Repeating Decimals to Fraction Conversion
Repeating decimals are decimal expansions that do not terminate. These decimal numbers have an infinite number of digits after the decimal points, digits in these expansion have a repetitive pattern.
Any decimal can be easily converted to fraction if it has terminating digits, such that, 2.3 = 23/10, 3.45 = 345/100. But in case of repeating decimals, it is impossible to count the number of decimal places as it is infinite and thus it is impossible to add power of 10 as denominator to convert it into fraction. So, we must follow the steps added below to convert repeating decimals to fractions.
Step 1: Identify the repeating digits pattern in the given decimal number.
Step 2: Equate the decimal number with a variable (generally x)
Step 3: Place repeating digits to left of decimal point by multiplying equation obtained in step 2 by a power of 10 equal to the number of repeating digits.
Step 4: Subtract equation obtained in step 2 from the equation obtained in step 3.
Step 5: Simplify to get your answer.
This can be explained using the example added below,
Example: Convert 2.3333.. to a fraction.
Step 1: 3 is repetitive digit in the given decimal number
Step 2: Lets take x = 2.3333…
Step 3: There is only 1 repetitive digit, thus multiplying equation by 10. 10x = 23.3333…
Step 4: Subtract x = 2.333… from 10x = 23.333…
9x = 21
Step 5: Simplifying
x = 21/9 = 7/3
Thus, 2.3333… = 7/3
Terminating and Non-Terminating Decimals
A terminating decimal is a decimal that has an end digit. While non terminating decimal is a decimal that doesn’t have an end term. In mathematics, we have various types of numbers, like natural numbers, whole numbers, integers, rational and irrational numbers, etc. A decimal number is one of them and is used to represent a whole number and a fraction together. In algebra, decimal numbers are a set of numbers that lie between integers on a number line.
A decimal number consists of a whole number part and a fractional part, which are separated by a point called a decimal point. A decimal number is classified into different types depending on the type of digits that come after the decimal point. Terminating decimals and non-terminating decimals are the two types of decimals, where a non-terminating decimal is further classified into non-terminating recurring decimals and non-terminating non-recurring decimals.