To find the answer to this question first we would have to understand the meaning of the words Non-terminating repeating decimals and rationals.
Once when we will find the meaning of these then it will be very easy to describe the answer in the best manner. So let’s Explore the meanings of these words.
Non-Terminating Repeating Decimals
A decimal fraction that will never come to an end, but will repeat one or more numbers after the decimal point in a predictable manner.
Let's Take Some Examples and Make This Definition Easier
Usually, we simply write pi as 22/7. This is not correct; it changes from the true value of pi just after a few decimal places, but let's look at the number we get by solving out this ratio as a decimal fraction:
3.142857 142857 142857 142857 142857 …
This fraction will also never come to an end, but it will go on and on repeating those six fix decimals.
Again, the decimal fraction will keep ongoing, but it is repeating the same number over and over again.
So, 22/7 and 10/3 never terminate, but keep on repeating their digits of decimals consisting of 6 numbers or just one respectively. Thus they are called non-terminating repeating decimals.
In mathematics, a rational number is a number which can be defined as a fraction or a quotient, i.e., in the form of p/q where p and q are the two integers and p will be the numerator and q will be the non-zero denominator and p/q is in the lowest form, i.e. p and q have no common factors.
Integers also can be expressed in this form of p/q with q=1. So, all the integers and fractions are rational numbers.
Rational Numbers in Terms of Terminating and Non-Terminating Decimals
Rational numbers can be expressed in the terms of decimal fractions. These rational numbers when converted into decimal fractions, can be both terminating and non-terminating decimals.
Terminating Decimals: Terminating decimals are the numbers which end themselves just after a few repetitions after the decimal point.
Example: 0.5, 2.456, 123.456, etc. are all examples of terminating decimals.
Non-Terminating Decimals: Non-terminating decimals are the ones which keep on continuing after decimal point (i.e. they go on forever), They don’t come to an end or if they do, it is after a long interval.
Example: 10/3 which equals 3.33333333…
Before discussing non-terminating decimals, let us consider that terminating decimals are rational. This is obvious because terminating decimals can be changed to the fractions (and all fractions are rational as per the definition). For example, 0.842 can be shown as
Further, terminating decimals can be expressed as the sum of the fractions also. For example, 0.842 can be shown as
8/10 + 4/100 + 2/1000.
Since rational numbers are closed under addition, the sum of any number of fractions is also a fraction. This shows that all terminating decimals are fractions.
Now, for the non-terminating decimals, a specific strategy will be needed to show that it is a rational number. The strategy will be to multiply all the decimals to powers of 10 and then subtract them so that the repeating decimals will be eliminated.
For example, to show that 0.7345345 (with 345 repeating indefinitely) as rational, we assume x = 0.7345345…
Now, 10x = 7.345345… and 10000x = 0.7345.345…
Now, subtract both sides of the equations, we have
9990x = 7338
10000x - 10x = 7345.345….-7.345...
Now, x = 7338/9990 which will be a fraction. Therefore,
0.7345345 = 7338/9990
Ways to Convert Non-Terminating Repeating Decimals Into Rational Numbers
Let us now learn the concept of the conversion of repeating decimals into the fractional form. Now, let's discuss the two different forms of the repeating fraction.
Form 1: Fraction of the Type 0.abcd
The formula or method to convert this type of non-terminating repeating decimal to a fraction is given by:
0.abcd = Repeated term / Number of 9’s for the repeated term
Convert the number 0.125125125… to the fractional form.
The decimal 0.125125125….., we can write as 0. 125.
Here, 125 contains three terms, and it is repeated again and again. Thus, the number of times 9 should also be repeated in the denominator three times.
0.125 =125/999 answer
Form 2: Fraction of the Type of 0.ab..cd
The formula or method to convert this type of non-terminating repeating decimal to the fractional is given by:
0.ab..cd =(ab….cd…..)–ab / (Number of time 9′s the term which is repeating, followed by the number of times 0′s which are the non−repeated terms)
Convert the number 0.1234 to the fractional form.
In the given problem, 12 is a non-repeated decimal value, and 34 is in the repeated form. Thus the denominator will become 9900.
0.1234 = (1234–12)/9900=1222/9900 answer.