RECURRING DECIMALS

INTRODUCTION

Recurring decimals can be defined as the repeating decimals in which several digits or sequences of digits after the decimal point are repeated. The decimal representation depends upon the digits, whether they are terminating or repeating. The decimal numbers are categorized into three types. Terminating decimals where the decimals come to an end after certain digits. Secondly, there are recurring decimals in which a sequence of digits repeat itself an infinite number of times. For example, 1 / 3 is represented as 0.3333… Lastly, there are irrational decimals, these decimals also repeat their digit after the decimal point, but they are not into any sequence or pattern.

DECIMALS

Decimals are the mathematical expression that denotes a combination of a whole number and a fractional number. It is represented by partitioning the numbers by a dot. The left side of the dot denotes the whole number whereas the right side of the dot denotes the fractional number. A whole number can be from 0 to infinity, however, the fraction number is always less than 1. The decimals are further categorized as terminating, the decimals that have a limited number of digits after the dot, for example, ½ will be represented as 0.5, and non-terminating decimals, the decimals that do not have any limit i.e., infinity, for example, ⅓ will be represented as 0.3333… which are non-terminating. The non-terminating decimals are further divided as recurring decimals, the decimals that repeat themselves in a pattern and non-recurring decimals that do not have any pattern or sequence but are extended till infinity, for example, the value of pi is extended till infinity but there is no sequence or pattern. 

WHAT ARE RECURRING DECIMALS?

The recurring decimals meaning states that these are these decimals that are defined as repeating decimals. A decimal is known as a recurring decimal when the digits after the decimal point start repeating themselves in a sequence for an infinite number of times. An equation related to recurring decimals may become very complex to solve as the digits reach up to infinity. It is said that the recurring decimal took its birth during Ancient Egyptian time. However, the recurring decimal is used to represent various fractions as well as the square roots of several numbers in decimal form to ease out the calculations.

REPRESENTATION OF RECURRING DECIMALS:

The recurring decimal is represented in a different way as per the adaptation of several countries. The most common representation of the recurring decimal is the Vinculum, i.e., a bar over the digit or the sequence of the digit.

If the fraction is 1/9, its recurring decimal will be 0.11111111, which can be represented as 0.1

Several other representations of the recurring decimal are:

• Dots: The recurring decimal can be represented by putting a dot over the digit in the case of single repeating digits. Whereas, in sequential repetition, the dots are put above the pattern’s first and last digit.
• Parentheses: A Parentheses is also used to represent a recurring decimal. If the recurring decimal is 0.111111, it could be represented as 0. (1)
• Arc: An arc could also be used to represent the recurring decimal. The arc is extended between the first and the last digit in the case of multi-digit repetition.
• Ellipses: It is the most common method of representation. An ellipsis is placed at the end of the digits or sequence of digits to represent a recurring decimal like 0.111… or 0.868686…

RECURRING DECIMAL INTO FRACTION:

Converting a recurring decimal into a fraction is quite easy. Questions related to the recurring decimals converting into fractions are quite basic in the mathematical quantitative aptitude exams. In order to convert a recurring decimal into a fraction, follow the given steps:

• Assume X represents the recurring decimal and consider it as equation 1.
• Carefully notice the recurring decimal pattern and place that sequence of the digit to the left side of the decimal point.
• According to the number of the digits placed through the left side of the decimal point, represent it with tens of x, considering it as equation 2. For example, if the recurring decimal is 0.7777… place 7 towards the left of the decimal and as it is a single digit, denote it with 10x, i.e., 10x=7.7777… Similarly, if the recurring decimal is 0.353535…. place the sequence 35 towards the left of the decimal and, as it is double-digit, represent it with 100x, i.e., 100x=35.353535…
• Subtract equation 2 from equation 1.
• A simple equation will be formed, which could be solved to find out the fraction of the recurring decimal.

RECURRING DECIMAL EXAMPLES:

One of the recurring decimals examples that might help in practice:

Considering a recurring decimal 0.3333…

x= 0.444…

10x= 4.444…

10x-x=4.4444-0.4444…

9x= 4

X= 4/9

Therefore, the fraction for the recurring decimal is 4/9.

CONCLUSION

The recurring decimal numbers are the repeating decimals, which are composed of a digit or a pattern of digits that repeat themselves after the decimal point for infinity. The recurring decimal is categorized into three types, i.e., The Terminating decimal, The Recurring decimal, and the Irrational decimal. Bars, dots and even ellipses represent these decimals according to the easement. The recurring decimals help in simplifying various fractional numbers as well as the square root of digits to simplify the overall equation for easy calculation. To understand the recurring decimal better, it is required to have the basic knowledge of fraction and fractional numbers.