# What does this highlighted formula means?

Can anyone tell me what the highlighted formula means, how to read this, how to prove it and if possible share a simpler version of this formula?

It seems like it has something to do with infinite periodic decimal fraction. I don't know what it means.

Any help would be appreciated.

I think the author means the following:

Given a rational number $$r=p/q$$ it has a decimal expansion. And the fact that $$r$$ is rational is equivalent to the decimal representation being periodic after a certain point. That is if $$r=1/3$$, then the decimal representation is $$r=0.333\cdots$$. As another example if you take $$r=9/13$$, we get $$r=0.692307692307\cdots$$. It is not necessary that the first few digits after the decimal place should repeat, say for example $$r=52/225=0.23111\cdots$$. Also for decimal expansions which terminate, we assume that $$0$$ repeats infinitely, that is if $$r=1/2=0.5$$, we consider $$r=0.5000\cdots$$.

In the book, $$a$$ denotes the integer part of $$p/q$$, $$(\alpha_1,\cdots,\alpha_n)$$ denotes the first $$n$$ non-repeating digits and $$(\beta_1\cdots\beta_m)$$ denotes repeating digits.

Now given a rational number we can find the corresponding $$a$$, $$\alpha_i$$'s and $$\beta_j$$'s. The reverse representation might mean that the reverse construction is also possible. That is given an integer $$a$$, $$\alpha_i$$'s and $$\beta_j$$ can we find the corresponding $$r$$.

They have given the formula as $$r=\pm a \pm \frac{\overline{\alpha_1\cdots\alpha_n\beta_1\cdots\beta_m}-\overline{\alpha_1\cdots\alpha_n}}{\overline{\underbrace{99\cdots 9}_{m}\underbrace{0\cdots 0}_{n}}}$$

Here by $$\overline{123}$$, let's say, I am guessing the author means the number 123 (One hundred and twenty three). I have also made a slight edit to the formula which I think is correct. Hope this helps.

• Thanks, that was really helpful but can you please elaborate the RHS of the formula like what is a and why is there a , (comma) after ±a±0? Nov 19, 2019 at 10:00
• The first formula (not highlighted one) is a representation. It doesn't necessarily have to make arithmetic sense. Like a vector $(1,1)\in\mathbb{R}^2$. As for the $\pm$ even I am not sure Nov 19, 2019 at 10:03
• $p$ can be positive or negative whereas $a$ I assume is taken to be positive, so the $\pm$ accounts for either possibility. Nov 19, 2019 at 16:29

This is the rule of how to convert an infinite periodic decimal fraction into an ordinary fraction.

The result is a fraction where in the numerator one has to subtract a number before the second period and a number before the first period and in the denominator one writes down $$m$$ nines ($$m$$ is the number of digits in the period) and $$n$$ zeros (n is the number of digits before the period). There is a typo in the formula you are providing. The number of nines should be $$m$$.

For example,

$$3{,}28(123) = 3 + 0{,}28(123) = 3 + \frac{28123-28}{99900} = 3 + \frac{28095}{99900} = 3\frac{1873}{6660}.$$

Here,

• $$n = 2$$:
• $$\alpha_1\alpha_2 = 28$$,
• $$m = 3$$:

• $$\beta_1\beta_2\beta_3 = 123$$.
• A number before the second period is 28123.

• A number before the first period is 28.

• According to your example, the number of nines should be $m$, the number of zeros should be $n$, not the other way round as you imply at the end of your first paragraph. Can you please double check? Nov 19, 2019 at 16:34
• Fix it. Thanks for the comment Nov 20, 2019 at 13:42