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A better number notation?

ceptimus

puzzler
C
Joined
May 20, 2003
Messages
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In the thread discussing the 1 = 0.999... topic, many posters seem to think that the symbols 0,1,2,3,4,5,6,7,8,9,- and the decimal point actually are numbers. This seems wrong to me. The base-10 decimal notation, using arabic numerals, is just a clumsy way of representing numbers; roman numerals are another, even clumsier, way. The posters in the other thread are really arguing about the inadequacies of the representational system, rather than the numbers themselves.

So does a better system exist, or can you invent one? By a better system, I mean one that can represent numbers at least as well as our current system, but without the ambiguity of being able to write some numbers down in two different ways.
 
Sure just work in base infinte alpeph infinite (for most perposes including the argument alpeph 2 would be big enough)
 
geni said:
Sure just work in base infinte alpeph infinite (for most perposes including the argument alpeph 2 would be big enough)
Click to expand...

I tried that, but I didn't seem to have enough fingers.
 
ceptimus said:
...The base-10 decimal notation, using arabic numerals, is just a clumsy way of representing numbers;...
Click to expand...

Woah, steady neddy, I'd take issue with that. Number representation using positional notation solves the problems of scalability (to a usable limit), negatives (to a usable limit in the other direction), reals (neat, just a dot and you're done) and zero (bang on, a gold piece for that man) in a general and consistent way and its arithmetic holds true for all bases. Its cracks only start to show when problems as exquisite as 0.9~=1 are discussed. Not a bad invention all told. As for a better notation, ya got me, I guess you have to make a judgement about the utility of the maths you're going to apply to the symbols, hence the need for differing number systems.
 
A better number notation is still the base 10 system, but lose the decimal, use fractions.
 
As an American, I would say a better number notation would involve not having numerals that look like letters (1 and 0).
 
Yahweh said:
A better number notation is still the base 10 system, but lose the decimal, use fractions.
Click to expand...

Naah. That works with nice and clear fractions, but when you get to stuff like comparing fractions like 617/907 and 173/251, you can't quickly say which one is bigger than the other.
 
If we had to choose a different base for numbers, I'd probably go with base 12 numbers. This because there are more ways to divide 12 evenly than 10, as 12 has factors: 1, 2, 3, 4, and 6 while 10 has only: 1, 2, and 5. But this is not a big thing.

The most important think is that the notation should allow representations that are logarithmic to the value of the number.

For example, in base 10 the representation of the number 1000 is only four characters long. This both saves space and removes a potential source of erros (for example, like miscounting the number of 'I's in a Roman numeral).

As for other number systems that have been in use, the ancient Egyptians had one that was mightly weird and cumbersome.

They used fractional numbers, but they didn't have fractions like 3/5, but instead their notation allowed only figures of the form 1/x. So, they had to normalize every single fractional number that occurred somewhere. For example, they would write 3/8 as 1/4 + 1/8. As you can guess, this led to rather difficult arithmetic.
 
DickK said:
(bang on, a gold piece for that man)
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Hopefully not the kind of gold piece they use in D&D. Prices in D&D are so hyperinflated that it costs 7 gold pieces just to buy an ordinary lantern.
 
By the way, my favourite number system is the one presented in Barendregt's The lambda calculus : its syntax and semantics. It was truly a mind-altering experience to read it.

Barendregt's system is basically an unary one, that is, there is the number 0 and then number one is s(0), two is s(s(0)), three, s(s(s(0))), and so on. This is pretty standard stuff thus far, but the way how he constructs the successor function s is not.

I won't go into all details, but the basic idea is that he starts by defining the truth values as two lambda terms (I use Lisp-syntax for them):

T = (lambda (x y) x)
F = (lambda (x y) y)

Then, he constructs a pairing function ("cons") that combines two lambda terms u and w into a pair [u, w]. The function is defined so that if M is a pair, then the term M T acts as 'car' (gives first element of pair) and M F as 'cdr' (gives the second element of pair).

The numerals are then formed by having 0 be the identity term that doesn't alter its argument in any way:

0 = (lambda (x) x)

and then if x is a numeral, then its successor is formed by "consing" the truth value false in front of it:

s(x) = (lambda (x) [F, x])

Then, you can check whether a number x is zero by applying it to the truth value T:

zero?(x) = (lambda (x) (x T))

If x > 0, then T acts as 'car' and returns the first element in the number, that is the truth value false. If x = 0, then the identity term "drops" T through it unaltered, so the answer is T.

Further on, Barendregt goes on to define a Turing-complete programming system that utilizes those numerals. Fascinating stuff.
 
That is why I tend to leave the maths deparment alone.:D
 
ceptimus said:
In the thread discussing the 1 = 0.999... topic, many posters seem to think that the symbols 0,1,2,3,4,5,6,7,8,9,- and the decimal point actually are numbers. This seems wrong to me. The base-10 decimal notation, using arabic numerals, is just a clumsy way of representing numbers; roman numerals are another, even clumsier, way. The posters in the other thread are really arguing about the inadequacies of the representational system, rather than the numbers themselves.

So does a better system exist, or can you invent one? By a better system, I mean one that can represent numbers at least as well as our current system, but without the ambiguity of being able to write some numbers down in two different ways.
Click to expand...

I like binary.

1 = 0.1111111111111111111111111111...
 
Abdul Alhazred: I like binary.
1 = 0.1111111111111111111111111111...
Click to expand...
Yep, which is why I like hexadecimal if we were to choose a different number base then ten.

Another interesting number base is the irrational number PHI, also called the Golden Mean, or Golden Ratio.

PHI = (1 + sqrt(5)) / 2 = 1.618...

The decimal integer 2 is written in base PHI as 10.01 exactly (no repeating digits after the radix point). I posted more about that in another thread:

http://www.randi.org/vbulletin/showthread.php?s=&postid=1869898369#post1869898369
 
Paul C. Anagnostopoulos said:

How do you represent irrational numbers?
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Irrational numbers are a tool of the Devil, so there's no need to represent them.
 
Paul C. Anagnostopoulos said:
Yahweh said:
How do you represent irrational numbers?

~~ Paul
Click to expand...

You can't represent irrational numbers (exactly) in the current decimal system either, so it wouldn't be any worse.

One idea that gets a lot of useful irrational numbers in is to represent a number by a polynomial with integer coefficents with a zero at that value. Then just write the coefficents in descending order. For example,
2 = "1,-2",
phi="1,-1,-1",
cos(pi/7) = "64,0,-80,0,24,0,-1"
There are some problems with this of course; there are still multiple ways to write a number, and each numeral actually expresses several different numbers. But I'm sure someone could make it work, and even come up with the rules for teaching kiddies to add and multiply...

One final idea is to represent a number by an encoding of a Turing machine that computes it. With this system, it would be possible to "exactly" represent pi, e, and some others. I suspect that doing simple operations like multiplying and dividing would be horrendous.

By the way, it's impossible to have a system that can represent any arbitrary real number using a finite number of symbols from a finite alphabet. There are uncountably infinite real numbers, and countably infinite ways of representing them in any useful scheme.
 
Phildonnia said:
You can't represent irrational numbers (exactly) in the current decimal system either, so it wouldn't be any worse.
Click to expand...
Well, we'd need two ugly numbers to represent the typical irrational, rather than one ugly number. I'd say that's worse. But not a lot worse.

~~ Paul
 
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