Question

How can you find natural numbers `a, b, c, d` in the range `20`-`100` to match a given decimal approximation in the form `a/b * c/d` ? For example: `0.71428571 ~~ 24/32 * 64/56`. Is there a general method to match `8` decimal places, e.g. `0.23687546` ?

Answer 1

Here's one way you can try...

Explanation:

Note that in the example you give `24/32 = 3/4` and `64/56 = 8/7`, resulting in `3/4*8/7 = 6/7`

So given a decimal to `8` decimal places, the first step may be to try to find a fraction that rounds to that representation.

We can do that by using continued fractions: Find a more-or-less terminating continued fraction for the given decimal, then resolve it to a fraction.

We can find the coefficients of the continued fraction using the steps:

• Write down the whole number part of the given number and subtract it.

• If the remainder is `0` (or near enough) then stop.

• Otherwise take the reciprocal and repeat.

Attempting to do this with `0.23687546` we find:

• `color(red)(0).23687546 rarr 4.22162769`

• `color(red)(4).22162769 rarr 4.51207157`

• `color(red)(4).51207157 rarr 1.95285202`

• `color(red)(1).95285202 rarr 1.04948090`

• `color(red)(1).04948090 rarr 20.20981833`

• `color(red)(20).20981833 rarr 4.76602783`

• `color(red)(4).76602783 rarr 1.30543560`

• `color(red)(1).30543560 rarr 3.27401259`

• `color(red)(3).27401259 rarr 3.64946735`

• `color(red)(3).64946735 rarr 1.53972328`

Hmmm: By this stage we are running into rounding errors and have not encountered a very clear stopping point.

Let us just work with what we have got here:

`0.23687546 ~~ [0;4,4,1,1,20,4,1,3,3]`

`color(white)(0.23687546) = 0+1/(4+1/(4+1/(1+1/(1+1/(20+1/(4+1/(1+1/(3+1/3))))))))`

`color(white)(0.23687546) = 11587/48916`

Note that `48916 > 100^2`, so this fraction is no good for our objective.

To simplify, we can use fewer terms.

`0.23687546 ~~ 0+1/(4+1/(4+1/(1+1/(1+1/(20+1/(4+1/1)))))) = 934/3943`

This fraction does not match all `8` decimal places and still does not help much, since `934 = 2 * 467` and `3943` is prime.

So it seems that we cannot achieve the desired objective with an arbitrary `8` place decimal representation.

About the best we can do with `0.23687546` seems to be:

`0.23687546 ~~ 0+1/(4+1/(4+1/(1+1/(1+1/20)))) = 185/781 = (5 * 37)/(11 * 71) = 20/44 * 37/71`

`~~ 0.23687580`