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Sometimes you encounter numbers likes 1.3332 in mathematics. Is there a more concise notation for representing a number with the same repeating digit after the decimal point that exists in common mathematical notation?

Of course if you encounter this problem with zeroes in something like 0.0001 you can use standard Scientific Notation, but say you have a number such as 11.0001?

When doing math by myself I often use my own notation for this, but it is not formal mathematical notation.

As an example, I can pretend this is the notation I need for a number like 1.3332:

$1.\text{[3 zeroes]}2$

Can this be done is formal math?


Perhaps my example number, $1.3332$ isn't necessarily applicable to the question. Say you had a number with more repeated digits, which I have seen quite often, like:

$1.11111 ... 12 \text{ ones}... 2$

Disclaimer: I do not do mathematics formally, it is a hobby of mine.

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I'm sure you could say $11 + 1\times 10^{-4}$ and have it understood. With $1.3332$ I don't really see it as necessary since there isn't that much repetition, but when you have something like $1.33333...$ or $0.142857142857...$repeating infinitely then I've seen $1.\overline 3$ or $0.\overline{142857}$ respectively. But I still wouldn't take it as a given that a reader would understand the overline notation. It's always best to define your notation beforehand (as mathlove says below). – andraiamatrix Dec 24 '13 at 7:19
+1 I was taught the overline notation in grade school. I think I've also seen a dot used? – J Trana Dec 24 '13 at 7:32
@andraiamatrix $1.3332$ is just a simple example. Overline notation is usually used for the infinite repitition, but I'm talking about a finite repitition. – beakr Dec 24 '13 at 16:07
@beakr Ok, ya, I've not seen a standard for finite repetition, sorry. – andraiamatrix Dec 24 '13 at 23:17
up vote 3 down vote accepted

I don't know such notation.

When you want to use your notation for your numbers in your paper(?), you can use it if you define it beforehand.

Note that you can make any notation if you define it strictly. In my opinion, simpler notation is better for both you and 'readers'.

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Warning: I'm tired and slightly under the influence of alcohol, so the following should be checked carefully: \begin{align*} 0.\underbrace {00\dots0} _{m\times} \underbrace {dd\dots d}_{n\times} &=10^{-m-1}\cdot d\cdot 1.\underbrace{11\dots 1}_{(n-1)\times} \\ &=10^{-m-1}\cdot d\cdot \sum_{k=0}^{n-1} \left(\frac 1 {10}\right)^{\!k} \\ &=10^{-m-1}\cdot d \cdot \frac {1-1/10^n}{1-1/10} \\ &=10^{-m}\cdot d \cdot \frac{1-1/10^n}{9} \\ &=\frac{1}{9}(10^{-m}-10^{-m-n})d \end{align*} So special notation may not be entirely necessary. If there are no leading zeros, the form is particularly simple.


I guess I actually did give some special notation there! I would certainly feel comfortable writing $$1.\underbrace{33\dots3}_{27\times}2$$ or similar if I had the need. For just a few digits, there clearly is no need.

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Alcohol ruins brain cells, the most valuable thing in the world for a mathematician. – GinKin Dec 24 '13 at 17:18
@GinKin It doesn't directly damage your brain cells but it damages the dendrites, which will slow response time to problems. – beakr Dec 26 '13 at 2:51

I'm not aware of any standard notation. But one can extend in a straighforward way the standard overline notation to also handle finite repetitions as follows

$$ 1\,.\overset{\underline{\phantom a}\!\!2}{a\,\ } \overset{\underline{\phantom c}\!\!3}{c\,\ } \overset{\underline{\phantom e}\!\!4}{e\,\ }\overline{m}\,\ :=\ 1.aaccceeeemmmmm\cdots $$

I'm fairly certain that I have seen this notation used by others too. Key is to typeset it so that it cannot be confused for anything else, e.g. if the overline meets the repeat count at half height then it looks like a minus sign, and if the count is placed too far rightward it looks like an exponent.

Of course one should always define any nonstandard notation before using it.

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