Terminating Decimals

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.

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.

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 2^p5^q or 2^p or 5^q, 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.

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.

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 2^p5^q or 2^p or 5^q, 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 2^p5^q or 2^p or 5^q, where p,q∈N, then the rational number has a non-terminating recurring decimal expansion.

Related Topics

Check these articles related to the concept of terminating decimal numbers.

• Decimal Representation of Rational Numbers
• Decimal Representation of Irrational Numbers
• A Rational Number as a Decimal
• How to convert decimal numbers to p by q form?
• Decimals

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 2^p5^q, 2^p, or 5^q. 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.