How to convert an infinite loop decimal to a fraction (algorithm)

Repeating decimal How to score
As we all know, the finite fraction is another form of the decimal score, so any finite fraction can be directly written into a few, a few percent, a few thousand ... The number. So can the infinite number of fractions be converted into fractions?

First we want to make it clear that infinite decimals can be divided into two categories according to whether the fractional part loops: Infinite loop decimals and infinite non-cyclic decimals. Infinite not repeating decimal the score, which will be explained in detail in the middle school, and the fractional number of infinite loops can be converted into fractions. So, how does an infinite loop decimal number turn into fractions? Because its fractional part is infinite, it is obviously impossible to write a very few, a few percent, a few thousand ... The number. In fact, it is difficult to repeating decimal fractions in an infinite number of decimal digits. So I'm going to start here and find a way to "cut off" the "big tail" of an infinite loop of decimals. The strategy is to enlarge the infinite loop by 10 times times, 100 times times, or 1000 times times with the multiplication method. Make the enlarged infinite loop decimal with the original infinite loop decimal "big tail" exactly the same, and then subtract the two numbers, "big tail" is not cut off! Let's take a look at two examples:

⑴ the 0.4747 ... and 0.33 ... Into fractions.

Want to 1:0.4747......x100=47.4747 ...

0.4747......x100-0.4747......=47.4747 ... -0.4747 ...

(100-1) x0.4747......=47

That is 99x0.4747 ... =47

So 0.4747......=47/99

Want to 2:0.33......x10=3.33 ...

0.33......x10-0.33......=3.33 ... -0.33 ...

(10-1) x0.33......=3

namely 9x0.33......=3

So 0.33......=3/9=1/3

Thus, the pure repeating decimal fraction, its fractional part can be written such a fraction: pure repeating decimal of the minimum number of cycles is a few, the denominator is composed of several 9 of the number, the molecule is a pure repeating decimal in a circular section of the number.

⑵ the 0.4777 ... and 0.325656 ... Into fractions.

Want to 1:0.4777......x10=4.777 ... ①

0.4777......x100=47.77 ... Ii

Use ②-① to get:

0.4777......x90=47-4

So, 0.4777......=43/90

Want to 2:0.325656......x100=32.5656 ... ①

0.325656......x10000=3256.56 ... Ii

Use ②-① to get:

0.325656......x9900=3256.5656 ... -32.5656 ...

0.325656......x9900=3256-32

So, 0.325656......=3224/9900

Induction: The fractional part of a mixed repeating decimal can be converted into fractions, the numerator of which is the difference between the number of fractional parts of the second cycle section and the number of non-cyclic parts in the fractional part. The first few digits of the denominator are 9, and the last few are 0. The number of 9 is the same as the number of bits in the Loop section, and the number of 0 is the same as the number of bits in the Loop section.

How to convert an infinite loop decimal to a fraction (algorithm)