Convert decimal into fraction

Answer 1

you say the decimal and write it as you say it. for example: you do .456, you would write 456/1000 then of corse you would reduce. Answer improved by Robert Richter 1-11-09 Here's a common technique used in computers. Lets take two random numbers between 1 and 1000, say 73/191. We get 0.38219895287958115183246073298429. We don't very often get fractions whose denominator is "nice" in a decimal system (ones with only 2 or 5 as factors). We seldom get numbers that repeat. For example, A/B may have B digits in it before repeating. So, what we do is subtract off integers leaving a value between 0 and 1, invert, and subtract off integers, invert, etc., until we get down to (almost) an integer.

Doing this, we get

0.38219895287958115183246073298429,

2.6164383561643835616438356164384,

1.6222222222222222222222222222222,

1.6071428571428571428571428571429,

1.6470588235294117647058823529412,

1.5454545454545454545454545454545,

1.8333333333333333333333333333333,

1.2,

4.9999999999999999999999999999999 (Actually I got 5, but the result will be the same either way, and I wanted to show that either value will work)

1.0000000000000000000000001

You want a threshold that is "close enough" to an integer to stop. If I subtract off the integer and get a value less than 0.00001, I am probably good enough for most rational numbers.

Now, some facts about fractions.

(1) 1/X, where X is an integer, is always reduced.

(2) A+B/C=(A*C+B)/C, is also reduced if B/C is reduced (although it may be an improper fraction)

(3) 1/(A/B)=(B/A) is also reduced if A/B is reduced.

These three facts mean that I am reduced at all steps; I don't need to worry about reducing the final answer.

Our technique is find each sub-fraction. What is 1.2? It is 6/5. What is 1.5454545454? It is 17/11. What is 1.6470588...? It is 28/17. Each of these are found using the fraction representation below, which initially starts with only our integers.

0.38219895287958115183246073298429 almost equals ...

0+1/(2+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(4+1/1))))))))=

0+1/(2+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(5))))))))=

0+1/(2+1/(1+1/(1+1/(1+1/(1+1/(1+1/(6/5)))))))=

0+1/(2+1/(1+1/(1+1/(1+1/(1+1/(1+5/6))))))=

0+1/(2+1/(1+1/(1+1/(1+1/(1+1/(11/6))))))=

0+1/(2+1/(1+1/(1+1/(1+1/(1+6/11)))))=

0+1/(2+1/(1+1/(1+1/(1+1/(17/11)))))=

0+1/(2+1/(1+1/(1+1/(1+11/17))))=

0+1/(2+1/(1+1/(1+1/(28/17))))=

0+1/(2+1/(1+1/(1+17/28)))=

0+1/(2+1/(1+1/(45/28)))=

0+1/(2+1/(1+28/45))=

0+1/(2+1/(73/45))=

0+1/(2+45/73)=

0+1/(191/73)=

0+73/191=

73/191

Answer 2

Can be done.