Terminating decimals are the numbers that have a fixed or a finite number of digits after the decimal point. Decimal numbers are used to represent the partial amount of whole, just like fractions. In this lesson, we will focus on the type of decimal numbers, that is, terminating decimal numbers. The word 'terminate' means to bring to an end. In terms of decimal, it is a number that ends. In this article, we will learn what are terminating decimals and the ways to recognize these numbers.
|1.||Definition of Terminating Decimal|
|2.||How to Recognize Terminating Decimal?|
|3.||Terminating Decimal Examples|
|4.||FAQs on Terminating Decimal|
Terminating Decimal Definition
The number which has a finite number of digits after the decimal point is referred to as a terminating decimal. Decimals are used to express the whole number and fraction together that is separated from each other by inserting a "." i.e. a decimal point. For example, 10.4, 10 is the whole number and 4 is the decimal fraction. There are different categories of decimals depending on what type of digits occur after the decimal point, they are:
- Terminating decimal expansion
- Non-terminating recurring decimal expansion
- Non-terminating non-recurring decimal expansion
A number has a terminating decimal expansion if the digits after the decimal point terminate or are finite. The fraction 5/10 has the decimal expansion of 0.5, which is a terminating decimal expansion because digits after the decimal point end after one digit. A rational number has either a terminating decimal expansion or a non-terminating recurring decimal expansion. For example, 23.5 is a terminating decimal number because it has 1 digit after the decimal point.
How to Recognize a Terminating Decimal?
Here are a few points that will help you to recognize a terminating decimal number.
- A number that is not rational is never a terminating decimal number.
- If you can express the denominator of a simplified rational number in the form 2p5q or 2p or 5q, where p, q ∈ N, then the number has a terminating decimal expansion.
- A terminating decimal number always has a finite number of digits after the decimal point.
Terminating Decimals Examples
To differentiate whether a given decimal is terminating or non-terminating decimal, it is necessary to understand their basic differences like:
- Terminating decimal has finite digits and non-terminating decimals do not have finite digits.
- It is easy to represent a terminating decimal in the form of p/q but it is difficult to express a non-terminating decimal (non-repeating) in p/q form, where q is not equal to 0.
The table given below shows examples which will help you in identifying terminating decimals better.
|Number||Terminating or Non-terminating Decimal|
|2.675||Since there are 3 digits after the decimal point, it is a terminating decimal number.|
|3/8||We can write 3/8 as 3/8= 3/(23). Clearly, the denominator is of the form 2p. So, it is a terminating decimal number.|
|√2||This is not a rational number. So, it is a non-terminating decimal number.|
Tips to Remember
- Terminating decimal numbers has a finite number of digits after the decimal point.
- A number with a terminating decimal is always a rational number.
- If the denominator of a rational number can be expressed in form 2p5q or 2p or 5q, where p,q∈N, then the decimal expansion of the rational number terminates.
- If the denominator of a rational number cannot be expressed in form 2p5q or 2p or 5q, where p,q∈N, then the rational number has a non-terminating recurring decimal expansion.
Check these articles related to the concept of terminating decimal numbers.
Terminating Decimal Examples
Example 1: The length and breadth of a rectangle are 7.1 inches and 2.5 inches respectively. Determine whether the area of the rectangle is a terminating decimal or not.
Solution: Given, the length of rectangle is 7.1 inches and the breadth of rectangle = 2.5 inches.
Area of Rectangle = Length × Breadth = 7.1 inches × 2.5 inches =17.75 inches2
As the number of digits is finite after the decimal point, the area of rectangle is a terminating decimal expansion.
Example 2: Look at the following pie charts. Which one of the pie charts represents a terminating decimal number?
Solution: From the above figures we understand:
a)The shaded portion of the first pie chart represents the number 4/6. 4/6 can be simplified as 2/3. The decimal expansion of 2/3 is 0.66... which is non-terminating and repeating decimal expansion.
b)The shaded portion of the second pie chart represents the number 2/8. 2/8 can be simplified as 1/4. The decimal expansion of 1/4 is 0.25 which is terminating decimal expansion.
Therefore, b) pie chart represents the terminating decimal expansion.
Example 3: Mary's teacher wrote 4 fractions on board: 2/7, 8/20, 10/30, and 5/32. Help Mary to find which among them is a terminating decimal?
Solution: The fractions can be expressed as:
2/7 = 0.285714....
8/20 = 0.4
10/30 = 0.333...
5/32 = 0.15625
Therefore, the fractions which are terminating decimals are 8/20 and 5/32.
FAQs on Terminating Decimal
What does Terminating Decimal Mean?
Terminating decimal numbers are decimals that have a finite number of decimal places. In other words, these numbers end after a fixed number of digits after the decimal point. For example, 0.87, 82.25, 9.527, 224.9803, etc.
What is an Example of Terminating Decimal?
A decimal number that has finite number of digits after the decimal point is known as a terminating decimal. One example of terminating decimal is 0.5.
How to Find Terminating Decimal Without Division?
If we have to find whether a number is a terminal decimal expansion or not, we can use prime factorization method. Here, we factorize the denominator and see if the prime factorization results in the form of either 2p5q, 2p, or 5q. If this condition is satisfied it means that the decimal expansion of the given rational number would be terminating. This is how we can find terminating decimals without division.
How to know if a Fraction is a Terminating Decimal?
To identify whether a fraction is a terminating decimal or not, we can use the division operation and divide numerator by denominator. For example, 1/8 is a terminating decimal expansion as the quotient after dividing 1 by 8 is 0.125.
Is 7/8 a terminating decimal?
The decimal representation of 7/8 is 0.875. So, it is a terminating decimal.
Is 11/20 a terminating decimal?
The decimal representation 11/20 is 0.55. So, it is a terminating decimal.
Is 3.3 a terminating decimal?
Yes, 3.3 is a terminating decimal because the digits after the decimal point come to an end. The number of digits after the decimal point is finite which is counted as 1.
Is Pi a terminating decimal?
No, pi is not a terminating decimal as the value of pi is 3.141592653589793238... It does not terminate and is not repetitive either which makes it an irrational number.
Is 0.75 a terminating decimal?
Yes, 0.75 is a terminating decimal because the digits after the decimal point come to an end. The number of digits after the decimal point is 2.