A Natural Numeral System
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A Natural Numeral System
In physics there are natural nonarbitrary units of measurement, based on things like the speed of light or planck length, which made me wonder if it is possible to invent such numeral system.
Us humans use base 10, which is comfortable because we have 10 fingers. One might say that base 2 is "more natural", but it is still not as natural as things like Pi and e. The problem is I've never seen a numeral system that is not based on an arbitrary natural number. Can you use things like base 1 or base Pi? Any other Ideas?
Us humans use base 10, which is comfortable because we have 10 fingers. One might say that base 2 is "more natural", but it is still not as natural as things like Pi and e. The problem is I've never seen a numeral system that is not based on an arbitrary natural number. Can you use things like base 1 or base Pi? Any other Ideas?
Re: A Natural Numeral System
yuvi777 wrote:In physics there are natural nonarbitrary units of measurement, based on things like the speed of light or planck length, which made me wonder if it is possible to invent such numeral system.
Us humans use base 10, which is comfortable because we have 10 fingers. One might say that base 2 is "more natural", but it is still not as natural as things like Pi and e. The problem is I've never seen a numeral system that is not based on an arbitrary natural number. Can you use things like base 1 or base Pi? Any other Ideas?
I think multiples of i have been suggested as good bases because they can represent complex numbers with i's, +s or s but they are cumbersome and, in order to make them work, it can be necessary to add symbols for fractions instead of just integers (e.g. ?=1/2).
Anyway, the natural numeral system (for integers at least), is base 1 (with an additional symbol for 0). The natural numbers are also called the counting numbers for a reason.
Still, this natural base has disadvantages, not least of which is that there's no equivalent of decimal notation in it so it isn't good for science.
So yeah, there is a natural base (for a positional number system), but it sucks.
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Re: A Natural Numeral System
yuvi777 wrote:One might say that base 2 is "more natural", but it is still not as natural as things like Pi and e.
What makes you say that? 2 is the ratio of a circle's diameter to it's radius. It's the only nonzero number which yields the same result whether you multiply it by itself or add it to itself.
It does a lot of other cool things as well  for example; it's the number of {protons + electrons} in Hydrogen (the most abundant element in the Universe), and the number of hydrogen atoms in a hydrogen molecule. It's also the first prime number, and a number of other things too.
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Re: A Natural Numeral System
As a base, I don't think Pi is "natural" at all. It shows up in a lot of seemingly disparate places in mathematics, but usually not as the base of exponents. There's more going for e in that regard (and phi), but I'd say 2 is in some sense "more natural" or at least "more basic" than both of them, especially if you're treating mathematics as built up from set theory. (Something is in a set or it isn't, for example, which gives 2 possibilities, and hence the result that the size of the power set of A is 2^{A}, and for A at most countably infinite, members of P(A) can be represented as numbers in binary notation.)yuvi777 wrote:One might say that base 2 is "more natural", but it is still not as natural as things like Pi and e.
Re: A Natural Numeral System
e is terrible as the base of a positional numeral system though because it is, at best, very unclear what set of numbers you would have symbols for that would allow you to span the space of real numbers.
Or more precisely, it is not clear what set (or even if a finite one exists) we should choose our a_{n}s from that will allow the sum of a_{n}e^{n} over n to span the real numbers.
If not finite set does exist then we need to come up with some system of representing our coefficients in some other positional numeral system and now it's turtles all the way down.
Edited to fix the broken math tag workaround (is there currently any properly working way to get math tags working again? I know they were overused and could slow machines down, but on this and the science board, they were exceedingly useful at times).
Or more precisely, it is not clear what set (or even if a finite one exists) we should choose our a_{n}s from that will allow the sum of a_{n}e^{n} over n to span the real numbers.
If not finite set does exist then we need to come up with some system of representing our coefficients in some other positional numeral system and now it's turtles all the way down.
Edited to fix the broken math tag workaround (is there currently any properly working way to get math tags working again? I know they were overused and could slow machines down, but on this and the science board, they were exceedingly useful at times).
Last edited by eSOANEM on Wed Apr 10, 2013 11:37 am UTC, edited 2 times in total.
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Re: A Natural Numeral System
Base 1 is probably the most simple option, and base i sounds interesting, but ideally I want a system that can be used for all real numbers.
That is a good point. I guess I just think of Pi and e as more natural because they are so specific, yet nonarbitrary.
The problem is, like eSOANEM said, that using e or Pi or Phi requires strange rules and new symbols, which are all arbitrary.
Philosophically speaking, I guess choosing any set of rules (even basic logic) is arbitrary, so we are left with what's useful or interesting.
Another option that comes to mind though is using prime numbers somehow, but I can't think of anything that doesn't require more rules than base 2 has.
dudiobugtron wrote:yuvi777 wrote:One might say that base 2 is "more natural", but it is still not as natural as things like Pi and e.
What makes you say that? 2 is the ratio of a circle's diameter to it's radius. It's the only nonzero number which yields the same result whether you multiply it by itself or add it to itself.
It does a lot of other cool things as well  for example; it's the number of {protons + electrons} in Hydrogen (the most abundant element in the Universe), and the number of hydrogen atoms in a hydrogen molecule. It's also the first prime number, and a number of other things too.
That is a good point. I guess I just think of Pi and e as more natural because they are so specific, yet nonarbitrary.
The problem is, like eSOANEM said, that using e or Pi or Phi requires strange rules and new symbols, which are all arbitrary.
Philosophically speaking, I guess choosing any set of rules (even basic logic) is arbitrary, so we are left with what's useful or interesting.
Another option that comes to mind though is using prime numbers somehow, but I can't think of anything that doesn't require more rules than base 2 has.
Re: A Natural Numeral System
Prime bases are very annoying because they have no nontrivial divisors. This means that almost all rational numbers will not have terminating pary expansions.
This is a bigger problem for large prime bases than small ones because over any reasonable region of the number line, the smaller prime base will have a greater density of terminating expansion rationals (because it's powers are closer together).
So, whilst base 2 does suffer this problem, it doesn't suffer it anywhere near as much as base 1021.
This is a bigger problem for large prime bases than small ones because over any reasonable region of the number line, the smaller prime base will have a greater density of terminating expansion rationals (because it's powers are closer together).
So, whilst base 2 does suffer this problem, it doesn't suffer it anywhere near as much as base 1021.
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Re: A Natural Numeral System
The golden ratio is used as the base of a numeral system.
http://en.wikipedia.org/wiki/Phinary/
I'll leave it up to you to decide how natural phi is though.
http://en.wikipedia.org/wiki/Phinary/
I'll leave it up to you to decide how natural phi is though.
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Re: A Natural Numeral System
How are e and Pi any more "specific" than 2? Is saying "two" somehow not infinitely specific, since that completely specifies the exact value of the number?
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Re: A Natural Numeral System
I don't know what's the problem with 2. It is the smaller natural integer which can build a meaning numerical system with the rules of positional notation (because with 1, it is not possible). Also, 2 is full of properties : the first prime number, 2=faceedges=summuits in convex polyhedron... Why is it no more naturan than building a system with e, π or φ ?
But the problem with binary is that it is writingconsuming. That's why a bigger number as a basis should be better. I love to count in 12basis, because 12 is full of interesting properties, and 12basis makes factorization easier than with 10basis.
But the problem with binary is that it is writingconsuming. That's why a bigger number as a basis should be better. I love to count in 12basis, because 12 is full of interesting properties, and 12basis makes factorization easier than with 10basis.
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Re: A Natural Numeral System
What if your increment when counting isn't 1? I can see why it needs to be 1 normally, since 1 is the multiplicative identity. But if you don't mind breaking multiplication, what would a numeral system look like if it counted up in steps different from 1?
Re: A Natural Numeral System
Voekoevaka wrote:I don't know what's the problem with 2. It is the smaller natural integer which can build a meaning numerical system with the rules of positional notation (because with 1, it is not possible). Also, 2 is full of properties : the first prime number, 2=faceedges=summuits in convex polyhedron... Why is it no more naturan than building a system with e, π or φ ?
But the problem with binary is that it is writingconsuming. That's why a bigger number as a basis should be better. I love to count in 12basis, because 12 is full of interesting properties, and 12basis makes factorization easier than with 10basis.
As I pointed out earlier, being a prime number is a bad thing for the base of a positional number system whilst the geometry is simply irrelevant to its suitability.
Systems with a base between 12 and 8 (inclusive) are quite good for counting human scale numbers of things because most such values will be representable in only a couple of digits. Unfortunately, many with so many absurdly large numbers in today's society (how big is the national debt?), arguably we ought to change to a larger base still such as going to the Babylonian
The ultimate problem is that, unlike physics where there are fundamental arguments for one system of units being natural (because they remove constants from most of the fundamental equations), no such base exists for positional notations and so, any choice of base depends purely on convenience. In practice, we will stick with base10 for all the daytoday stuff and use hex and binary only for computerbased stuff.
dudiobugtron wrote:What if your increment when counting isn't 1? I can see why it needs to be 1 normally, since 1 is the multiplicative identity. But if you don't mind breaking multiplication, what would a numeral system look like if it counted up in steps different from 1?
The two systems (under addition only) are isomorphic.
Say your new increment is 2 (in the original system). Now your new system counts:
0, 2, 4, 6, 8, 10, etc.
and adds exactly as if it were
0, 1, 2, 3, 4, 5, etc.
all you've done is map n to 2n. Likewise for any increment k, you'll always just end up mapping the integer n to a pseudointeger kn. Multiplication would be broken (or at least, quite interesting (and wouldn't allow inverses for instance)) though.
Last edited by eSOANEM on Thu Apr 11, 2013 9:45 am UTC, edited 1 time in total.
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Re: A Natural Numeral System
Babylonians used base60, and iirc counted up to it in base10. We could accomplish exactly the same thing with existing numerals, by simply writing a pair of decimal digits vertically in each position up to 59, and moving to the next position after that.
Socalled "natual units" in physics also require some arbitrary decisions. You can't simultaneously get *all* the fundamental constants to be 1 because some of them are related by dimensionless constants. So you have to start out by picking which ones you don't want to worry about any more.
c=G=1 is typical for relativity, for example, but if you don't particularly care about small things you can let h and k be whatever corresponds to the most convenient base unit of mass or length or distance. Doing black hole stuff, a convenient base unit is the mass of the Sun, and then the base length is half its Schwarzschild radius and the base duration is how long light takes to go that distance. Calculating time effects from relativistic space travel, on the other hand, is something I like to do in years and lightyears, and then not worry that my base unit of mass is now the absurd amount required to make a black hole 4 lightyears in diameter. If you also want to work with Hawking radiation, then you might like h and k to have convenient values as well, and if you want hbar=k=1, you get Planck units and suddenly most of your manageablysized thing have ridiculous exponents in scientific notation.
Socalled "natual units" in physics also require some arbitrary decisions. You can't simultaneously get *all* the fundamental constants to be 1 because some of them are related by dimensionless constants. So you have to start out by picking which ones you don't want to worry about any more.
c=G=1 is typical for relativity, for example, but if you don't particularly care about small things you can let h and k be whatever corresponds to the most convenient base unit of mass or length or distance. Doing black hole stuff, a convenient base unit is the mass of the Sun, and then the base length is half its Schwarzschild radius and the base duration is how long light takes to go that distance. Calculating time effects from relativistic space travel, on the other hand, is something I like to do in years and lightyears, and then not worry that my base unit of mass is now the absurd amount required to make a black hole 4 lightyears in diameter. If you also want to work with Hawking radiation, then you might like h and k to have convenient values as well, and if you want hbar=k=1, you get Planck units and suddenly most of your manageablysized thing have ridiculous exponents in scientific notation.
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Re: A Natural Numeral System
I'm not against a 64base counting, but we have to find the 64 symbols that don't look like each other. The only drawback I can think about this is that it would be very inefficient with the divisions by 3 or 5 (and all the other prime numbers except 2, and obviously 0).
For divisibility, I recommand base 60 : it makes easier division by 2,3,4,5 and 6, it seems the babylonians have understood how the divisors of the base are important. But, it makes a lot of symbols too (more than all the letters and numbers of classical alphabet).
About physical units, I don't see the problem about setting c=G=h=1. Exposants are made to compare the size of things that are very different in order.
For divisibility, I recommand base 60 : it makes easier division by 2,3,4,5 and 6, it seems the babylonians have understood how the divisors of the base are important. But, it makes a lot of symbols too (more than all the letters and numbers of classical alphabet).
About physical units, I don't see the problem about setting c=G=h=1. Exposants are made to compare the size of things that are very different in order.
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Re: A Natural Numeral System
gmalivuk wrote:Babylonians used base60, and iirc counted up to it in base10. We could accomplish exactly the same thing with existing numerals, by simply writing a pair of decimal digits vertically in each position up to 59, and moving to the next position after that.
Socalled "natual units" in physics also require some arbitrary decisions. You can't simultaneously get *all* the fundamental constants to be 1 because some of them are related by dimensionless constants. So you have to start out by picking which ones you don't want to worry about any more.
c=G=1 is typical for relativity, for example, but if you don't particularly care about small things you can let h and k be whatever corresponds to the most convenient base unit of mass or length or distance. Doing black hole stuff, a convenient base unit is the mass of the Sun, and then the base length is half its Schwarzschild radius and the base duration is how long light takes to go that distance. Calculating time effects from relativistic space travel, on the other hand, is something I like to do in years and lightyears, and then not worry that my base unit of mass is now the absurd amount required to make a black hole 4 lightyears in diameter. If you also want to work with Hawking radiation, then you might like h and k to have convenient values as well, and if you want hbar=k=1, you get Planck units and suddenly most of your manageablysized thing have ridiculous exponents in scientific notation.
Oh, quite right with base 60, it's fixed now.
IIRC the babylonian system was a base60 positional system it's just that the symbols used for base60 were written in a nonpositional base10 (or base5)like system (it worked more like Roman numerals IIRC).
Yes, physically natural units do require some arbitrary decisions however those are only really which equations you consider to be the fundamental ones and then what value for the constants gives the simplest equation.
c=1 is obvious from SR.
G=1 is obvious from Newtonian gravity, but looking at the EFE without already thinking Newtonianly, setting 8G or 4G or 4*pi*G or even 8*pi*G to 1 would seem more natural.
Setting the value for ε_{0} can be done by analogy to G.
ħ=1 is the obviously choice from the Schrödinger equation.
That leaves some thermodynamic constant to be normalised. I suppose you could normalise the molar gas constant instead of Boltzmann's, but I think it's clear that the molar gas constant is less fundamental being a constant relevant to ensembles of particles rather than individual ones.
That leaves us a few choices, mainly relating to G and ε_{0} (setting ε_{0}=1 is quite nice because it also normalises μ_{0} and removes factors of 4*pi from the EFE and Gauss' law) but the others are fairly natural.
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Re: A Natural Numeral System
Factorial base is fun. Instead of the positions to the left of the "decimal" point being successive powers of some fixed base, they are successive factorials. Such factorial base integers are useful for numbering permutations.
Of course, negative factorials are not defined (I guess we could use the gamma function, but that's going to get messy), but we can assign the reciprocals of factorials to the positions of the right of the "decimal" point. Obviously, all rational numbers terminate when written with this notation, and various irrational / transcendental numbers with simple Taylor series make nice simple patterns when written in factorial base; in fact, a simple way to calculate e to many decimal places is to simply write it in factorial base & then convert it to decimal.
Of course, negative factorials are not defined (I guess we could use the gamma function, but that's going to get messy), but we can assign the reciprocals of factorials to the positions of the right of the "decimal" point. Obviously, all rational numbers terminate when written with this notation, and various irrational / transcendental numbers with simple Taylor series make nice simple patterns when written in factorial base; in fact, a simple way to calculate e to many decimal places is to simply write it in factorial base & then convert it to decimal.
Code: Select all
#! /usr/bin/env python
"""
ECalc  Big e calculator
Generate digits of e, using factorial base notation
Inspired by Dik T. Winter
Created by PM 2Ring 2003.12.26
Converted from C to Python 2007.12.10
"""
import sys, math
#Global constants
N = 50 #Default number of digits
W = 5 #Digits per block
A = 10 ** W
LR2P = .5 * math.log(2.*math.pi) #Log Root 2 Pi
L10 = math.log(10.) #Log 10
def invlfact(n):
""" Inverse Stirling's approx for log n factorial, using Newton's method """
x = y = n * L10  LR2P
for i in xrange(3):
x = (y + x) / math.log(x)
return int(round(x))
def bigE(n):
""" Generate n digits of e using factorial base notation """
kmax = invlfact(n) + 1
print "Calculating e to %d places, using %d cells" % (n, kmax)
#Table for factorial base digits, initialized with series for e2 = 1/2! + 1/3! + ...
#We ignore first two entries
d = [1] * kmax
print "e ~= 2."
j = 1
while n>0:
# Get the next W digits by firstly multiplying d by A,
# propagating carries back down the array,
# then printing the integer part & removing it.
kmax = invlfact(n) #Number of cells needed for this loop
c = 0 #Clear carry
for k in xrange(kmax, 1, 1):
d[k] = d[k] * A + c
c = d[k] // k
d[k] = c*k
#Print block & field separator. May need modifying if you make W large
jj = (j%10 and 1 or j%50 and 2 or j%200 and 3 or 4)  1
print "%0*d%s" % (W, c, "\n" * jj),
j += 1; n = W
def main():
#Get number of digits
n = len(sys.argv) > 1 and int(sys.argv[1]) or N
bigE(n)
if __name__ == '__main__':
main()
Re: A Natural Numeral System
gmalivuk wrote:How are e and Pi any more "specific" than 2? Is saying "two" somehow not infinitely specific, since that completely specifies the exact value of the number?
Maybe "specific" isn't the best word to describe what I mean. I'll try to explain myself better:
2 is a definition, a name we gave to having a thing and another thing. 3,4,5 and so on can be defined in the same way. These definitions don't require any geometry, or even 2D thinking.
Pi is also a definition, a name we gave to the ratio of a circle's circumference to its diameter. But this definition won't explain why it's value is between 3 and 4, who both have an independent unrelated 1D definition.
Therefore the only thing you can answer is that Pi came that way from the Universe Factory.
(All of this might be more of a feeling than a logical argument, though, so I am sorry in advance if these arguments offended anyone.)
Also, thanks for all the odd bases I now need to check out..
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Re: A Natural Numeral System
So, collecting from everyone else...
[list]
[*]You need an integer base, so there's a sensical notion of "digit symbols". Base e is technically welldefined, it doesn't have any reasonable answer for what you write in each position.
[*]You need a positional notation, because additive notations grow too quickly and are inconvenient for multiplication.
[*]You need a system with an easy successor function (which implies that addition is simple), which rules out some more exotic multiplicationbased systems.
[*]You'd like a system with a reasonable number of digit symbols, so they're easy to remember and distinguish for humans.
[*]You'd like a system where the base has a high density of divisors, as that means more fractions will terminate.
[*]You'd like a system divisible or neardivisible (a small multiple of the base is divisible) by as many low numbers as possible, as that means more *useful* fractions will terminate.
[/quote]
#1, #2, and #3 basically leave us with a normal integerbase system where each position is worth an increasing power of the base.
For the rest, base 2 makes #3 really easy, but falls down on #5 and #6. Base 8/10/12 pass all of them reasonably, with varying degrees of passing #6. Base 30/60 pass #5 really well (and 60 passes #6 great, as it's divisible by 2/3/4/5/6/10/12/15/20/30, and neardivisible by 8 and 9), but are hitting the upper limits of what #4 allows.
[list]
[*]You need an integer base, so there's a sensical notion of "digit symbols". Base e is technically welldefined, it doesn't have any reasonable answer for what you write in each position.
[*]You need a positional notation, because additive notations grow too quickly and are inconvenient for multiplication.
[*]You need a system with an easy successor function (which implies that addition is simple), which rules out some more exotic multiplicationbased systems.
[*]You'd like a system with a reasonable number of digit symbols, so they're easy to remember and distinguish for humans.
[*]You'd like a system where the base has a high density of divisors, as that means more fractions will terminate.
[*]You'd like a system divisible or neardivisible (a small multiple of the base is divisible) by as many low numbers as possible, as that means more *useful* fractions will terminate.
[/quote]
#1, #2, and #3 basically leave us with a normal integerbase system where each position is worth an increasing power of the base.
For the rest, base 2 makes #3 really easy, but falls down on #5 and #6. Base 8/10/12 pass all of them reasonably, with varying degrees of passing #6. Base 30/60 pass #5 really well (and 60 passes #6 great, as it's divisible by 2/3/4/5/6/10/12/15/20/30, and neardivisible by 8 and 9), but are hitting the upper limits of what #4 allows.
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))
Re: A Natural Numeral System
eSOANEM wrote:e is terrible as the base of a positional numeral system though because it is, at best, very unclear what set of numbers you would have symbols for that would allow you to span the space of real numbers.
Or more precisely, it is not clear what set (or even if a finite one exists) we should choose our a_{n}s from that will allow the sum of a_{n}e^{n} over n to span the real numbers.
Sure, irrational bases can be a bit messy, and they are certainly not practical for basic arithmetic, but they do have their uses. The usual convention for digits in positive integer bases can be easily extended to irrational bases: the digits are simply all the nonnegative integers that are less than the base. True, many numbers under this scheme will have multiple representations, but that's also true of many rationals in integer bases. And we can use the greedy algorithm to construct a standard representation for any given number.
See http://en.wikipedia.org/wiki/Noninteger_representation
yuvi777 wrote:Pi is also a definition, a name we gave to the ratio of a circle's circumference to its diameter. But this definition won't explain why it's value is between 3 and 4, who both have an independent unrelated 1D definition.
Therefore the only thing you can answer is that Pi came that way from the Universe Factory.
We discovered pi via the circle, but that's just the tip of the pi iceberg. The circle stuff is ultimately a consequence of the fact that pi is a divisor of the period of any function that solves the differential equation y'' + y = 0.
And pi is intimately related to the integers: all infinite sums of the form 1/i^{2n}, with i running from 1 to infinity, are multiples of π^{2n} and a rational number, eg the sum of the reciprocals of the squares is π^{2}/6, the sum of the reciprocals of the 4th powers is π^{4}/90.
See http://en.wikipedia.org/wiki/Zeta_constant#Positive_integers for more examples and the general formula.
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Re: A Natural Numeral System
PM 2Ring wrote:And pi is intimately related to the integers: all infinite sums of the form 1/i^{2n}, with i running from 1 to infinity, are multiples of π^{2n} and a rational number, eg the sum of the reciprocals of the squares is π^{2}/6, the sum of the reciprocals of the 4th powers is π^{4}/90.
See http://en.wikipedia.org/wiki/Zeta_constant#Positive_integers for more examples and the general formula.
It is interresting how π can appear where you don't expect it. For example, 44/3+4/54/7+4/94/11+...=π. One other appearence of π is in the Gamma function (Γ(1/2)=√π).
But we get offtopic.
The question I ask is what is the more reasonable base for you (choose in {2,3,4,6,10,12,16,30,60}). For me, 12 is a good compromise between rule #4 and rule #5.
I'm a dozenalist and a believer in Tau !
Re: A Natural Numeral System
PM 2Ring wrote:eSOANEM wrote:e is terrible as the base of a positional numeral system though because it is, at best, very unclear what set of numbers you would have symbols for that would allow you to span the space of real numbers.
Or more precisely, it is not clear what set (or even if a finite one exists) we should choose our a_{n}s from that will allow the sum of a_{n}e^{n} over n to span the real numbers.
Sure, irrational bases can be a bit messy, and they are certainly not practical for basic arithmetic, but they do have their uses. The usual convention for digits in positive integer bases can be easily extended to irrational bases: the digits are simply all the nonnegative integers that are less than the base. True, many numbers under this scheme will have multiple representations, but that's also true of many rationals in integer bases. And we can use the greedy algorithm to construct a standard representation for any given number.
See http://en.wikipedia.org/wiki/Noninteger_representation
Ooh, good point.
I suppose what was making me dislike that notion is that sufficiently large integers will have to be expressed with symbols after the piimal point making it harder to identify them.
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Re: A Natural Numeral System
Is it possible to have a base of less than 1, but still count up in steps of 1? Like base 1/2? What would the digits look like?
Re: A Natural Numeral System
Voekoevaka wrote:It is interresting how π can appear where you don't expect it. For example, 44/3+4/54/7+4/94/11+...=π. One other appearence of π is in the Gamma function (Γ(1/2)=√π).
Well, I kinda expect pi to turn up as the sum of arctan series, but I get your point. OTOH, the pi  Gamma connection is a good example, as is the appearance of pi in the Stirling approximation for factorial / gamma.
Voekoevaka wrote:But we get offtopic.
Yeah. ok.
Voekoevaka wrote:The question I ask is what is the more reasonable base for you (choose in {2,3,4,6,10,12,16,30,60}). For me, 12 is a good compromise between rule #4 and rule #5.
It depends. The base divisor issue isn't so important these days, IMHO, since most people who need to perform calculations have access to computers & cheap calculators.
If we're pretending that we're back in the olden days without calculators, I'd probably go with base 12, or maybe a base 60 system that uses base 10 to represent the base 60 digits, in other words, the kind of notation used for hours, minutes, seconds and degrees, minutes, seconds. FWIW, that sort of base 60 was a pretty common way to write rational numbers in things like trigonometric function tables before decimal fractions became popular.
But if we're not worried about our base not having lots of divisors, then I'd have to vote for a binary power base like 8 or 16. Base 16 is mathematically nicer in some ways, because 16 is a binary power of a binary power, but it's a lot easier to memorize the addition and multiplication tables in base 8. Numbers written in base 2 are easy to manipulate, but they get too long too quickly, which makes them hard to read and errorprone.
As I indicated in my earlier post, I'm rather fond of factorial base. Converting numbers between factorial base and a fixed base is no harder than converting between fixed bases. But I guess a major downside is that they require an arbitrarily large supply of digit symbols.
eSOANEM wrote:I suppose what was making me dislike that notion is that sufficiently large integers will have to be expressed with symbols after the piimal point making it harder to identify them.
Sure. They are a messy way to write arbitrary integers, but they can be useful if you want to focus on pi power series. OTOH, most real numbers look messy in any base; integer bases have the edge because we like integers to look neat. But when we want to play with subsets of reals that are power series of some irrational with rational coefficients, then using numerals based on that irrational can be a sensible way to proceed.
dudiobugtron wrote:Is it possible to have a base of less than 1, but still count up in steps of 1? Like base 1/2? What would the digits look like?
If we do that, we have to break the rule that all our digits are less than (the absolute value of) our base. To write arbitrary reals as power series of 1/2 would require arbitrarily high numbers for our digits (because of the convergence properties of power series of the form a_{k}/2^{k}), so it'd be a very messy basis for a numeral system.
Re: A Natural Numeral System
dudiobugtron wrote:Is it possible to have a base of less than 1, but still count up in steps of 1? Like base 1/2? What would the digits look like?
Base 1/2 can function identically to base 2 but lsb first instead of msb first (also with unity is the other side of the point).
So:
4_{10}=100_{2}=0.01_{1/2}
17.75_{10}=10001.11_{2}=111.0001_{1/2}
PM 2Ring wrote:eSOANEM wrote:I suppose what was making me dislike that notion is that sufficiently large integers will have to be expressed with symbols after the piimal point making it harder to identify them.
Sure. They are a messy way to write arbitrary integers, but they can be useful if you want to focus on pi power series. OTOH, most real numbers look messy in any base; integer bases have the edge because we like integers to look neat. But when we want to play with subsets of reals that are power series of some irrational with rational coefficients, then using numerals based on that irrational can be a sensible way to proceed.
but we can do that anyway by expanding it as a power series explicitly. Seeing as integers (and other numbers with one symbol as well simpe functions thereof including the rationals, surds, logs etc.) crop up way more often than power series of some messy irrational, it makes sense (for a standard notation) for it to be an integer base and to write such power series explicitly.
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Re: A Natural Numeral System
I think its odd that no one has mentioned balanced ternary.
http://en.wikipedia.org/wiki/Balanced_ternary
Its found its uses in the past, and naturally represents both positive and negative numbers.
http://en.wikipedia.org/wiki/Balanced_ternary
Its found its uses in the past, and naturally represents both positive and negative numbers.
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Re: A Natural Numeral System
jestingrabbit wrote:I think its odd that no one has mentioned balanced ternary.
http://en.wikipedia.org/wiki/Balanced_ternary
Its found its uses in the past, and naturally represents both positive and negative numbers.
Aah, good ol' balanced ternary! I like the idea of using negative digits mostly because the negative sign disrupts the continuity between the positive and negative numbers. Whether this is a good or bad thing is up to you to decide. Speaking of which, the padic way of doing things is another interesting approach to represent negative numbers. In base 2, the number 1 would be represented as ...1111 (the infinite sum of all 2^n, where n is a nonnegative integer). Yet another option is just to use a negative base. Counting takes some getting used to (try it in base 2 and base 10), but I don't know how much of a dealbreaker that is. I've even seen people propose using a combination of negative base with negative digits! I fail to see what advantages THAT sort of system would have, but it's an option at least!
Re: A Natural Numeral System
jestingrabbit wrote:I think its odd that no one has mentioned balanced ternary.
http://en.wikipedia.org/wiki/Balanced_ternary
Its found its uses in the past, and naturally represents both positive and negative numbers.
I was going to mention it, but I forgot. I used it years ago in some Tower of Hanoi programs; I think it's odd that the Wikipedia article on balanced ternary doesn't mention that application.
Re: A Natural Numeral System
cyanyoshi wrote:Yet another option is just to use a negative base. Counting takes some getting used to (try it in base 2 and base 10), but I don't know how much of a dealbreaker that is.
An imaginary base is even better. Then you get +ve and ve reals and imaginaries as well as all Gaussian integers for free without the need of decimal points.
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Re: A Natural Numeral System
eSOANEM wrote:cyanyoshi wrote:Yet another option is just to use a negative base. Counting takes some getting used to (try it in base 2 and base 10), but I don't know how much of a dealbreaker that is.
An imaginary base is even better. Then you get +ve and ve reals and imaginaries as well as all Gaussian integers for free without the need of decimal points.
It all depends on what you mean by "better", of course. When going into imaginary bases, things get slightly more difficult. Representing Gaussian integers with only the digits 0 and 1 neatly isn't an option from what I can tell, but it can be done if you are willing to accept nonrepeating representations (e.g. writing 3i in base sqrt(2)i). Base 2i with the digits 0,1,2,and 3 works pretty well, though. The whole base system where we write an arbitrary number as an infinite sum of a_{n}b^{n}, where a_{n} is a member of a finite set does have its limits. Complex numbers may be the most general set I can think of that can be completely covered by this system, but luckily complex numbers in arrays fits most practical purposes just fine. It might get annoying in an imaginary base that positive real numbers will have a bunch of zeros to separate the contributing digits, but the same can be said by using a negative base so...meh. What is or isn't "natural" is completely subjective anyway.
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Re: A Natural Numeral System
I love balanced ternary, and I consider this as a mathematical wonder. But I don't think everybody could agree about using it.
What I love with padic binary is that it can represent negative numbers, but also some rationnal numbers.
Take the number :
...1010101
Multiply it by two :
...1010101+
...1010101=
...0101010
Add itself yet :
...0101010+
...1010101=
...1111111
So ...1010101=1/3
Even √2 can be written in padics numbers of base 7.
cyanyoshi wrote:Speaking of which, the padic way of doing things is another interesting approach to represent negative numbers. In base 2, the number 1 would be represented as ...1111 (the infinite sum of all 2^n, where n is a nonnegative integer).
What I love with padic binary is that it can represent negative numbers, but also some rationnal numbers.
Take the number :
...1010101
Multiply it by two :
...1010101+
...1010101=
...0101010
Add itself yet :
...0101010+
...1010101=
...1111111
So ...1010101=1/3
Even √2 can be written in padics numbers of base 7.
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Re: A Natural Numeral System
cyanyoshi wrote:Base 2i with the digits 0,1,2,and 3 works pretty well, though.
This is because base 2i is actually just two base 4 numbers interleaved digitbydigit.
This is easy to see because of the fact that it's *impossible* to hit the real digits just by counting in the imaginary digits, and vice versa. You have to have two distinct "successor" functions to be able to enumerate all the numbers. As well, the numbers lose a natural ordering, unless you explicitly specify a diagonalization or the like.
So, it's easier and more natural to just keep treating complex numbers as pairs, I think.
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