# How do you calculate how many decimal places there are before the repeating digits, given a fraction that expands to a repeating decimal?

If you have a fraction such as $$\frac{7}{26}=0.269230\overline{769230}$$ where there are a number of digits prior to the repeating section, how can you tell how many digits there will be given just the fraction?

I believe I could run through the standard long division algorithm until I come across the same remainder for the second time and then use the location of the first instance of this remainder to calculate the number of digits before the repeating section, but this feels very cumbersome.

After lots of reading online, I came across what looks like a formula for it from Wolfram MathWorld:

When a rational number $\frac{m}{n}$ with $(m,n)=1$ is expanded, the period begins after ${s}$ terms and has length ${t}$, where ${s}$ and ${t}$ are the smallest numbers satisfying $10^s\equiv10^{s+t}\pmod{n}$.

I know how to calculate the length of the period of a fraction, and so in the case of my original fraction we have $10^s\equiv10^{s+6}\pmod{26}$, but I don't know how to solve for ${s}$ in this equation!

So there are really two questions here - the one in the title, and a sneaky one about how to take logs in a modulo arithmetic equation.

• Nobody else knows how either; it's an open problem. You can use trial-and-error, and there are some things you can do with Fermat's theorem to cut down the number of exponents you have to check, but I don't think anything significantly better than brute force is known.
– MJD
Mar 26, 2014 at 0:37
• @MJD That's not quite true; the algorithm is quite simple, but there isn't a constant-time closed form AFAIK. Mar 26, 2014 at 0:38
• @BobbyOcean You are correct that they are the best rational approximation, but that's very different than the best decimal approximation. Mar 26, 2014 at 0:44
• I disagree with the point you chose to claim as beginning repetition in $7/26$. It can be considered as repeating with the second decimal place. Mar 26, 2014 at 0:45
• @BobbyOcean Note that the continued fraction representations for the numbers $\frac1n$ all have the same length; each is $[0; n]$. But they have completely different base-10 periods, for example the period of $[0; 3]$ is 1 and the period of $[0; 17]$ is 16 and the period of $[0; 21]$ is 9.
– MJD
Mar 26, 2014 at 1:15

Rewrite the fraction as $$\frac{m}{n}=\frac{p}{10^sq}$$ where $p,q$ are coprime and $q$ is not divisible by $2$ or $5$ while $p$ is not divisible by $10$. Computing $s$ (the pre-period) is easy; it is the larger of the number of times $2$ divides $n$ and the number of times $5$ divides $n$. Then we want the smallest $t$ such that $10^t\equiv 1\;(\bmod\;q)$. By Fermat's little theorem, we have $10^{\varphi(q)}\equiv 1\;(\bmod\;q)$, thus $\;t|\varphi(q)$ so it suffices to check the divisors of $\varphi(q)$.

• It would be $\varphi(q)$, not generally $q-1$. And if I read the question right, the OP is more interested in the length of the pre-period than of the period, that would be $s$, provided $10 \nmid p$. Mar 26, 2014 at 0:49
• This is similar to what I have done to calculate the length of the period but, as @DanielFischer said, it's the length of the pre-period I am interested in. Ultimately I am trying to custom-render a repeating decimal and work out how many digits I need in total and where to draw the overbar, all given a simple fraction. Mar 26, 2014 at 1:25
• @DanielFischer Thanks, I tricked myself with my own notation. Mar 26, 2014 at 1:46
• @AlexBecker you seem to have confirmed and formalised the pattern I noticed (see my comment on the original post). Thanks, I'll mark this as the correct answer. Mar 26, 2014 at 1:50

'Cumbersome' is what computers are meant for! Here is an algorithm I converted into a python program to do something similar for Project Euler 26.

In the $$2^{nd}$$ section of this answer we get the answer using the high-school algorithm that the OP mentioned, but eschewed as being too cumbersome. But by studying the theory behind the algorithm and using elementary number theory, we will be able to understand, directly, the wolfram theory on decimal expansions at formulas/examples (7) - (9).

For the zero divisor $$[10] \in {\textstyle \mathbb {Z} /26\mathbb {Z}}$$, calculations show that

$$\quad x \pmod{26} \text{ where } x \in [10^0, 10^1, 10^2, 10^3, 10^4, 10^5, 10^6, 10^7] = [1, 10, 22, 12, 16, 4, 14, 10]$$

The numerator $$7$$ of the fraction $$\large \frac{7}{26}$$ carries over to a unit, $$[7] \in {\textstyle \mathbb {Z} /26\mathbb {Z}}$$.

So the unit $$[7]$$ 'goes for a ride' over the 'exponential graph' of zero-divisors generated by $$[10]$$,

$$\quad [1\cdot7,10\cdot7,\, 22\cdot7,\, 12\cdot7,\, 16\cdot7,\, 4\cdot7,\, 14\cdot7]\quad \text{(not necessary to to calculate any of these residues)}$$

Conclusion: In the decimal expansion of $$\large \frac{7}{26}$$, the repeating block of digits is of length $$6$$ and begins at the $$2^{nd}$$ fractional decimal digit ($$10^7 \equiv 10^1 \pmod{26}$$).

With this behind us, we can easily get the explicit answer - multiply the numerator by $$10^7$$ and keep the 7 quotient digits (padding $$0s$$ after the decimal point might be necessary) after dividing,

$$\quad 7 \cdot 10^7 = 26\cdot2692307+18$$

and (we've got the seven digits in the quotient),

$$\quad \large \frac{7}{26} \approx 0.2\overline{692307}$$

The same pattern (1 plus 6 block) occurs whenever the numerator of $$\large \frac{n}{26}$$ satisfies

$$\quad 1 \le n \lt 26 \land n \text{ is odd } \land n \ne 13$$

Analyzing the high-school algorithm (see some theory here), there are at most $$26$$ divisions (neophyte estimate) that have to be performed. So let us just jump into it!

Expand $$\large \frac{7}{26}$$:

Divide Approximate (append $$q$$ digit)
$$7\cdot 10 = 26 \cdot 2 + 18$$ $$\large \frac{7}{26} \approx 0.2$$
$$18\cdot 10 = 26 \cdot 6 + 24$$ $$\large \frac{7}{26} \approx 0.26$$
$$24\cdot 10 = 26 \cdot 9 + 6$$ $$\large \frac{7}{26} \approx 0.269$$
$$6\cdot 10 = 26 \cdot 2 + 8$$ $$\large \frac{7}{26} \approx 0.2692$$
$$8\cdot 10 = 26 \cdot 3 + 2$$ $$\large \frac{7}{26} \approx0.26923$$
$$2\cdot 10 = 26 \cdot 0 + 20$$ $$\large \frac{7}{26} \approx 0.269230$$
$$20\cdot 10 = 26 \cdot 7 + 18$$ $$\large \frac{7}{26} \approx 0.2692307$$

Now the residue $$18$$ has already appeared and was used after calculating the first fractional decimal; the answer is summarized as follows:

$$\quad \large \frac{7}{26} \approx 0.2\overline{692307}$$

• Want another example? Calculate $\frac{1}{84}$ using this work. Dec 27, 2020 at 19:50