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Is it ever possible to have multiple decimal points in a number? If so, how?

For example is the value 1.1.2 possible?

This is a question posed purely out of curiosity.

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Kind of. Consider numerical representation in other languages, for example some countries in Europe you write numbers using a comma for the decimal seperator, ie: 1.000.000,95 –  VolatileDream Jul 8 '13 at 21:01
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But, in those instances, the 'dots' are not "decimal points", they are "thousands separators". Indeed, the generic term is a "decimal mark". In some countries it's a "point", in others, it's a "comma" –  Dancrumb Jul 8 '13 at 23:10
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See my answer below for further generalisation of the decimal mark as a radix character –  Dancrumb Jul 8 '13 at 23:22
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None of the current answers are sufficiently humorous. –  dfeuer Jul 8 '13 at 23:38
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When I was a child just learning about decimal numbers, I actually used to ponder this; does "3.5.2" mean "three and 5.2 tenths" or "3.5 and two tenths"? –  David Zhang Jul 9 '13 at 6:51

13 Answers 13

up vote 65 down vote accepted

Mathematical notation, like other aspects of human language, is a human creation - we decide what things mean. The system that is most commonly used at the moment is decimal notation:

  • A "digit" is one of the following integers: $0,1,2,3,4,5,6,7,8,9$.

  • If $a_n,\ldots,a_0$ and $b_1,\ldots,b_m$ are digits, and $a_n\neq 0$, then the expression $$\Large\color{red}{a_na_{n-1}\,\underset{\substack{\small\strut\,\uparrow\\\small\mathsf{ellipsis}}\,}{{\scriptsize\ldots}}\; a_1a_0\;\underset{\substack{\small\uparrow\,\strut\\\small\mathsf{decimal}\,\\ \small\mathsf{point}\,}}{.}\;b_1b_2\,\underset{\substack{\small\strut\,\uparrow\\\small\mathsf{ellipsis}}\,}{{\scriptsize\ldots}}\; b_m}$$ denotes the number $$a_n10^n+a_{n-1}10^{n-1}+\cdots+a_110+a_0+b_110^{-1}+\cdots+b_m10^{-m}$$

The purpose of the single decimal point in this notation (ignore the ellipses) is to clarify which parts of the expression correspond to the powers of $10$ where the exponent is $\geq 0$, and those where the exponent is $<0$.

In this system of notation, there is no meaning of adding a second decimal point. That is not to say that it is impossible to come up with a meaning - perhaps there is some usage in mathematics where a second decimal point would be a convenient and clear way of indicating something, and if there were, perhaps it would then be adopted as a part of our system of notation. But currently, it does not mean anything to write $1.1.2$.

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In fact, we come up with such meanings every day; see for example Section 3.4.2(a)(ii) of the Needless Bureaucracy and Governmentese Convention Proceedings. –  Arkamis Jul 8 '13 at 18:36
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@timothy.s.lau: No, my example is not an ellipsis - I tried to make the ellipses nearly invisible specifically to avoid that confusion. There is one, and only one decimal point when I write the notation $$a_na_{n-1}\,{\tiny\ldots}\; a_1a_0\;.\;b_1b_2\,{\tiny\ldots}\; b_m$$ and I explain what it does. If it makes you more comfortable, consider a specific example, like $$\mathsf{579}\,{\Large.}\,\mathsf{ 23}$$ corresponding to $$(5\times 10^2)+(7\times 10^1)+(9\times 10^0)+(2\times 10^{-1})+(3\times 10^{-2})$$ There is exactly one decimal point. –  Zev Chonoles Jul 8 '13 at 20:10
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@timothy.s.lau, when a decimal point is used in a number to distinguish between the powers of 10 greater than or equal to 0, the decimal point is used as a delimiter, so I feel that Arkamis example is perfectly relevant. –  zzzzBov Jul 8 '13 at 20:20
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@zzzzBov The part left of the decimal point are the positive exponents and the part to its right are the negative ones. What exactly would happen with a second decimal point? Obviously for sectioning, there is no such thing as negative exponents so you can have as many separators as you like but I'm just to unimaginative to continue the series "positive, negative, ..." so please help me out. –  Christian Jul 9 '13 at 10:29
    
@Christian, I was merely pointing out that the decimal point is already used as a delimiter, not claiming that I knew a usage where multiple decimal points were helpful. I could point out that the period is not always used as a decimal mark and point out that some countries use it as the thousands separator (e.x. 1.234.567'89), but I'm sure that OP will say that that usage doesn't count either. –  zzzzBov Jul 9 '13 at 13:08

I'm surprised no one has mentioned this yet.

The answer is Yes. Case in point: IP addresses. Example: 192.168.1.1

:)

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Those aren't decimal points, though. The character may look the same, but the function is different. –  Billy Jul 9 '13 at 2:54
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Not significantly different, the '.' character in an IP address represents an increase in multiplier: $192.168.1.1 = 192 \times 256^3 + 168 \times 256^2 + 1 \times 256^1 + 1 \times 256^0 = 3232235777$ (try it: [1249763622/][http://1249763622/] ) –  Mitch Jul 9 '13 at 4:22
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Please bear in mind the smiley face is supposed to convey the slightly tongue-in-cheek nature of my answer. –  aidan Jul 9 '13 at 5:13
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Not decimal points, but octbitqual points. Or dihectopentacontakaihexal(?) points. –  Dan Jul 9 '13 at 5:29
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@Mitch But the IP would function fine without them right? 192168001001 The dots are just delimiters to make it look pretty and get rid of useless leading zeroes. –  Doorknob 冰 Jul 9 '13 at 17:09

Think about time in hours, minutes and seconds - 8:12:56 - that is a form which uses a double separator.

The point is not whether it can be done, but when it is useful and coherent to do it.

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That's an example of the use of a colon as a delimiter. –  timothy.s.lau Jul 8 '13 at 20:04
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@timothy.s.lau - indeed yes. But what the delimiter is, is a convention. Some people use "." others "," for decimals. The interesting point is that the double delimiter represents a division between base $12/24$ and base $60$ - the first - and a kind of "decimal point" in base $60$ - the second one. I wanted to give an example which was both common and mathematically coherent. –  Mark Bennet Jul 8 '13 at 20:09
    
So are you saying it would be the same if the convention was understood to be 8.12.56? –  timothy.s.lau Jul 8 '13 at 20:25
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@timothy.s.lau Bible references sometimes use "." in this way. You wouldn't use the time version with "." in my culture with time, because the convention is to use ":". If the convention were different, the notation would be different. I interpreted the question as "could you have a convention of this kind?" I saw some answers. I think this illustrates a possibility which wasn't included in the answers I read. No more and no less. –  Mark Bennet Jul 8 '13 at 20:42
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This system isn't decimal, though. Old British currency notation (£2 9s 11d) is another example. –  reinierpost Jul 9 '13 at 15:19

It is of course possible to define whatever you like.

Hewlett-Packard defined an interesting and very handy interpretation of two decimal points for some of their later calculators, starting with the HP-32SII.

If I enter 1.2, I see 1.2. If I enter 1.2. I see 1 2/. Now if go on with 1.2.3, I see 1 2/3. It interprets a number with two decimal points as a fraction, with the three numbers being the whole part, the numerator, and the denominator.

When I hit ENTER, I get 1.66666666667, where it gives me the resulting approximate decimal value.

I love this handy use of two decimal points. I use it all the time.

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Is typing two .s really that much easier than a + and a /? –  user7530 Aug 2 '13 at 8:57
    
Your alternative is incomplete, and yes, it is really that much easier. And it is less error prone since I get to see the number in fractional form before it is reduced, for valuable visual feedback. And it doesn't take up stack space. –  Mark Adler Aug 2 '13 at 14:52
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The keystrokes are a . b . c vs. a ENTER b ENTER c / + or b ENTER c / a +. –  Mark Adler Aug 2 '13 at 14:52

The truth is that you can do whatever you want. The point of creating a decimal system and languages for that matter is so that people can have a general framework in which to work and communicate with others. In the decimal system we only use on decimal point. That doesn't mean you can't do whatever you like if it makes you understand it better.

But no, no one uses that because it has no meaning in the decimal system everyone uses nowadays.

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The "decimal point" is a specific type of "decimal mark", being that it is a dot, period or 'point' that acts as a decimal mark. In other countries, the comma character "," is used as the decimal mark.

A "decimal mark" is a specific type of "radix character", being that it is a radix character for a base-10 number. For a base-2 number, for instance, it might be called the "binary mark", or "binary point".

A "radix character" is a character that separates the integer part of a number from its fractional part in whatever base-notation you are using. Since there are only two parts to this number, there's no need for a further radix character. Since the significance of a digit in a number is defined by its position, things would get very complicated if you allowed the use of multiple radix characters. For instance

1234.56 = 1234.560 = 1234.5600

But does

1234.56 = 1234.56. = 1234.56.0 = 1234.560. = 1234.560.0

?

Since the position of a number in a multi-radix-character notiation system can now be determined with respect to multple points, it would be hard to definitively parse a number, which would kind of defeat the point of a notation system.

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As far as I know, nobody uses such notations. For example, if you interpret $1.2.3$ as $(1.2).3=\left(1+2\cdot 10^{-1}\right)+3\cdot10^{-1}=1.5$ or as $1.(2.3)=1+\left(2+3\cdot 10^{-1}\right)=3.3$

Of course, what I just wrote isn't useful. To use such symbols, you must define them first and see if they have any use in mathematics maybe in a totally ordered field where you have three types of elements not two types like $\mathbb{R}$ (positive and negative numbers).

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Yes, I think decimals aren't typically interpreted as interpunct's –  timothy.s.lau Jul 8 '13 at 20:06
    
Software developers do use such notation. –  Tymek Jul 9 '13 at 1:37

IIRC, Babylonian numbers were sometimes written this way, but their base was 60.

For example, 3.34.5 might mean 3*3600+34*60+5.

I have a recollection of there being "digits" to the right of the sexigesimal point, so that number might mean 3*60 + 34 + 5/60.

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The Babylonians did not really have a place value notation. There could be unseen zeroes between the other given numbers, and in fact tablets of arithmetic problems were designed to avoid them. –  Eric Jablow Jul 8 '13 at 23:48
    
They did mostly have a place value notation, but the lack of symbols for 0 and . created ambiguity. Also, they didn't use Arabic numerals, this number would have been written more like "YYY <<<YYYY YYYYY" (where < and Y are ASCII approximations of the cuneiform wedges for ten and one). –  Dan Jul 9 '13 at 0:23
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They really did have a place-value notation, and while it was base 60, it was a division system. That is 3.30 means 3 1/2, not 210. They did have medial and final zeros, but not trailing zeros. So they could write 0.0.1, for one second, (unit = hour), but not the other way around. Although they got 'large numbers', eg 44.26.40, they apparently struggled with 1/7, and sqrt 2 to three places. The big numbers are like saying in decimal (1/16 = 625), since 1.21 * 44.26.40 = sixty. Neugebauer is the authority over this. –  wendy.krieger Jul 9 '13 at 10:09

I think in the notation it is written, this is ambiguous. In a decimal system, a.b = a + b*(a/10), where .=decimal point. So if you interpret 1.1.2 as 1.(1.2), this is 1 + (1/10)*1.2 = 1 + 0.12 = 1.12. However if 1.1.2 means (1.1).2 then it is 1.1 + (1.1/10)*2 = 1.1 + 0.22 = 1.32.

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Again, there is a totally different meaning between decimals and interpunct's en.wikipedia.org/wiki/Middle_dot and I am interested in the use of decimals –  timothy.s.lau Jul 8 '13 at 20:07

In software engineering it's very common to use 2 or 3 'decimal points' to provide information about the version of an application.

For example the version of Skype I have installed is '5.10.32.116'.

For more details see Software versioning on Wikipedia.

Also if you look at the very bottom of this page you should notice the revision number e.g. 'rev 2013.7.8.823' which uses decimal points to divide the information to easily readable chunks.

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Again, no answer to the question. Those are points, not decimal points (i.e. decimal marks that happen to be points). –  Christian Jul 9 '13 at 10:35

We could if we only decide to ! (even an infinity in this $10$ protuberances alien notation :-)) $$\pi=3.7.15.1.292.1.1.1.2.1.3.1.14...$$ $$e=2.1.2.1.1.4.1.1.6.1.1.8.1.1.10...$$ $$\phi=1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1...$$ $$171/372=0.2.5.1.2.3$$

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We could also define π = 3+7+1+5+1... Yeah sure, we'd need a new symbol for addition but with your idea we'd need a new symbol for the decimal mark so it's makes about equal sense. –  Christian Jul 9 '13 at 10:40
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@Christian: the decimal point would have no interest in this representation : it is a replacement (any real value may be written this way) ! Further all the components are simply integers that don't depend of the basis used : this is a more universal representation of numbers that aliens should appreciate ! –  Raymond Manzoni Jul 9 '13 at 11:24
    
Good point that it's more universal. Although I wouldn't want to write the 292 without using some kind of place-value notation ;) –  Christian Jul 9 '13 at 17:34

Some calculators, like the latest HP ones, accept a number with two decimal points as a fraction. Typing $a.b.c$ gives $a \frac bc$.

Another use for multiple radices is where 'the point of conversion' is different to the quoted unit. It's easier to do with weights annd measures, than with number. For example, there are $1440$ minutes in a day. The conversion of this into duodecimal, would take to account that the preferred division of the day would be $12$ hours of $144$ minutes.

One could write this $1440$ as $144;0.$, where the conversion actually takes place at the semicolon, but the radix is placed at the point (in both cases). So $1.$ minite is converted as $;1.$, gives $;1.2497\dots$ minutes.

One could use mutiple radices to demonstrate conversion of metres to millimetres, or similar in a like fashion. eg 1;000. metres (;) = 1;000. millimetres (.).

Added fractions are fractions with continued numerators, for example the following. They arise quite naturally out of the old measure, eg feet, inches and eights would be $f \frac i{12} \frac e8$. $$a \frac bc \frac de = a \frac {b\frac de}c$$

They're also pretty nifty for added fractions, where one might use several bars to represent radices at spaced units. For example, amounts of monye (in £sd), could be written as $$£l/s/d\frac f4 = l \frac s{20} \frac d{12} \frac f4$$

One then uses vertical bars as the radix of several units (eg pounds and pence), to rearrange the units, while still keeping the $240$ pence in the pound ratio, eg these South-german currencies, derived from the frankish pound.

$$ l \mid \frac s{20} \frac d{12} \mid \frac f{4} = l \mid \frac {xr}{60} \frac d{4} \mid \frac f4$$

One sees that the product of the denominators is the ratio of the units, must remain the same, but intermediate units can be reorganised. The bars represent the radix or unitpoint for the pound and penny. There are still 240 pence in the pound, but the shilling (12d) is being replaced by kreuzer or groat (4d). Once this rearrangement occurs, one could then rearrange the groat of 16 fathings, from an intermediate unit of a penny of four fathings, to a heller of two fathings.

But in the main, there is really little end to having mutlople radices.

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Yes.

Some examples.

  1. While writing a date, we write $ 13/7/2013 $. This "slash" ("$\bf{/}$") is actually a decimal point (or really, the recihexal - recitricentahexadecapental point, to be more accurate) because

$$13/7/2013=13+7\left(60\right)+2013\left(365\right)$$

  1. While writing a time (admittedly, this is the same as the above), we write something like $6:15:40\operatorname{pm}$, the "colon" ("$\bf{:}$") is really the decimal point (or really, the hexacental point) because:

$$6:15:40=6\left(60\right)^2+15\left(60\right)+40$$

  1. While (insane) people are writing angles. Admittedly, all sane people use radians, so this isn't a great argument, but whatever, the same argument as the previous point.

$$$$

  1. While writing nonsense. For example, $djsnmsmcommax.cmcmsdsdmskds.sdsdkssnkskndksdnsnd.dmsdmdsosa.dmaddmefjdc.s.snkfdk.smkdkdmkskss$. Notice that after each decimal point (or really, the quadal point), your urge to edit my answer and get rid of this point quadruples. Another example:

$$spammers.monsters.stupidity.human beings.humanbeingswhodon'tknownwhatthey'rearguingabout .$$

At each "." The harmfulness, quadruples.

  1. While writing numbers.

$$ \mbox{ Ok, others have discussed this. } $$

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Can you give a reference for the terms recitricentahexadecapental & recihexal, a cursory Google search didn't come up with anyone having used it before. –  timothy.s.lau Jul 15 '13 at 20:22
    
@timothy.s.lau: Sure, the reference is this answer. : ) . I made those terms up. If you analyse the words, you 'll see that those makes sense. –  dimensio1n0 Jul 15 '13 at 20:23

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