terminating and repeating decimals

 Repeating and Terminating Decimals: An Overview

A fraction refers to a portion of a whole number in pure mathematics. On the other hand, the decimal system is where a decimal point separates the entire number and fractional parts. There are different varieties of decimals, including terminating, non-terminating decimals and repeating and non-repeating decimals. If a number is finite and has a fixed number of digits, that is known as a terminating decimal. This type of decimal always has an endpoint.

While solving different mathematical problems, the conversion of decimal to fractional values is a favoured method. That is because fractional values are easier to simplify. On that note, let us get an overview of repeating and terminating decimals.

An Introduction to Repeating and Terminating Decimals

Non-terminating decimals are divided into two types of decimals:

• Repeating Decimals
• Terminating Decimals

Repeating Decimal

Repeating decimals refer to non-terminating decimals that are repetitive in nature. A repeating decimal has an endless number of digits after the decimal point. However, all the digits following the decimal are known. Although, if the digits after the decimal point are all zero, it cannot be considered a repetitive decimal.

Terminating Decimal

If the digits after the decimal point end, the number is referred to as a terminating decimal expansion. Since the digits after the decimal point end after one digit, the fraction 5/10 is represented as the decimal expansion 0.5. This is an example of terminating decimal expansion. A terminating decimal expansion or a non-terminating recurring decimal expansion can be derived for a rational number.

Types of Repeating and Terminating Decimals

There are two types of Repeating Decimal Numbers –

• Non-terminating recurring decimal
• Non-terminating non-recurring decimals

However, one cannot distinguish between various types of terminating decimals. Whenever the number ends after a decimal, it is involuntarily categorised under Terminating decimals.

Long Division and Factors

Long Division is a method that is used for dividing large numbers. The long division method follows a sequence and breaks any division problem into multiple steps. Like any other division problem, the dividend also gets divided by the divisor, which gives us the quotient. You can also get a remainder sometimes from long divisions.

A student needs to follow the same steps of a division to do long division –

• Divide
• Multiply
• Subtract
• Bring down the quotient and repeat or find the remainder.

Here's an example of long division with decimals.

Factors:

In mathematics, Factors refer to a numerical or algebraic expression that evenly divides another number or expression. You cannot get a remainder in Factors, unlike long division. For example, 2 and 4 are factors of 12 because 12 ÷ 2 = 6 and 12 ÷ 4 = 3. Here you are getting absolute value with no remainders. The other factors of 12 are 1, 3, 6, and 12.

Factoring Denominators

The denominator is the number of equal parts that make one unit, whole. You can use a factoring denominator to get the exact value of a factor without a remainder. For example, if you divide 10 by 5, the five come below ten in the division, generating an exact value of 2. Hence, here 5 is the factoring denominator.

Some Examples of Repeating and Terminating Decimals

If the decimal numbers have infinite numbers of digits following the decimal point, and the digits are repeated at equal intervals, it is known as the repeating or recurring decimal numbers. For example, if the numbers end like 0.111…, 4.444444…, 5.232323…, 21.123123…etc., those are repeating decimals.

Formula of Repeating Decimals:

Write an equation where x = the original number. For example, x = 0.4444. Then, multiply both sides of the equation by 10^1. Since there is only 1 repeating digit in the original number, you will get 10x = 4.4444.

On the other hand, terminating decimals are numbers that end eventually after some time post decimals. For example, 0.5, 4.123, 123.789 are some examples of terminating decimals.

The formula of Terminating Decimal:

You need to divide the numerator by the denominator. If the number doesn’t have a remainder, then it is a terminating decimal. For example: If you divide 3 by 5, 3/5 is terminating, and 0.6 is a terminating decimal.

A Decimal system is a crucial part of basic mathematics. Although decimals seem simple, knowing the formulas and knowing how to calculate one may seem complex to many students. Read this blog to get a detailed idea of what is repeating and terminating decimals and ace your assessments.