How to Identify and Use Terminating Decimals in Math
Before learning about terminating decimal meaning, let us learn about decimals. Decimals are an alternate way to write fractions. In a decimal number, the whole numbers are written on the left side of the decimal point ‘.’ and the remainders/ fractional part is written on its right. For example, 310/100 can be written as 0.31.
Image: An example of a decimal number.
Decimals are of Three Types:
Terminating decimals
Non-Terminating Non-Repeating decimals
Non-Terminating Repeating decimals
In this article, we will learn about terminating decimals in detail.
What is Terminating Math Definition?
Terminating decimal definition is a decimal number with a finite number of digits after the decimal point. A terminating decimal like 5.65 can be represented as the repeating decimal 5.6500000000..., but when the repeating digit is zero, the number is usually labelled as terminating. Terminating decimals are rational numbers, which when converted into fractions have 0 as a remainder. Any terminating decimal representation can be written as a fraction with a power of ten in the denominator.
Recognising a Terminating Decimal
For a number to have a terminating decimal expansion, you must check for the following points:
It should always have a finite number of digits after the decimal point.
It should be a rational number.
It shouldn’t have a bar over the finite digits occurring after the decimal point.
When expressing the number in the reduced fractional form, its denominator should not contain any prime numbers other than two or five.
A terminating decimal number An example is given below.
Image: Example of a terminating decimal.
In this, if we factorise 8, it’s 23. This shows that 0.125 is terminating.
What is Non-Terminating Meaning?
Non-Repeating Decimal
A non-terminating, non-repeating decimal is a decimal number that lasts indefinitely with no repeating digits after the decimal point. In short, there is no uniform repetition of a number in non-repeating decimals. This sort of decimal can't be expressed as a fraction because it is an irrational number. Therefore, it can't be represented in the p/q format.
If the fraction form of this type of decimal expansion does not result in a remainder equal to zero, the decimal is referred to as a non-terminating in Maths. Examples of non-terminating non-repeating decimal numbers are: pi (π), √2, 157.8000619…, etc.
Repeating Decimals
A decimal representation of some periodic digits and an infinitely repeated number that is not zero is known as a repeating decimal. In certain cases of non-terminating decimals, a digit or a set of digits in the decimal portion repeats itself indefinitely, which are also called the pure repeated decimals.
Image: A non-terminating decimal expansion with recurring digits.
Contrary to the non-repeating decimals, repeating decimals are rational numbers and can be written in the form of p/q, where q ≠ 0. A bar is mounted on the shortest sequence of repeating digits to represent these decimal numbers. For example,
Image: A repeating decimal expansion with a bar.
Take another example, 68.5940000… with 0’s as repeating digits. In this, since the zeros can be omitted and the decimal terminates before these zeros, this decimal representation is considered a terminating decimal rather than a repeated decimal where the repeated number is a zero.
Theorem
Let x be a rational number with a decimal expansion that ends. Then prime factorization of q is of form 2m5n, where n and m are non-negative integers, and x can be expressed in the form p/q, q ≠ 0, where p and q are co-prime.
Let x = p/q, q ≠ 0 be a rational number with prime factorization 2m5n, and non-negative integers n and m. As a consequence, x has a decimal extension that is terminating.
Let x =p/q, q ≠ 0 be a rational number with a q prime factorization that is not 2m5n, and n and m be non-negative integers. Then x has a non-terminating and repeating decimal extension (recurring).
Therefore, terminating and non-terminating repeating decimals are rational, while only the non-terminating non-repeating decimals are irrational.
Also, these different decimal expansions of rational numbers make our daily lives quite convenient. We can use fractions or terminating/ repeating decimals in different ways depending on the context of a situation.
For example, you will have to convert the weight of an item in terminating decimal form to check it on a digital scale, whereas while dividing a pizza among friends requires the knowledge of fractions.
Solved Examples
1. Transform 8/11 to decimal.
Ans:
Thus, 8/11 = 0.727272…
2. Identify 8/3 as terminating or non-terminating.
Ans: Method 1: We can write 8/3 in decimal form using the long division method.
Thus, 8/3 = 2.6666…. Which means it is a repeating decimal.
Method 2: We can factorise the denominator into the simplest form.
As, denominator 3 cannot be written in the form of 2m5n, 8/3 is a non-terminating repeating decimal.
3. Convert 0.7777777... to fraction.
Ans: Let us start by equating 0.7777777… to x,
x = 0.777777...
10x = 7.777777
10x - x = 7.777777 - 0.7777777
9x = 7
Thus, x =7/9
4. State whether 798/500 will have a terminating or a non-terminating decimal form without performing division.
Ans: Factorising the denominator, 500 = 2 x 2 x 5 x 5 x 5
It can be written as 22 x 53
Since it can be factorized in the form of 2p5q, it is terminating decimal.
FAQs on What Is a Terminating Decimal?
1. What is a terminating decimal? Give an example.
A terminating decimal is a decimal number that has a finite, or limited, number of digits after the decimal point. The digits do not continue forever. For instance, common examples include 0.5, 1.75, and 0.625, all of which end after a certain number of decimal places.
2. How can you tell if a fraction will result in a terminating decimal without performing long division?
To determine if a rational number, expressed as a fraction p/q in its simplest form, has a terminating decimal expansion, you must analyse the prime factors of its denominator (q). If the prime factorization of 'q' consists only of the factors 2 and/or 5 (i.e., it can be written in the form 2ⁿ5ᵐ, where 'n' and 'm' are non-negative integers), the fraction will convert to a terminating decimal.
3. Using the rule, is the fraction 13/80 a terminating decimal?
Yes, the fraction 13/80 represents a terminating decimal. To verify this, we find the prime factors of the denominator, which is 80. The prime factorization of 80 is 2 × 2 × 2 × 2 × 5, or 2⁴ × 5¹. Since the denominator's prime factors are only 2 and 5, it satisfies the condition for a terminating decimal expansion.
4. What is the key difference between a terminating decimal and a non-terminating repeating decimal?
The primary difference lies in the nature of the digits after the decimal point.
- A terminating decimal has a finite number of digits. It stops. For example, 1/4 = 0.25.
- A non-terminating repeating decimal has an infinite number of digits that follow a recurring pattern. For example, 1/3 = 0.333... (the digit 3 repeats infinitely).
5. Are all terminating decimals considered rational numbers? Explain why.
Yes, every terminating decimal is a rational number. This is because any terminating decimal can be rewritten as a fraction with a denominator that is a power of 10 (such as 10, 100, 1000, etc.). For example, 0.85 can be expressed as 85/100. Since a rational number is any number that can be written in the p/q form, all terminating decimals fit this definition.
6. Why is 3.14 a terminating decimal, while the value of Pi (π) is not?
This highlights the difference between a number and an approximation. The number 3.14 is a terminating decimal because it has exactly two digits after the decimal point and is equal to the fraction 314/100. In contrast, Pi (π) is an irrational number, meaning its decimal representation is both non-terminating and non-repeating (3.14159...). We frequently use 3.14 as a convenient, rounded-off value for π in calculations, but it is not the true, complete value of Pi.
7. Is a number like 0.777 a terminating or non-terminating decimal?
As it is written, 0.777 is a terminating decimal because it has a finite (three) number of digits. However, if the notation was 0.777... (with an ellipsis) or had a bar over the 7, it would represent a non-terminating repeating decimal. The way a number is written is crucial; without any indication of infinite repetition, it is considered to be finite and thus terminating.
