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. Are non-terminating repeating decimals rational?
Yes, non-terminating repeating decimals are rational numbers. By definition, a rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. A non-terminating repeating decimal, such as $0.666\ldots$, can always be represented as a fraction. For example, $0.\overline{3}$ (repeating 3) equals $\frac{1}{3}$. Vedantu’s expert maths teachers can help you understand the process of converting these decimals into fractions with step-by-step explanations.
2. Is 0.7777777 rational or irrational?
The decimal $0.7777777\ldots$ (with repeating 7s) is a rational number. Since the digit 7 repeats indefinitely, it is a non-terminating repeating decimal that can be written as the fraction $\frac{7}{9}$. This makes it rational according to the definition. If you want more examples or a guided method of conversion, Vedantu's maths curriculum covers this topic in detail.
3. Is 0.333333333 rational or irrational?
The decimal $0.333333333\ldots$ is rational because it is non-terminating but repeating. The repeating digit 3 means it can be expressed as $\frac{1}{3}$, which is a ratio of two integers. For students learning number systems, Vedantu provides live classes and practice problems to master such concepts.
4. Is 0.333 repeating a rational number?
Yes, $0.333\ldots$ (where the digit 3 repeats) is a rational number. This is because it can be written in the fraction form as $\frac{1}{3}$. Since both 1 and 3 are integers and the denominator is not zero, this confirms its rational nature. Vedantu provides comprehensive tutorials explaining such conversions for all maths grades.
5. How do you convert a non-terminating repeating decimal into a fraction?
To convert a non-terminating repeating decimal into a fraction, follow these steps:
- Let $x$ represent the repeating decimal (e.g., $x = 0.\overline{6}$).
- Multiply both sides by a power of 10 that moves one full repeat to the left of the decimal (for $0.\overline{6}$, multiply by 10: $10x = 6.\overline{6}$).
- Subtract the original equation: $10x - x = 6.\overline{6} - 0.\overline{6}$.
- This simplifies to $9x = 6$, so $x = \frac{6}{9} = \frac{2}{3}$.
6. What distinguishes a non-terminating repeating decimal from a non-repeating decimal?
Non-terminating repeating decimals have a sequence of digits that repeats indefinitely (like $0.727272\ldots$), while non-terminating non-repeating decimals (like $\pi = 3.14159\ldots$) have no pattern of repetition.
- Repeating decimals can be written as fractions (rational numbers).
- Non-repeating decimals cannot be expressed as fractions (irrational numbers).
7. Are all recurring decimals rational numbers?
Yes, all recurring (repeating) decimals are rational numbers. This is because any decimal with a repeating pattern, no matter how long, can be converted into a fraction. Vedantu’s maths experts focus on teaching these concepts through visual and practical examples, making it easier for students to identify rational numbers in decimal form.
8. Can you give examples of non-terminating repeating decimals and their fractional representations?
Here are some examples of non-terminating repeating decimals and their equivalent fraction forms:
- $0.\overline{1} = \frac{1}{9}$
- $0.\overline{36} = \frac{4}{11}$
- $0.\overline{123} = \frac{123}{999}$
9. Why are non-terminating repeating decimals considered rational numbers in maths?
In mathematics, a rational number is any number that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Non-terminating repeating decimals fit this definition because their repeating patterns allow them to be exactly written as a fraction. Vedantu’s maths teachers help clarify these concepts through illustrations and real-life examples in their lessons.
