Recurring Decimal Represented with Bar

Decimal notation is basically a numeral system that expresses a number as a part of 10 or 10n. Any integer or a non-integer can be expressed in this system. 

In general, decimals are numbers that represent fractions of the type p/q, where q is in the power of ten; and any equivalent fraction. Let us understand this with examples,

1. 23/10 is a fraction that can be written as 0.23. 0.23 is the decimal representation of the fraction.
2. 0.25, likewise, is nothing but the decimal form of the fraction 25/100 or its equivalent fraction ¼.

Types of decimals numbers:

Decimal numbers are of two principal types, viz., terminating and non-terminating.

1. Terminating decimals: decimals of the types 0.25, 0.23 are terminating decimals.
2. Non-terminating decimals: decimals of the type 0.3333…, 0.1242424…., 0.13415926…. are non-terminating decimal numbers. 

All three of these non-terminating decimal numbers represent three different subgroups. 

1. Pure recurring decimals: Number 0.333333….is called a pure recurring number since ‘3’ is repeated infinitely. This decimal can be written as 0.33. The bar above a number or a group of the number indicates its recurring nature. 

The number which is repeating itself is known as a period, whereas periodicity is the number of digits that form a repeating group. Here in the above example, ‘3’ is the period, and 1 is the periodicity (3 is the only number that is getting repeated). 

1. Mixed recurring decimal: A decimal number in which there is at least one digit after a decimal point that is not part of the repeating unit is known as a mixed recurring decimal number. Among the above three examples, 0.1242424…is the mixed recurring decimal and can be written as 0.124. Here ’24’ is the period, and 2 is the periodicity.

1. Non-recurring non-terminating decimal: a decimal number that is non-terminating with none of the numbers after the decimal point is repeated is called a non-recurring non-terminating decimal. The number 0.13415926….is an example of such a type of decimal. Such a number, in fact, can be represented in the form of a fraction!

Are these numbers rational numbers…?

By definition, any number which can be represented as p/q. Where q is not zero, it is a rational number. Thus terminating and non-terminating but recurring decimal numbers are rational numbers. At the same time, non-terminating non-recurring numbers are irrational. 

Could you identify the above non-recurring number0.13415926….? Yes, it’s Pi.

Converting recurring decimals to fractions:

Above it is described how we can find out a decimal number for a given fraction. Now let’s learn to calculate a fraction from a corresponding decimal.

Steps to be followed: 

• Let’s suppose the given decimal number equals x. This is your equation (1)
• Multiply the given number by the power of 10 so as to get the recurring unit just after the decimal point. This is your equation (2)
• Multiply the given number by a power of 10 so as to get the recurring unit just left of the decimal point. This is your equation (3)
• Subtract equation (2) from (3) and solve for x
• Now simplify the obtained fraction.

Here we will consider some examples:

1. Convert the decimal 2.444… in the fraction.

2.44…=  x                                    (1)

24.44…= 10x   (2)

Now subtract equation (1) from (2)

10x-x = 24.44… – 2.44….

9x = 22

X = 22/9

• Convert decimal 4.34545… into a fraction

Let’s take,    

X = 4.34545… (1)

10x = 43.4545…      (2)

1000x = 4345.4545…  (3)

Subtracting equation (2) from (3)

1000x – 10x = 4345.45… – 43.4545…

990x = 4302

X = 4302/990

X = 239/55

Conclusion:

This article intended to explain the recurring decimal in an elaborated way. The examples given in this article will help the students to understand the concepts in a detailed way.