Recurring Decimal Calculator
This calculator converts recurring decimal to fraction.
-
Please enclose the recurring part in parentheses ().
- \( 0.333\cdots\rightarrow \) 0.(3)
- \( 0.0909\cdots\rightarrow \) 0.(09)
- \( 0.142857142857\cdots\rightarrow \) 0.(142857)
What is recurring decimal?
Recurring decimals are decimals that repeat the same number of times, as follows
\[ 0.33\cdots,0.1212\cdots,0.123123\cdots \]
Recurring decimals are expressed as follows
\[ 0.\overline{3},0.\overline{12},0.\overline{123} \]
Example
Example 1
Convert \( 0.\overline{3} \) to a fraction.
Solution
Let \(x\) be 0.3.
\[ x=0.\overline{3}\tag{1} \]
Multiply both sides by \( 10 \) since the circular clause is one digit.
\[ 10x=3.\overline{3}\tag{2} \]
Subtracting equation (1) from equation (2) yields
\[ 9x=3 \]
Solving this yields the following fractions
\[ 0.\overline{3}=\frac{1}{3} \]
Example 2
Convert \( 0.\overline{142857} \) to a fraction.
Solution
Let \(x\) be \( 0.\overline{142857} \).
\[ x=0.\overline{142857}\tag{3} \]
Multiply both sides by \(10^6\) since the circular clause is six digit.
\[ 1000000x=142857.\overline{142857}\tag{4} \]
Subtracting equation (3) from equation (4) yields
\[ 999999x=142857 \]
Solving this yields the following fractions
\[ 0.\overline{142857}=\frac{142857}{999999}=\frac{1}{7} \]