Mixed Recurring Decimal

To begin with, a mixed decimal number refers to numbers that contain fractional components. The recurrent term is derived from the Latin term phrase that implies a ‘return to’ feeling. The term “recurring” in mathematics pertains to the regular and periodic repeating of a digit. A recurring decimal, in this context, is a decimal fraction in which a single digit or set of digits is repeated endlessly. Understanding every aspect of mixed recurring decimals is the subject of this article. But before that, let us take a better look at the basic terms. 

What is a Fraction?

If I have seven out of twenty apples, I will write 7/20. The whole is written below a line (which can be horizontal or diagonal), while the fraction’s number of pieces (numerator) is written above. Vulgar fractions, often known as simple fractions, are such fractions.

What is a Decimal?

A decimal number is one in which the numerical and decimal parts may be clearly distinguished or recognized due to the decimal point. A decimal number could be represented in decimal form in general. A decimal point is commonly indicated by a dot “.” The numerical values of the digits prior to the decimal point are larger than those of the digits following the decimal point, which are smaller. 

It may be stated as 14.68, where 14 is a whole number with its significant value and digits after the decimal point, and 68 is a decimal component with smaller values.

Types of Decimals:

Following is the broad classification of decimals:

1. Nonrecurring decimal or terminating decimal: Non-recurring implies that they do not recur or repeat, i.e., they stop or come to an end, such as 0.3, 0.35, and 0.135.
2. Recurring decimal or non-terminating decimal: Recurring means the decimals that recur or repeat, i.e., do not cease or come to an end, such as 0.222…, 0.616161…., and so on. A numerator and denominator are used to express a number. It’s a rational number that usually has a digit pattern that continues forever beyond the decimal point.

A decimal point can be used to distinguish repeated or recurring sections from the total number if a digit or set of digits repeats eternally after the decimal point. In repeating decimals, digits are also seen to be repeated at regular intervals following decimal points in general.

These are further two types: 

1. Pure: This is a repeating decimal in which all digits or groups of digits (after the decimal point) repeat. 0.8888…0r 2.454545…
2. Mixed: A mixed recurring decimal is a repeating decimal in which one or maybe more digits after the decimal point do not repeat, but the remaining digits or set of digits (after the decimal point) do. 

It can be supported by citing some relevant cases, such as:

77/600 = 0.1283333333……. (Where 3 repeats up to infinity, but 128 doesn’t)

Note: Dots are placed over the beginning and last digits of repeated patterns to denote a digit or set of digits that is determined to be repeating. To show the recurring character of a line, it is sometimes presented across a number or collection of digits.

Converting a mixed recurring decimal into a fraction:

The decimal numbers in which certain digits after the decimal are fixed, and some digits are repeated indefinitely after that digit are known as mixed recurring decimals. The following steps can be used to convert mixed recurring decimals to a fraction.

Step 1: First, make a mixed recurring decimal number by eliminating the top bars, and then equate it to any variable x.

Step 2: Determine which digits following the decimal do not include a bar or are recurring.

Step 3: Multiply either side by 10b, where b is a non-recurring decimal number, and the equation is the result (i).

Step 4: At least two times, write a repeating decimal.

Step 5: If the n-digits have a bar, multiply both sides by 10n, and get equation (ii).

Step 6: Subtract equation (i) from (ii)

Step 7: Finally, divide either side of the equation by the x coefficient.

An example of how to convert mixed recurring decimal to fraction can be used to justify the preceding steps:

Suppose, x = 0.2555…

10x = 2.555…                     -equation (i)

10 * 10x = 10 * 2.555…

100x = 25.55…                   -equation (ii)

Subtracting equation (i) from equation (ii)

100x – 10x = 25.55… – 2.555…

90x = 23

x = 23/90

Hence, the value of 0.2555 (which is a mixed recurring decimal) is 23/90 (which is a vulgar fraction).

Conclusion:

A decimal number with the repeating of certain digits after a decimal point is referred to as a mixed recurring decimal. Converting a mixed recurring decimal into a vulgar fraction necessitates the use of straight methods that aid in the better resolution of problems with mixed recurring decimals. Mixed recurring decimals are observed to be a pure periodic type of decimal numbers. After a decimal point, it refers to the recurrence of numbers or digits. A bar sign (¯) on top of repeating numbers or digits can also be used to signify repeated digits of a recurring decimal number. Apart from addressing mathematical difficulties, recurring decimals may also be used to calculate weight and height and to deal with money.