Talk:Radix/Archive 1

Archive 1

Numbers vs. integers?

And on a completely unrelated matter, I want to ask any mathematicians something. I was reading Mathematical Toolchest (A.W. Plank and N.H. Williams, AMT Publishing) the part on it about bases refers to integers only.

Now, when I think about it, we don't really use bases for any numbers apart from integers, do we? I mean, most multi-base calculators that I've seen can only work with integers in bases other than ten, and there isn't a true application for fractions in bases other than ten — we just don't use them.

That last paragraph, however, is my observation, and is backed only by (the evidence in it and) my reference to Mathematical Toolchest, and I do not by any means claim to be a knowledgeable mathematician. So, can I ask for views on this matter — are bases ever used for non-integer numbers? Neonumbers 09:55, 25 January 2006 (UTC)

Well, some digital calculators and most computer applications work in base 2 (binary) internally, also when calculating fractions or floating point. They won't always display them to the user in this representation, but for floating point, some clever manipulation can reveal which representation is used (for example you might use the fact that 0.1 can be represented exactly in base 10, but not in base 2). Rasmus (talk) 10:56, 25 January 2006 (UTC)

Bascically what Rasmus said; you can use "bases" for reals as well, but there's the caveat that in a particular base, the radical expansion may not terminate (just like 0.333... = 1/3 in decimal). Dysprosia 11:12, 25 January 2006 (UTC)

If you are curious about this subject, I recommend taking a look at Continued fraction. As you will see, continued fractions have some properties which are interesting because they allow circumventing some problems with representing reals in various bases.

On a side note, this question suggests possible improvements to the article. AdamSmithee 11:53, 25 January 2006 (UTC)

So what's your question exactly? I see a lot of paragraphs, but no question. Is it whether you can use a radix other then 10 to represent noninteger numbers? The answer is of course yes. There's nothing special about radix 10, except for its approximate equality to the average number of digits of humans. -lethe ^talk 03:39, 26 January 2006 (UTC)

The question is at the end of my post. And the question is not can a base other than ten be used, it is if a base other than ten used in practice. I've touched on bases in studying number theory, where only integers are dealt with, and nowhere else. Yes, this can imply changes to the article, whether it does I'm not sure.

If I want to shift my focus to the article, I can re-word the question: would it be inaccurate to focus mostly on integers in this article? (though I'd imagine the matter would be far more complex than just that, maybe separate section, I don't know, I'm not an expert.)

Thanks for the responses, everyone. Much appreciated. (More welcome.) Neonumbers 03:59, 26 January 2006 (UTC)

There is some interest in exploring bases that are not based on integers. The only example I am aware of, though, (and its possibly the only example), is the rational zeta series, in part because various common numbers have an interesting form when expressed in "base zeta". linas 14:32, 26 January 2006 (UTC)

Some nerdy geeks also play around with phinary numbers; that is, numbers with the golden ratio as radix. Now, the notion "in practice" is a bit subjective, but for the most part, everyone in the world uses 10 as a radix in practice, except for computer scientists, who use 2 or sometimes 8 or 16 in practice. So not only do most people not use non integer numbers "in practice"; most people never use anything but 10 in practice. -lethe ^talk 18:29, 26 January 2006 (UTC)

Wow, phinary... that's interesting (sincerely, no sarcasm, honestly)

Anyway, my question was intended not to refer to non-integer bases, but non-integer numbers in bases other than ten. To clarify, I'll quote from the book:

Let b be an integer greater than 1. Every integer a can be expressed as

$a_{0}b^{0}+a_{1}b^{1}+a_{2}b^{2}+...+a_{n}b^{n}$

${\mbox{such that for }}0\leq i\leq n{\mbox{, }}0<a_{i}<n$

This is in a number theory section, and so may be too limited in perspective for reference in this article — it only deals with integers. (Take note of the second bolded "integer", not the first.) I understand that in bases other than ten, non-integer numbers are dealt with. But by "in practice", I mean, well, I'll put it this way: When talking about bases, we generally deal with integers only, right? So would it be fair to start this article with integers and introduce non-integers in a later section, commenting on things such as what Dysprosia said earlier? Or are non-integer numbers a sufficiently important part for them to be given equal status to integers throughout this article (like they are now)?

(For anyone wondering, I ask heaps of questions when I'm trying to learn something... sorry if I'm getting annoying. And, I was considering changing the current version to one that did start out by focusing on integers as in the above quotation, and commenting on non-integers in a later part.) Neonumbers 11:27, 27 January 2006 (UTC)

So it's true that any integer can be expressed in base b. It's also true that any real number can be expressed. It's also probably true that people who mess around with alternate bases play with integers (or even natural numbers), and not so much real numbers. But non integral numbers present no difficulties (beyond those already present in base 10). -lethe ^talk 11:41, 27 January 2006 (UTC)

Cool. When I feel like it, I'm gonna change the article to mention integers first, but an acknowledgement of real numbers will of course be made (and not a minor one). You can run over the change when I've done it, but it won't be for a while... (getting lazy with actual article editing...) Thanks all for your assistance.  :-) Neonumbers 10:26, 28 January 2006 (UTC)

Actually, there is some interest in how decimals behave in other bases, particularly base 2. For example, we all know that 1/3 = 0.333333333..... and that 2/3 = 0.66666666..... But, when you have to truncate a decimal (for example, as a calculator does), then 2/3 becomes 0.66666667 and that final 7 creates an error. Well, in computers, which operate base 2, you have the same problem; for example 1/10 in base 2 is a repeating fraction just like our example 1/3 is in base 10. Eventually, the computer truncates the final digit and has to choose to round. What method one uses to round and/or truncate numbers can lead to instabilities in some calculations. For example, you may expect that some arithmetic series converges and eventually becomes constant (perhaps you're using an algorithm to find sin(pi/9)). But, because of precision errors, those last few digits may flop around and never converge. So, how decimals behave in other bases can be an important matter in some (admittedly picky) applications. - grubber 02:13, 15 February 2006 (UTC)

Non-standard positional numeral systems

I have addressed certain issues by creating the article Non-standard positional numeral systems, and making related changes to Unary numeral system, Golden ratio base, Quater-imaginary base, Positional notation, Base (mathematics), and Category:Positional numeral systems. I suggest further discussion of these issues takes place at talk:Non-standard positional numeral systems.--Niels Ø (talk) 14:36, 26 February 2006 (UTC)

Exceptions

The article states that a number x and its representations base-b are on a one-to-one ratio. First of all, it should be mentioned that is commonly known as the Basis Representation theorem. (I would cite "Number Theory" by George E. Andrews.) Also, the certain exceptions should be noted. For example, in base-10, 1 can be represented as 1 or as $0.{\bar {9}}$. He Who Is (talk) 19:09, 10 June 2006 (UTC)

Relationship between real numbers and their representations

Under this heading, user:VKokielov has added the following:

It has been demonstrated that there is a one-to-one correspondence between real numbers and their representations in any base except the trivial 0 or 1. That is, given a real number x and a base b, we can find only one function (or vector) f(x, k) which retrieves the kth digit, where k = 0 is the units digit, k = ...,-1,0 are the fractional parts, and k = 1, ... are the whole registers (10s, 100s, etc. for b = 10). In other words, each real number has exactly one infinite decimal representation in any base. Also, each such representation converges to a real number. (The latter fact is very easily justified using the so-called completeness axiom).

Another, stronger result states that every rational number has a repeating fraction representation in any base b: that is, for each rational number we can find a representation in any legitimate base such that the sequence of digits repeats with a fixed period after some nth digit. Every such representation converges to a rational number.

Some references would be nice, and so would a correction taking into account the facts discussed in Proof that 0.999... equals 1.--Niels Ø 21:35, 2 May 2006 (UTC)

I fixed the error, and added a section on infinite represenations to support the material. I favored finite representations over infinite representations when selecting a standard representation. Some references would still be nice. -DCary (talk) 01:25, 5 August 2006 (UTC)

Is this correct?

In base 7, third place is shown as 48 (for 7 to the power of 7). I think this is 49. Refer the image.

4\times 7^2 + 6\times 7^1 + 5\times 7^0 = 4\times 48 + 6\times 7 + 5\times 1

Thanks Subramanya 13:16, 7 November 2006 (UTC)

Fixed now. Thanks. — Arthur Rubin | (talk) 19:10, 7 November 2006 (UTC)

Misleading sentence

I recommend deleting or clarifying "Note that this means 10 b = b 10 is true for any base b." To me this sentence implies that 10 in any base = b decimal for any b, which is only true because we happen to express b in decimal. In other words, whatever base we use we would express as 10 in its own base, and the above equation applies. I know the sentence is technically correct and does not explicitly say what I inferred from it, but I found it quite misleading, which is why I brought it up. Thank you. 88.80.200.138 (talk) 11:06, 29 January 2008 (UTC)

I think it had several problems. I have moved and modified it in [1]. Is that better? PrimeHunter (talk) 12:16, 29 January 2008 (UTC)

Sole Reference

The only reference for the radix article is about babylonian numeral systems? — Preceding unsigned comment added by 24.5.102.57 (talk) 00:05, 26 December 2008 (UTC)

most commonly used

decimal, by humans directly certainly, but if you would expand that to include all data stored and processed you would have to conclude binary, since when we use a computer we process and store data in the binary system as well as every thing on flash drives mobile phones this text etc. the representation of those numbers as text or as decimal number strings is just a different way to display the same data. therefore i would suggest the removal of the decimal system as he most commonly used one, or change it to read "by humans" or even "by humans directly" —Preceding unsigned comment added by 217.166.119.28 (talk) 08:22, 27 January 2009 (UTC)