Understanding Terminating Decimals in Mathematics

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:

1. Terminating decimals

2. Non-Terminating Non-Repeating decimals

3. 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?

1. 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.

2. 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

1. 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.

2. 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.

3. 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.

$\frac{7}{9}$