change $0.684 210 526 3$ into a fraction

Question

I am a non-mathematician who quit with math after middle school. Now I face a practical problem which I cannot solve.

Suppose I want to turn $0.684 210 526 3$ into a fraction, how would I do that withou using my `convert to fraction button on my calculator'. To use other words, are there two number $a$ and $b$ less than $20$ such that $\frac{a}{b}=0.684 210 526 3?$

Thank you in advance,

Mary

Answer 1

Write the digits on the top and 1 followed by the same number of zeroes as you have digits on the bottom: $$ 0.23 = \frac{23}{100} $$ and for you $$ 0.6842105263 = \frac{6842105263}{10000000000} $$

To find if there is a simple fraction which indeed resembles yours (in other words, if the output of the calculator is an approximation and you are looking for something reasonably close to it) -- Wolfram Alpha does a very decent job at such things.

Your input gives 13/19, for example.

Answer 2

When you want to convert any decimal into a fraction, follow these steps:

1. Put the decimal over 1 i.e. $\frac{\text{decimal}}{1}$
2. Then multiply the top and bottom by 10 until all the decimals are gone.
3. Try to reduce the problem

So, in this case we have: $0.684 210 526 3$ and which as 10 decimals places so we will have:

$$\frac{0.684 210 526 3}{1}$$

then

$$\frac{684 210 526 3}{10000000000}$$

Now the question is, can we reduce it? The answer is no. Since, the only numbers that can divide denominator are either $2,5$ or a multiple of $5$ (there are other divisors i.e. $4,8,16 \ldots 1024$). This also has to be the case for the numerator but it isn't so, the problem cannot be reduced further. If you want an approximation for the decimal then as the other answer has suggested, $\frac{13}{19}$ is pretty good according to Wolfram|Alpha.

Answer 3

This is a try to make the continued fraction clearer (see comment above)

$$\frac{1}{1+\frac{1}{2+\frac{1}{6}}}$$ is the fraction giving the approximation. Begin at the bottom at $2+\frac{1}{6}$ and work up to the top.

Here is how you get the numbers 1,2,6 :

Begin with x, calculate 1/x and subtract the number before the comma. Continue this process.

Answer 4

As others have pointed out, your specific question does not reduce to an exact fraction whose numerator and denominator are less than twenty.

To address the general question, most software and hardware implementations convert decimal numbers to fractions by traversing (as a type of binary search) over the Stern-Brocot tree until either an exact or approximate match (to some specified precision) is found.

The general procedure is as follows:

1) Given a positive decimal input $x$ and precision $p$, let $a$ and $b$ be fractions representing the boundaries of the search with initial values $a \leftarrow \frac{0}{1}, b \leftarrow \frac{1}{0}$.

2) Calculate the mediant of $a$ and $b$ as a fraction, $m$, then its decimal representation, $y$.

3a) If $y = x$ (to precision), then terminate

3b) If $y < x$, then let $a \leftarrow m$ and repeat step 2

3c) If $y > x$, then let $b \leftarrow m$ and repeat step 2

After terminating, $m$ now contains the fractional representation of $x$ with precision $p$.

(To address $x < 0$, let $x \leftarrow \lvert x\rvert$, perform the above and instead report $-m$.)