increment the base 60 digits. It’s kinda like counting seconds and minutes: 58 seconds, 59 seconds, 1 minute 0 seconds, 1 minute 1 second, etc. And hours. And degrees. We still do it, it’s not a nightmare at all.
And, Whack-a-Mole, were you trolling about that base 10 being easier stuff 'cause you can just move the decimal?
But your are writing the 12 based numbers using 10 based notation. You see, the symbol 10 means that in counting you have counted up to 1 base and 0 more. The number 12 means that you have counted up to 1 base and 2 more etc.
So in any base 10*10 = 10^2.
Number systems are used to keep track of how many items we have counted. For example this many ||||||| items would be written at 111 in the binary system, 21 in a 3 based system because in counting the items we have ||| ||| | or 2 bases plus 1. In 4 based system we have |||| |||, or 13, 1 base plus 3, in a 5 based system ||||| ||, 12 or 1 base plus 2. By this time I’m sure you see the pattern.
To repeat, there is nothing special about 10 except that we are accustomed to it.
Because without a base system it would be difficult to count things. Or you would have to use a system which is difficult to do arithmetic with, such as Roman numerals.
A base system is simply a convenient method for representing quantities. There is no inherrant difference between 42[sub]b10[/sub], 2A[sub]b16[/sub], 101010[sub]b2[/sub]. and 1120[sub]b3[/sub].
42 in base ten means two “ones” and four “tens.”
2A in base 16 means ten “ones” and two “sixteens”
101010 in base 2 means zero “ones,” one “two,” zero “fours,” one “eight”, zero “sixteens” and one “thirty-two.”
1120 in base 3 means zero “ones,” two “threes,” one “nine,” and 1 “twenty-seven.”
They all represent the same quantity are it is equally easy to do arithmetic on them within their base. Base ten seems more natural only because it is what we’re used to. We have spent our whole lives counting things in base 10 so it’s much easier for us to deal with quantities expressed in that base. But mathematically, there is no difference at all between them.
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.
I guess my issue is (to me) a base numbering system represents a grouping and that grouping is arbitrary. Pick whatever you like and you can work with it.
So I go back to the example of 10[sup]2[/sup] vs. 12[sup]2[/sup]. In this case I am working with discrete ‘bundles’ of numbers…basically counting how many of those bundles I have. At this point in my life I can answer what 12[sup]2[/sup] equals as easily as 10[sup]2[/sup]. However, move to 10[sup]23[/sup] vs. 12[sup]23[/sup] and I think a base-10 system is far simpler to work with.
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.