WikiProject Mathematics (Rated Start-class, Mid-importance)
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Field:  Basics

## Sole Reference

The only reference for the radix article is about babylonian numeral systems?

Gerson Chicareli Gerson Chicareli 17:23, 20 August 2017 (UTC) — Preceding unsigned comment added by COMMON.EXE (talkcontribs)

## 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)

## Incorrect Example

The aboriginal base 2 counting example is completely incorrect binary counting. So wrong is the counting that I doubt the validity of the statement that the number system of the mentioned language was base 2. Furthermore, Kala Lagaw Ya doesn't discuss the number system and no reference is cited in this article. If a reliable reference cannot be found, I think that this should be removed from the article. The counting example is frivolous anyway.

Kala Lagaw Ya "Numbers": urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar
binary: 0, 1, 10, 11, 110, 111
decimal: 0, 1, 2, 3, 6, 7

Nope! It means urapon = 1, ukasar = 2. Then it is 1, 2, 2+1=3, 2+2=4, 2+2+1=5, 2+2+2=6. Double sharp (talk) 10:39, 9 October 2015 (UTC)

## Merge

Wow, duplicate information.

Even though the history of radix is longer, this base (mathematics) article is longer in terms of article length. Hence I conclude it would be easier to merge radix into here. I have changed the templates accordingly, and started merging.

There is very little in radix to merge; the only part that remains is the part about notation. I don't udnerstand this part, and looking at talk:radix, I'm not alone. I've already merged the first paragraph, which was almost the same as the first paragraph here.

Any help in the merge would be appreciated.

After the merge is completed, I propose that this page be moved (back) to radix and the text be changed accordingly, because "radix" is an unambiguous name whereas "base" is not. This final step is open to discussion, and will only be done after the merge is complete (we still start by merging radix into here beacuse it's easier to do so.) Neonumbers 11:50, 13 January 2006 (UTC)

Thanks for doing this. I would have done it right away but the previous merge I did someone complained about not giving warning time between tagging. I couldn't believe it either when I saw it -- that duplication of such basic concepts has gone unnoticed as for so long. I also thought about which name would be better and that 'radix' is easier because it's unambiguous, but "Base" is used much more commonly. So despite the extra typing I would go for "Base (mathematics)". Quarl (talk) 2006-01-13 12:36Z
I've merged Radix to Base (mathematics). Quarl (talk) 2006-02-06 09:16Z

There is still a lot of redundancy between this and related articles. --DCary 01:25, 5 August 2006 (UTC)

### Requested move

#### Survey

Add *Support or *Oppose followed by an optional one-sentence explanation, then sign your opinion with ~~~~
1. Support on the basis of ambiguity. -lethe talk + 21:00, 6 May 2006 (UTC)
2. Support to remove ambiguity. Once done, I assume this would become a dab. Confusing Manifestation 12:23, 8 May 2006 (UTC)
3. Support to remove ambiguity. It might be a pseudo-dab, (list of short alternative meanings, not long enough for even a stub), rather than a dab. — Arthur Rubin | (talk) 13:35, 9 May 2006 (UTC)
4. Support - grubber 17:22, 9 May 2006 (UTC)
Done. —Nightstallion (?) Seen this already? 10:44, 11 May 2006 (UTC)

I know this has been discussed before, but it was a binary choice between two options, one too ambiguous, the other too technical.

I would like to propose moving this page to Base (radix), to reflect the most common term and also the more precise disambiguation. Geometry guy 19:07, 9 April 2007 (UTC)

### Merge (again)

I have marked both articles to be merged, again, since it seems like there was a general consensus to merge in 2006... Perhaps this time Radix needs a note at the top about the redirect from Base. LobStoR (talk) 01:09, 20 March 2009 (UTC)

### Merge results

Agreed, it is much more focused on its topic now - good stuff. I was surprised that someone actually jumped up to fix these articles, I just slapped the merge template on in order to "get around to it when i have the chance." LobStoR (talk) 23:00, 8 April 2009 (UTC)

## 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
${\displaystyle a_{0}b^{0}+a_{1}b^{1}+a_{2}b^{2}+...+a_{n}b^{n}}$
${\displaystyle {\mbox{such that for }}0\leq i\leq n{\mbox{, }}0
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 14:36, 26 February 2006 (UTC)

## Relationship between real numbers and their representations

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 01:25, 5 August 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 ${\displaystyle 0.{\bar {9}}}$. He Who Is 19:09, 10 June 2006 (UTC)

## Fractional bases

This article makes no mention of non-integer/fractional bases. Any particular reason why, or is it just lack of knowledge/use?

Base pi can be fun to play with :)

some stuff on non standard bases on wikipedia

--Luke-Jr 05:10, 13 August 2006 (UTC)

There may be a lack of knowledge regarding a fractional radix, I'm not sure. But there DEFINITELY is a lack of interest. How does one create a fractional base anyway? I wasn't able to create one without creating something called a "step." The step is the number you count by. So if you count by fourths, you can have a base of 1/2. Another interesting thing about having a fractional radix is it can create fractional digits.

Counting with base 3/2 step 1/2 gives the following sequence, where each number is separated by a comma, and certain digits within a number are separated by parenthesis:

0, 1/2, 1, (1/2)0, (1/2)(1/2), (1/2)1, 10, 1(1/2), 11, (1/2)00, (1/2)0(1/2), (1/2)01, (1/2)(1/2)0, (1/2)(1/2)(1/2), (1/2)(1/2)1, (1/2)10, (1/2)1(1/2), (1/2)11, 100...

This sequence translates to the more familiar quantities:

0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9....

So the number two in the familiar base ten step one system is written in the base 3/2 step 1/2 system as (1/2)(1/2). And the number four would translate to 11. Four and a half (4.5) would be represented as (1/2)00, and nine would be 100.

Theboombody (talk) 17:29, 8 May 2013 (UTC)

I get different results:
 decimal: radix 3/2: 0½ 0¾ 1 1¼ 1½ 1¾ 2 2¼ 2½ 2¾ 3 3¼ 3½ 3¾ 4 4¼ 4¾ h h0 1 hh 10 h1 1h 100 11 10h 1h0 101 1hh 110 1h1 11h 111
where h (for half) is more concise than (1/2). —Tamfang (talk) 00:59, 1 January 2014 (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)

## List of Bases

Should there not be a category Category:Numeral Base or Category:Numeral Radix or something to put all articles into that are articles about a base? (eg. Base 7)... That would help alot, to find info on them I had to just type in "Base X" through the number 50... I finally just stopped when It looked like they were getting sufficiently less written about to be annoying to search for. Anyone know how to make something like this? Leif902 03:03, 11 March 2007 (UTC)

Done. We now have category: positional numeral systems. That should list all articles on "Base X" for any integer X, no matter what they are named, including base 60 aka sexagesimal and base64. (I see that "base 100" and "base 256" are occasionally mentioned at Wikipedia, but they aren't listed in those categories because we don't yet have an article about them).
The help page Help:Category lists everything I know about making categories like this. As described on that page, if there are any articles that should be in the list at category: positional numeral systems, but isn't, please add go to each of those articles and add [[category: positional numeral systems]] to the end of those articles.
(The same applies to articles on "Base X" for non-integer X -- such as quater-imaginary base -- which should all be listed at category: non-standard positional numeral systems). --DavidCary (talk) 18:09, 10 October 2012 (UTC)