Repeating Decimals as Fractions Calculator

Repeating Decimal to Fraction Converter

function gcd(a, b) { return b === 0 ? a : gcd(b, a % b); } function calculateFraction() { var integerPartStr = document.getElementById("integerPart").value.trim(); var nonRepeatingDigitsStr = document.getElementById("nonRepeatingDigits").value.trim(); var repeatingDigitsStr = document.getElementById("repeatingDigits").value.trim(); var resultDiv = document.getElementById("result"); // Input validation if (!/^\d*$/.test(integerPartStr)) { resultDiv.innerHTML = "Please enter a valid integer for the Integer Part."; return; } if (!/^\d*$/.test(nonRepeatingDigitsStr)) { resultDiv.innerHTML = "Please enter valid digits for Non-Repeating Decimal Digits."; return; } if (!/^\d+$/.test(repeatingDigitsStr)) { // Repeating digits must not be empty resultDiv.innerHTML = "Please enter at least one digit for Repeating Decimal Digits."; return; } var integerPart = parseInt(integerPartStr || "0"); // Default to 0 if empty var nonRepeatingLength = nonRepeatingDigitsStr.length; var repeatingLength = repeatingDigitsStr.length; var numerator; var denominator; if (nonRepeatingLength === 0) { // Pure repeating decimal, e.g., 0.(3) numerator = parseInt(repeatingDigitsStr); denominator = parseInt(Array(repeatingLength + 1).join("9")); // e.g., "999" for 3 repeating digits } else { // Mixed repeating decimal, e.g., 0.12(34) var fullDecimalDigits = nonRepeatingDigitsStr + repeatingDigitsStr; var nonRepeatingValue = nonRepeatingDigitsStr === "" ? 0 : parseInt(nonRepeatingDigitsStr); numerator = parseInt(fullDecimalDigits) – nonRepeatingValue; denominator = parseInt(Array(repeatingLength + 1).join("9") + Array(nonRepeatingLength + 1).join("0")); } // Add the integer part if (integerPart !== 0) { // Only add if integerPart is not zero numerator = numerator + (integerPart * denominator); } // Simplify the fraction var commonDivisor = gcd(numerator, denominator); var simplifiedNumerator = numerator / commonDivisor; var simplifiedDenominator = denominator / commonDivisor; resultDiv.innerHTML = "The fraction is: " + simplifiedNumerator + " / " + simplifiedDenominator + ""; }

Understanding Repeating Decimals and Their Fractional Equivalents

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)). A fundamental concept in mathematics is that all rational numbers (fractions) can be expressed as either terminating or repeating decimals.

This calculator provides a straightforward way to convert any repeating decimal into its simplest fractional form. This conversion is a key skill in number theory, algebra, and pre-calculus, helping to deepen understanding of rational numbers.

How the Conversion Works

The method to convert a repeating decimal to a fraction relies on algebraic manipulation. The general formula used by this calculator is highly efficient and covers all types of repeating decimals:

Fraction = (Whole number formed by non-repeating and repeating digits - Whole number formed by non-repeating digits) / (Number of 9s equal to repeating digits followed by number of 0s equal to non-repeating digits)

Let's illustrate this with a few examples:

Example 1: Pure Repeating Decimal (e.g., 0.(3))

Consider the decimal 0.333… (or 0.(3)).

  • Integer Part: 0
  • Non-Repeating Decimal Digits: (empty)
  • Repeating Decimal Digits: 3

Applying the formula:

The "whole number formed by non-repeating and repeating digits" is 3. The "whole number formed by non-repeating digits" is 0. The denominator consists of one '9' (for the single repeating digit '3') followed by zero '0's (because there are no non-repeating digits).

Numerator = 3 - 0 = 3

Denominator = 9

Fraction = 3/9

Simplifying this fraction by dividing both by their Greatest Common Divisor (GCD), which is 3, gives:

Simplified Fraction = 1/3

Example 2: Mixed Repeating Decimal (e.g., 0.12(34))

Consider the decimal 0.12343434… (or 0.12(34)).

  • Integer Part: 0
  • Non-Repeating Decimal Digits: 12
  • Repeating Decimal Digits: 34

Applying the formula:

The "whole number formed by non-repeating and repeating digits" is 1234. The "whole number formed by non-repeating digits" is 12. The denominator consists of two '9's (for the two repeating digits '34') followed by two '0's (for the two non-repeating digits '12').

Numerator = 1234 - 12 = 1222

Denominator = 9900

Fraction = 1222/9900

To simplify, we find the Greatest Common Divisor (GCD) of 1222 and 9900, which is 2.

Simplified Fraction = (1222 / 2) / (9900 / 2) = 611 / 4950

Example 3: Repeating Decimal with an Integer Part (e.g., 5.1(6))

Consider the decimal 5.1666… (or 5.1(6)).

  • Integer Part: 5
  • Non-Repeating Decimal Digits: 1
  • Repeating Decimal Digits: 6

First, we convert the decimal part 0.1(6) to a fraction:

Numerator = 16 - 1 = 15

Denominator = 90 (one '9' for '6', one '0' for '1')

Fractional Part = 15/90

Simplifying 15/90 by dividing by their GCD (15) gives 1/6.

Now, we add the integer part to this fractional part:

Total Fraction = 5 + 1/6 = (5 * 6 + 1) / 6 = 31/6

This calculator automates these calculations, providing you with the simplified fractional form of any repeating decimal you input quickly and accurately.

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