# Silly to count beyond 10?

1. Sep 7, 2008

### tgt

Silly to count beyond base 10?

If so then you'd have things like '234' as a 'single digit' in base 300 for example. That would be very weird?

In the binary system, representing decimals seem like a real pain. i.e 2/3 seems very complicated to represent.

Last edited: Sep 7, 2008
2. Sep 7, 2008

### gunch

Re: Silly to count beyond base 10?

Yeah, it would be just as weird as having the binary number '101' as a single digit in base 10 (it's 5). As long as you have symbols for all digits it's not weird at all. Consider hexadecimal (base 16) where A, B, C, D, E and F are the digits after 9.

2/3 is represented by 10/11 in binary. Yes the actual numerical expansion isn't exactly pretty, but most decimal expansions of fractions aren't pretty either.

The base you chose isn't really important, the important thing is that you are experienced in manipulating that base (and that others are too; no one will be able to follow your base 7 or 23 computations or get a good feel for the magnitude of numbers). There are of course some obvious exceptions, such as when dealing with computer hardware where binary or hexadecimal can be extremely useful (in some contexts; mostly we work on a higher level today).

3. Sep 7, 2008

### HallsofIvy

gunch said pretty much every thing I wanted to say. Just two more points:

Computer engineers use base 16 all the time. Nothing "silly" about that!

2/3 in binary is 10/11 as gunch said or, as a "decimal", 0.1010101010... I don't see that as any more complicated than 0.333333...

4. Sep 7, 2008

### arildno

Of course, decimals don't exist in the binary system; bimals do.

5. Sep 7, 2008

### Hurkyl

Staff Emeritus
Re: Silly to count beyond base 10?

And don't forget that when working with 'large' numbers, computers usually operate in radix 4,294,967,296 or 18,446,744,073,709,551,616... at least when working with algorithms for arithmetic that don't require even larger radixes!

6. Sep 7, 2008

### GTrax

I suppose using radix 10 was because we have 10 digits on our hands. I am not even sure if the word "digit" first meant fingers and thumbs. 10 can be inconvenient because it divides only by 2, 5. English tradition freely used 12 (inches per foot) and dozens, because it divided by 2, 3, 4, 6.

For me, the worst is representing decimals, each digit coded binary (as in BCD). The controllers I try to program have "long" double-word versions of multiply and divide in BCD, but the arithmetic truncates the answers because it is not floating point. Thus example 11/8 returns 1 instead of some version of 1.375.

One has to turn all numbers into integers with enough digits to represent the precision needed, and remember how much to throw away at the end, in effect keeping track of where the decimal point might be.
Therefore, I am one who has come to appreciate using hexadecimal, where the largest 32-bit number can be more than 4 billion. Once you get into it, 10-base seems less attractive.

Last edited: Sep 7, 2008
7. Sep 7, 2008

### HallsofIvy

The Kwakuitl indians of the Pacific northwest (Oregon, Washington, British Columbia) counted on the spaces between fingers and developed a base 4 numeration system. They are, I believe, the only people known to have done that.

8. Sep 9, 2008

### tgt

Interesting.

Actually when thinking about it all, I think I finally know why we use base ten. It's because we have 10 fingers! Nothing more to it then that.

9. Sep 9, 2008

### vanesch

Staff Emeritus
Yes, digitus is latin for finger...

10. Sep 9, 2008

### Kurret

The maya indians number base is also very interesting. They used base 20 if it was in a mathematical context, but in their calendar, they used a mixture where the first digit was worth 1, the second worth 20, and the third 20*18=360 and after that continuing multiplying with 20. I have made a converter to maya numerals on my comp if someone is interested, cant find one on the net though.
http://en.wikipedia.org/wiki/Maya_numerals

11. Sep 9, 2008

### CRGreathouse

Base 60 was also used in similar fashion in Mesopotamia.