Non-Terminating Decimal
Non-terminating decimals are decimals that have never-ending decimal digits and continue forever. We will be learning about the non-terminating decimal expansion and conversion of non-terminating decimal to fraction in this article.
| 1. | Non-Terminating Decimal Definition |
| 2. | Non-Terminating Decimal Expansion |
| 3. | Conversion of Non-Terminating Decimal to Rational Number |
| 4. | FAQs on Non-Terminating Decimal |
Non-Terminating Decimal Definition
Non-terminating decimals are defined as those decimal numbers that do not have an endpoint in their decimal digits and keep continuing forever. This happens when a dividend is divided by a divisor but the remainder is never 0 and hence the process keeps repeating and the non-terminating decimal is obtained in the quotient where the decimal digits keep occurring and never come to an end. A non-terminating decimal has an infinite number of decimal places and it is named as non-terminating because the decimal will never terminate. For example, 1.333333..., 4.65675747775..., etc.
Non-Terminating Decimal Expansion
A non-terminating decimal expansion has an infinite number of places and its expansion continues forever. We have two types of non-terminating decimal expansions and they are as follows:
- Non terminating recurring decimal expansion
- Non terminating non-recurring decimal expansion
Non-Terminating Recurring Decimal Expansion
Non-terminating recurring decimal is also known by the name non terminating repeating decimal. In this expansion, the decimal places will continue forever and never come to an end but since the name says repeating or recurring, it signifies that the repetition of the decimal values forms a specific pattern that can be easily identified. For example, 2/9 = 0.2222222…. is a non-terminating recurring decimal expansion. The repetition of the decimal can also be indicated by showing a bar on top of the numbers that are repeating i.e., 0.222222... can also be represented as \(0.\overline{2}\). Similarly, 1/7 = 0.142857 142857 142857... which can also be written as \(0.\overline{142857}\) is also a non-terminating repeating decimal expansion as the block of decimals 142857 is repeating after every 6 digits. Non-terminating recurring decimals can always be converted to a rational number.
Non-Terminating Non-Recurring Decimal Expansion
Non-terminating non-recurring decimal is also known by the name non terminating non-repeating decimal as the values after decimal do not repeat or terminate. For example, 1.4142135..., 2.35638745... Unlike non-terminating recurring decimal, the decimal places do not form any pattern. A non-terminating non-recurring decimal cannot be converted to a rational number. Hence, non terminating non-recurring decimals are also known as irrational numbers. Note that pi is an irrational number as its expansion is non-terminating non-recurring i.e., 3.1415926535 897...
Conversion of Non-Terminating Decimal to Rational Number
As seen in the previous section, a non-terminating recurring decimal can be converted into a rational number. A rational number is defined as the ratio of two integers p and q and is represented as p/q where q ≠ 0. Let us take an example to understand the conversion of a non-terminating recurring decimal to a rational number.
Steps to Convert Non Terminating Recurring Decimal to Rational Number
Let us understand the steps to convert a non-terminating recurring decimal to a rational number by taking an example.
- Step 1: Assume the repeating decimal to be equal to some variable x.
- Step 2: Write the number without using a bar and equal to x. (Bar is for digits that repeat in the same pattern)
- Step 3: Determine the number of digits having a bar on their heads or the number of digits before the bar for mixed recurring decimal.
- Step 4: If the repeating number is the same digit after decimal such as 0.2222... then multiply by 10, if repetition of the digits is in pairs of two numbers such as 0.7878... then multiply by 100 and so on.
- Step 5: Subtract the equation formed by step 2 and step 4.
- Step 6: Then find the value of x in the simplest form.
Let's take an example of a non-terminating recurring decimal number 0.777...
Let, x = 0.777... -------------- (1)
Multiplying 10 on both the sides, we get,
10x = 7.777.. ----------------- (2) (This has to be chosen in such a way that on subtracting we get rid of the decimal)
Subtracting the two equations,
10x - x = 7.777 - 0.777
9x = 7
x = 7/9
Let's take another example to understand this. Convert a non-terminating decimal 0.6565... to a rational number.
Let x = 0.6565... --------------------- (1)
Multiplying 100 on both the sides,
100x = 65.6565... -------------------- (2)
Subtracting the above equations, we get,
100x - x = 65.6565 - 0.6565
99x = 65
x = 65/99
Thus, we have understood the steps to convert a non-terminating recurring decimal to a rational number.
Related Articles
Check these articles related to the concept of the non-terminating decimal.
Non-Terminating Decimal Examples
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Example 1: What kind of decimal expansion does 10/3 have? Show using long division.
Solution: Let us divide 10 by 3 using the long division method,
We see that when 10 is divided by 3, the quotient is 3.333... in which 3 is repeating and does not come to an end. Therefore, we can say that 10/3 = 3.333... is a non-terminating decimal. To be more specific, since the block of numbers is repeating, it is known as a non-terminating recurring decimal.
FAQs on Non-Terminating Decimal
What is Non-Terminating Decimal?
A non-terminating decimal is defined as a decimal number that does not have an endpoint in its decimal digit and keeps continuing forever. For example, 3.12345... is a non-terminating decimal.
What is Non-Terminating Decimal Expansion?
A non-terminating decimal expansion has an infinite number of places and its expansion continues forever. For example, 1.232323...
What are the Types of Non-Terminating Decimals?
There are two types of non-terminating decimals that are: a) Non-terminating recurring decimal which has a pattern of digit repetition in the decimal number. Example, 1.55555, and b) Non-terminating non-recurring decimal which does not have any pattern of digit repetition in the decimal number. Example, 3.12567509...
What does Non-Terminating Decimal Mean?
A non-terminating decimal is defined as a decimal expansion that has an infinite number of decimal places. Example, 1.222222.....
How do you Write a Non-Terminating Decimal?
A non-terminating decimal is written with a bar on top of the block of numbers that is repeating. For example, 0.88888... can be written as \(0.\overline{8}\).
How to Convert Non-Terminating Decimal to Rational Number?
Non-terminating decimals are converted to rational number by following the steps below:
- Step 1: Identify the repeating digits in the given decimal number.
- Step 2: Equate the decimal number with x or any other variable.
- Step 3: Place the repeating digits to the left of the decimal point by multiplying the equation obtained in step 2 by a power of 10 equal to the number of repeating digits. This way you will get another equation.
- Step 4: Subtract the equation obtained in step 2 from the equation obtained in step 3.
- Step 5: Simplify to get the answer.
For example, let's convert 1.888... to a rational number. Let x = 1.888... be equation 1. Multiplying equation (1) by 10, we get,
10x = 18.888... (equation 2)
Subtract equation (1) from equation (2),
9x = 17
x = 17/9
Therefore, 1.888... = 17/9.