22 Nov 09
Originally posted by sonhousei know you were being cheeky, but i feel like this is an interesting point and would like to expound upon it!
Yeah but what are the other symbols for the rest of the set? Base 16, 0-9 and A, B, C, D, E, F makes up that set so what is the symbol set for base 49? You run out of letters and numbers pretty soon🙂
you can make up whatever "symbols" you want, but the concept of the position system will remain the same - so rather than continuing the "letters and numbers" concept which eventually fails thanks to there being a finite number of "familiar" symbols, we go back to the position-based arabic numeral system and denote "places" by commas in a vector (usually.) just ask the egyptians and the romans what happens when you run out of symbols for representing sufficiently large or small numbers!
so though it may seem ill-conceived to need base 10 to even DENOTE base 49, it actually makes the concept much clearer and more useful.
(1,5) [also sometimes just written as 15] = 1*49^1 + 5*49^0.
and often for fractions in different bases (i.e. negative exponents in base 49), analogous to the decimal system, i've often seen a semicolon used in between the base^0 place and the base^-1 place. for example, the base 10 mixed number 49 and 54/2401ths would be written in base 49 as such:
(1,0;1,5) = 1*49^1 + 0*49^0 + 1*49^-1 + 5*49^-2
23 Nov 09
Originally posted by AetheraelHave you ever heard of a book called "A number for your thoughts" by Malcom Lines?
i know you were being cheeky, but i feel like this is an interesting point and would like to expound upon it!
you can make up whatever "symbols" you want, but the concept of the position system will remain the same - so rather than continuing the "letters and numbers" concept which eventually fails thanks to there being a finite number of "familiar" sym ...[text shortened]... ritten in base 49 as such:
[b](1,0;1,5) = 1*49^1 + 0*49^0 + 1*49^-1 + 5*49^-2[/b]
It turns out he used to work at a start-up in New Jersey I helped build from scratch.
He gave me a copy, interesting stuff. He interested Isaac Asimov with that book and they became pen pals. It says among other things, you don't need a positive base to make a number system, it can be based on say, -15. Check it out, great little book! I just saw it on Amazon!
23 Nov 09
Originally posted by AetheraelI may be missing something here, but it's just notation, right? Notation can and should be changed at will when that brings clarity to the operations.
i know you were being cheeky, but i feel like this is an interesting point and would like to expound upon it!
you can make up whatever "symbols" you want, but the concept of the position system will remain the same - so rather than continuing the "letters and numbers" concept which eventually fails thanks to there being a finite number of "familiar" sym ...[text shortened]... ritten in base 49 as such:
[b](1,0;1,5) = 1*49^1 + 0*49^0 + 1*49^-1 + 5*49^-2[/b]
23 Nov 09
Originally posted by sonhouse... and fractional bases?
Have you ever heard of a book called "A number for your thoughts" by Malcom Lines?
It turns out he used to work at a start-up in New Jersey I helped build from scratch.
He gave me a copy, interesting stuff. He interested Isaac Asimov with that book and they became pen pals. It says among other things, you don't need a positive base to make a number system, it can be based on say, -15. Check it out, great little book! I just saw it on Amazon!
for example base 1/2
5(decimal) = 1.01 (base 1/2)
Any practical applications I wonder?
23 Nov 09
Originally posted by wolfgang59i think you may have miscalculated...
... and fractional bases?
for example base 1/2
5(decimal) = 1.01 (base 1/2)
Any practical applications I wonder?
a fractional base of (1/n) would just map any number in base (n) to the right of the "decimal point" (i'm not really sure what the name of the place-defining syntax element would be called in a non-base 10 system - the "n-thimal point?" lol)
consider that n^(-x) = (1/n)^x
so 5(decimal) = 11 (base 2) = .11 (base 1/2)
similarly, 14.5(decimal) = 1110.1(base 2) = 1.0111 (base 1/2)
but this begs the question: are we allowed integer placeholders in a fractional base? in base 2 only the digits 1 and 0 are allowed... and it stands to reason that only 0 and 1 would be required to represent any number in base(1/2). similarly, in base (1/n) we could restrict the digits to the list 0,1,2,...,(n-1) but then isn't it more practical and internally consistent to just invert the order of our digits and move the "decimal point" to represent the number in base (n) without "breaking" any of the "rules"? food for thought 🙂
i haven't, you may notice, considered what would happen in a base of (x/n) which could prove to be much more interesting and less straightforward than (1/n)! though the same problem presents itself: what digits would be allowed, and would they be sufficient to have a unique representation of all numbers?
24 Nov 09
Originally posted by AetheraelI was considering that the place values in any base x are;
i think you may have miscalculated...
a fractional base of (1/n) would just map any number in base (n) to the right of the "decimal point" (i'm not really sure what the name of the place-defining syntax element would be called in a non-base 10 system - the "n-thimal point?" lol)
consider that n^(-x) = (1/n)^x
so 5(decimal) = 11 (base 2) = .11 (ba ...[text shortened]... lowed, and would they be sufficient to have a unique representation of all numbers?
x^2 x^1 x^0 (decimal point) x^-1 x^-2
which in decimal gives hundreds, tens, units, tenths, hundredths
extend to base 1/2 we get place values of
1/4 1/2 1 (point) 2 4
so that 1.01 in base 1/2 is 1+4 = 5