Repeating Decimal to a Fraction Calculator

Repeating Decimal to Fraction Converter

Enter Repeating Decimal (e.g., 0.(3), 1.2(34), 0.123):

Understanding Repeating Decimals and Their Conversion to Fractions

A repeating decimal, also known as a recurring decimal, is a decimal representation of a number whose digits are periodic (eventually repeating the same sequence of digits indefinitely). For example, 1/3 is 0.333… (often written as 0.(3)), and 1/7 is 0.142857142857… (written as 0.(142857)). All rational numbers (fractions) can be expressed as either terminating or repeating decimals.

How to Convert a Repeating Decimal to a Fraction

The process of converting a repeating decimal to a fraction involves a bit of algebraic manipulation. Let's break down the general method:

Identify the parts: A repeating decimal can have an integer part, a non-repeating decimal part, and a repeating decimal part.
- Example: In 1.2(34), 1 is the integer part, 2 is the non-repeating decimal part, and 34 is the repeating decimal part.
- Example: In 0.(3), 0 is the integer part, there's no non-repeating part, and 3 is the repeating part.
Set up an equation: Let the repeating decimal be equal to x.
Shift the decimal: Multiply x by powers of 10 to move the decimal point past the non-repeating part and then past one full repeating block.
Subtract the equations: Subtract the equation where the decimal is just past the non-repeating part from the equation where it's past one full repeating block. This eliminates the repeating part.
Solve for x: Solve the resulting equation for x to get the fraction.
Simplify: Reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).

The Formula Used by This Calculator

This calculator uses a generalized formula to convert repeating decimals of the form I.N(R) (where I is the integer part, N is the non-repeating decimal part, and R is the repeating decimal part) into a fraction:

Let lenN be the number of digits in the non-repeating part N.

Let lenR be the number of digits in the repeating part R.

The fraction is given by:

Numerator = (I * 10^lenN + N) * (10^lenR - 1) + R

Denominator = 10^lenN * (10^lenR - 1)

After calculating the numerator and denominator, the fraction is simplified by dividing both by their Greatest Common Divisor (GCD).

Examples of Repeating Decimal to Fraction Conversion

0.(3)
- Here, I=0, N="" (empty), R="3".
- lenN=0, lenR=1.
- Numerator = (0 * 10⁰ + 0) * (10¹ - 1) + 3 = 0 * 9 + 3 = 3
- Denominator = 10⁰ * (10¹ - 1) = 1 * 9 = 9
- Fraction: 3/9, which simplifies to 1/3.
0.1(6)
- Here, I=0, N="1", R="6".
- lenN=1, lenR=1.
- Numerator = (0 * 10¹ + 1) * (10¹ - 1) + 6 = (1) * 9 + 6 = 15
- Denominator = 10¹ * (10¹ - 1) = 10 * 9 = 90
- Fraction: 15/90, which simplifies to 1/6.
1.2(34)
- Here, I=1, N="2", R="34".
- lenN=1, lenR=2.
- Numerator = (1 * 10¹ + 2) * (10² - 1) + 34 = (12) * 99 + 34 = 1188 + 34 = 1222
- Denominator = 10¹ * (10² - 1) = 10 * 99 = 990
- Fraction: 1222/990, which simplifies to 611/495.
0.123 (Terminating Decimal)
- This is treated as a simple decimal.
- Numerator = 123
- Denominator = 1000
- Fraction: 123/1000 (already simplified).

Why Convert Repeating Decimals to Fractions?

Converting repeating decimals to fractions is a fundamental concept in mathematics, particularly when dealing with rational numbers. It allows for:

Precision: Fractions provide an exact representation of a number, unlike repeating decimals which are often truncated for practical use.
Algebraic Manipulation: Fractions are easier to work with in algebraic equations and calculations, avoiding rounding errors.
Understanding Number Properties: It reinforces the understanding that all repeating decimals are rational numbers.

Use this calculator to quickly and accurately convert your repeating decimals into their simplest fractional form!

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Please use 0.(3) or 1.2(34).'; resultDiv.classList.add('error'); return; } var nonRepeatingPartStr = decimalPartStr.substring(0, decimalOpenParenIndex); var repeatingPartStr = decimalPartStr.substring(decimalOpenParenIndex + 1, decimalCloseParenIndex); if (repeatingPartStr.length === 0) { resultDiv.innerHTML = 'Repeating part cannot be empty.'; resultDiv.classList.add('error'); return; } var numNonRepeating = parseInt(nonRepeatingPartStr || "0"); var numRepeating = parseInt(repeatingPartStr); var lenN = nonRepeatingPartStr.length; var lenR = repeatingPartStr.length; // Formula: (I * 10^lenN + N) * (10^lenR – 1) + R // Denominator: 10^lenN * (10^lenR – 1) numerator = (integerPart * Math.pow(10, lenN) + numNonRepeating) * (Math.pow(10, lenR) – 1) + numRepeating; denominator = Math.pow(10, lenN) * (Math.pow(10, lenR) – 1); } else if (openParenIndex !== -1 || closeParenIndex !== -1) { resultDiv.innerHTML = 'Invalid parenthesis notation. Please use 0.(3) or 1.2(34).'; resultDiv.classList.add('error'); return; } else { // Terminating decimal or integer var decimalPointIndex = input.indexOf('.'); if (decimalPointIndex === -1) { // Integer var intValue = parseInt(input); if (isNaN(intValue)) { resultDiv.innerHTML = 'Invalid integer input.'; resultDiv.classList.add('error'); return; } numerator = intValue; denominator = 1; } else { // Terminating decimal var parts = input.split('.'); var integerPart = parseInt(parts[0] || "0"); var decimalPart = parts[1] || ""; if (isNaN(integerPart) || !/^\d*$/.test(decimalPart)) { resultDiv.innerHTML = 'Invalid decimal input.'; resultDiv.classList.add('error'); return; } numerator = integerPart * Math.pow(10, decimalPart.length) + parseInt(decimalPart || "0"); denominator = Math.pow(10, decimalPart.length); } } if (denominator === 0) { resultDiv.innerHTML = 'Cannot convert to fraction (denominator is zero).'; resultDiv.classList.add('error'); return; } var commonDivisor = gcd(numerator, denominator); var simplifiedNumerator = numerator / commonDivisor; var simplifiedDenominator = denominator / commonDivisor; if (isNegative) { simplifiedNumerator = -simplifiedNumerator; } resultDiv.innerHTML = 'Fraction: ' + simplifiedNumerator + ' / ' + simplifiedDenominator; }