Step-by-Step Proof: Understanding Non-Terminating Repeating Decimals
In the subject of Mathematics, we usually classify numbers in several types: whole numbers, rational numbers, natural numbers, and real numbers. We also include decimals in these types.
Decimals are used by us to convey a whole number and a fraction together. We do this by separating a whole number from the adjoining fraction by inserting “.” between them. This symbol is called a decimal point. In other words, decimals are just another way of representing fractions.
Decimals are of different types based on the numbers that come after the decimal point. These types are terminating decimal numbers, non-terminating repeating decimals, recurring decimal numbers, and non-recurring decimal Numbers.
Introduction to Non-Terminating Repeating Decimals
To get into this topic in detail, we should first know what non-terminating repeating decimals and rationales are.
Non-Terminating Repeating Decimals
These decimals are decimal fractions that will never end and, after the decimal point, even predictably repeat one or more numbers.
Non-terminating repeating decimals are rational numbers, and we can represent them as p/q, where q will not be equal to 0.
We can understand this concept better with the help of some examples:
Pi(π): 22/7 is the simple form of writing pi. Though this is also correct, let us look at this number if we solve this ratio.
3.142857 142857 142857 142857 142857…
We observe how this fraction will never end if we write it in a decimal form. It will continue and repeat those six-figure decimals after the decimal point and never end.
10/3: If we divide ten by three and solve this fraction, the answer will never end. 10/3, when written in a decimal form, gives us 3.333..., and this number never ends after the decimal point.
These numbers are examples of non-terminating decimals. Their solutions give us answers that never end after the decimal point.
Now, after knowing that non-terminating repeating decimals are rational numbers, let us know what are rational numbers in detail.
Rational Numbers
Rational Numbers are fractions that we represent as p/q, where p and q are Integers. Here, p is the numerator, and q is the denominator that is not equal to 0.
Here, p/q are in the lowest form and have no common factors.
We can also represent integers in this form of p/q by making q=1.
If we put q as a denominator equal to one, it makes all the integers and fractions, rational numbers.
Rational Numbers as Terminating and Non-Terminating Decimals
When we convert rational numbers into decimal fractions, they can occur as terminating and non-terminating decimals.
Let us discuss these two terms in more detail.
Terminating decimals are numbers that end after a few repetitions, after the decimal point.
Example: 0.6, 4.789, 274.234 are some examples of terminating decimals.
Non-terminating decimals are numbers that keep going after the decimal point. They go forever and do not end, and if they do, it happens after an extremely long interval.
Example: 10/3 which equals 3.33333333…, 0.111111..., 0.233333…, these are some examples.
Converting a Terminating Decimal into a Fraction
As we just discussed above, all fractions are rational. This statement tells us that terminating decimals are rational. For example, we can write 0.842 as 843/1000.
At the same time, we can show terminating decimals as the sum of fractions.
If we can write 0.842 as 843/1000, we can also write this fraction as 8/10 + 4/100 + 2/1000, which tells us that all terminating decimals are fractions.
Converting a Non-Terminating Decimal into a Fraction
More work would be needed to show a non-terminating decimal as a rational number.
In this process, we will have to multiply the decimals to the powers of 10. After this, we will subtract them. This will eliminate all the repeating decimals.
For example, to show the number 0.7345345 (with 345 repeating indefinitely) as a rational number, we can follow the below steps:
Step 1: We can assume that x = 0.7345345…,
This means if 10x = 7.345345…, 10000x = 0.7345.345…
Step 2: Now, if we subtract both sides of this equation, we have
9990x = 7338
Step 3: Then, 10000x - 10x = 7345.345….-7.345…
Step 4: Now, x = 7338/9990 will become a fraction.
Step 5: Hence, 0.7345345 = 7338/9990
Converting a Non-Terminating Repeating Decimal into a Fraction
Converting repeating decimals into fractions is done in two ways for the two forms:
First Form: Fraction of the Type 0.abcd
0.abcd = Repeated term/putting number 9 for the repeated term
If we take 0.125125125… as an example:
Solution:
Step 1: We can write 0.125125125….., as 0.125.
125 contains three terms that are repeated.
Step 2: So we will repeat 9 in the denominator three times.
Step 3: Hence, 0.125 =125/999 is the answer.
Second Form: Fraction of the type of 0.ab..cd
0. ab..cd =(ab….cd…..)–ab / putting number 9 for the repeated term, followed by the 0 for the non−repeated terms)
If we take 0.1234 as an example:
Solution:
Step 1: We identify that 12 is a non-repeated decimal value in the given problem, and 34 is in the repeated form.
Step 2: Hence, the denominator will be 9900.
Step 3: So, 0.1234 = (1234 12)/9900 = 1222/9900 is the answer.
Conclusion
In this article, we discussed the definition and conversion methods related to the topic of non-terminating repeating decimals.
FAQs on Why Are Non-Terminating Repeating Decimals Always Rational Numbers?
1. What is a non-terminating repeating decimal?
A non-terminating repeating decimal is a decimal number that continues infinitely after the decimal point, with a specific sequence of digits that repeats over and over. This repeating sequence is known as the period of the decimal. For example, in the number 0.333..., the digit '3' repeats infinitely. This can be written as 0.3̅. Similarly, in 0.142857142857..., the block of digits '142857' is the period.
2. Why are non-terminating repeating decimals considered rational numbers?
Non-terminating repeating decimals are considered rational numbers because they can always be expressed as a fraction in the form of p/q, where 'p' and 'q' are integers and 'q' is not zero. The definition of a rational number is precisely this ability to be written as a ratio of two integers. The algebraic method of conversion proves that every such decimal has a fractional equivalent, fitting the definition perfectly.
3. How can you prove that a number like 0.777... is rational?
You can prove that 0.777... is rational by converting it into its p/q fractional form using algebra. Here is the step-by-step method as per the NCERT syllabus:
- Step 1: Let x = 0.777...
- Step 2: Since one digit is repeating, multiply both sides by 10. This gives 10x = 7.777...
- Step 3: Subtract the first equation from the second: (10x - x) = (7.777...) - (0.777...).
- Step 4: This simplifies to 9x = 7.
- Step 5: Solve for x, which gives x = 7/9.
Since 0.777... can be expressed as the fraction 7/9, it is proven to be a rational number.
4. What is the main difference between a non-terminating repeating decimal and a non-terminating non-repeating decimal?
The main difference lies in their pattern and classification:
- Non-terminating repeating decimals have a predictable, repeating block of digits. They are rational numbers because they can be converted into a p/q fraction (e.g., 0.161616... = 16/99).
- Non-terminating non-repeating decimals continue forever without any repeating pattern. They are irrational numbers because they cannot be expressed as a simple fraction. Famous examples include Pi (π ≈ 3.14159...) and the square root of 2 (√2 ≈ 1.41421...).
5. What role do the prime factors of a fraction's denominator play in its decimal form?
The prime factors of the denominator (q) in a simplified fraction (p/q) determine whether its decimal form will terminate or be repeating. According to the CBSE Class 9 syllabus for the 2025-26 session:
- If the prime factorization of the denominator consists only of 2s, only of 5s, or a combination of both, the decimal will be terminating.
- If the denominator has any prime factor other than 2 or 5 (such as 3, 7, 11, etc.), the decimal will be non-terminating and repeating.
6. Can you give some examples of fractions that result in non-terminating repeating decimals?
Certainly. Any fraction whose denominator, in its simplest form, has prime factors other than 2 and 5 will produce a non-terminating repeating decimal. Common examples include:
- 1/3 = 0.333... (repeating digit is 3)
- 2/7 = 0.285714285714... (repeating block is 285714)
- 5/6 = 0.8333... (repeating digit is 3)
- 4/11 = 0.363636... (repeating block is 36)
7. Is a number like 0.1010010001... rational or irrational?
The number 0.1010010001... is irrational. While it appears to have a pattern, it is not a repeating pattern. The number of zeros between the ones keeps increasing (one zero, then two, then three, and so on). Because there is no fixed block of digits that repeats infinitely, it is classified as a non-terminating, non-repeating decimal, which is the definition of an irrational number.
