On Sun, Nov 21, 2021 at 11:39 AM Avi Gross via Python-list <python-list@python.org> wrote: > > Can I suggest a way to look at it, Grant? > > In base 10, we represent all numbers as the (possibly infinite) sum of ten > raised to some integral power.
Not infinite. If you allow an infinite sequence of digits, you create numerous paradoxes, not to mention the need for infinite storage. > 123 is 3 times 1 (ten to the zero power) plus > 2 times 10 (ten to the one power) plus > 1 times 100 (ten to the two power) > > 123.456 just extends this with > 4 times 1/10 (ten to the minus one power) plus > 5 times 1/100 (10**-2) plus > 6 time 1/1000 (10**-3) > > In binary, all the powers are not powers of 10 but powers of two. > > So IF you wrote something like 111 it means 1 times 1 plus 1 times 2 plus 1 > times 4 or 7. A zero anywhere just skips a 2 to that power. If you added a > decimal point to make 111.111 the latter part would be 1/2 plus 1/4 plus 1/8 > or 7/8 which combined might be 7 and 7/8. So any fractions of the form > something over 2**N can be made easily and almost everything else cannot be > made in finite stretches. How would you make 2/3 or 3 /10? Right, this is exactly how place value works. > But the opposite is something true. In decimal, to make the above it becomes > 7.875 and to make other fractions of the kind, you need more and more As it > happens, all such base-2 compatible streams can be made because each is in > some sense a divide by two. > > 7/16 = 1/2 * .875 = .4375 > 7/32 = 1/2 * .4375 = .21875 > > and so on. But this ability is a special case artifact caused by a terminal > digit 5 always being able to be divided in tow to make a 25 a unit longer > and then again and again. Note 2 and 5 are factors of 10. In the more > general case, this fails. In base 7, 3/7 is written easily as 0.3 but the > same fraction in decimal is a repeating copy of .428571... which never > terminates. A number like 3/7 + 4/49 + 5/343 generally cannot be written in > base 7 but the opposite is also true that only a approximation of numbers in > base 2 or base 10 can ever be written. I am, of course, talking about the > part to the right of the decimal. Integers to the left can be written in any > base. It is fractional parts that can end up being nonrepeating. If you have a number with a finite binary representation, you can guarantee that it can be represented finitely in decimal too. Infinitely repeating expansions come from denominators that are coprime with the numeric base. > What about pi and e and the square root of 2? I suspect all of them have an > infinite sequence with no real repetition (over long enough stretches) in > any base! I mean an integer base, of course. The constant e in base e is > just 1. More than "suspect". This has been proven. That's what transcendental means. I don't think "base e" means the same thing that "base ten" does. (Normally you'd talk about a base e *logarithm*, which is a completely different concept.) But if you try to work with a transcendental base like that, it would be impossible to represent any integer finitely. (Side point: There are other representations that have different implications about what repeats and what doesn't. For instance, the decimal expansion for a square root doesn't repeat, but the continued fraction for the same square root will. For instance, 7**0.5 is 2;1,1,1,4,1,1,1,4... with an infinitely repeating four-element unit.) > I think there have been attempts to use a decimal representation in some > accounting packages or database applications that allow any decimal numbers > to be faithfully represented and used in calculations. Generally this is not > a very efficient process but it can handle 0.3 albeit still have no way to > deal with transcendental numbers. Fixed point has been around for a long time (the simplest example being "work in cents and use integers"), but actual decimal floating-point is quite unusual. Some databases support it, and REXX used that as its only numeric form, but it's not hugely popular. > Let me leave you with Egyptian mathematics. Their use of fractions, WAY BACK > WHEN, only had the concept of a reciprocal of an integer. As in for any > integer N, there was a fraction of 1/N. They had a concept of 1/3 but not of > 2/3 or 4/9. > > So they added reciprocals to make any more complex fractions. To make 2/3 > they added 1/2 plus 1/6 for example. > > Since they were not stuck with any one base, all kinds of such combined > fractions could be done but of course the square root of 2 or pi were a bit > beyond them and for similar reasons. > > https://en.wikipedia.org/wiki/Egyptian_fraction It's interesting as a curiosity, but it makes arithmetic extremely difficult. > My point is there are many ways humans can choose to play with numbers and > not all of them can easily do the same thing. Roman Numerals were (and > remain) a horror to do much mathematics with and especially when they play > games based on whether a symbol like X is to the left or right of another > like C as XC is 90 and CX is 110. Of course, there are myriad ways to do things. And each one has implications. Which makes it even more surprising that, when someone sits down at a computer and asks it to do arithmetic, they can't handle the different implications. ChrisA -- https://mail.python.org/mailman/listinfo/python-list
