A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p, and a non-zero denominator q. Since q cannot be zero, every integer is a rational number (for example, 7 can be written as 7/1). Numbers like 1.5 (which is 3/2) and 0.333… (which is 1/3) are also rational.
Operations with Rational Numbers
Our calculator performs basic arithmetic operations on two rational numbers, represented as fractions:
1. Addition and Subtraction:
To add or subtract two fractions, a/b and c/d, they must have a common denominator. The least common multiple (LCM) of the denominators is often used, but any common multiple will work.
Find a common denominator (e.g., b * d).
Convert each fraction:
a/b = (a * d) / (b * d)
c/d = (c * b) / (d * b)
Perform the addition or subtraction on the numerators:
Addition: (a*d + c*b) / (b*d)
Subtraction: (a*d - c*b) / (b*d)
Simplify the resulting fraction if possible.
2. Multiplication:
Multiplying two fractions, a/b and c/d, is straightforward:
Multiply the numerators together: a * c
Multiply the denominators together: b * d
The result is: (a * c) / (b * d)
Simplify the resulting fraction.
3. Division:
Dividing by a fraction is the same as multiplying by its reciprocal. To divide a/b by c/d:
Find the reciprocal of the divisor (c/d becomes d/c).
Multiply the first fraction by the reciprocal of the second: (a/b) * (d/c)
The result is: (a * d) / (b * c)
Ensure the original denominator c is not zero (as it becomes a denominator in the multiplication step).
Simplify the resulting fraction.
Use Cases:
Rational numbers and their operations are fundamental in many areas:
Mathematics: Core concepts in algebra, calculus, and number theory.
Computer Science: Representing data, algorithms, and financial calculations where exact precision is needed.
Engineering & Physics: Calculating ratios, proportions, and precise measurements.
Everyday Life: Recipes (scaling ingredients), budgeting, and understanding fractions in practical contexts.
This calculator helps visualize and compute these operations efficiently.
function gcd(a, b) {
var a = Math.abs(a);
var b = Math.abs(b);
while (b) {
var t = b;
b = a % b;
a = t;
}
return a;
}
function simplifyFraction(numerator, denominator) {
if (denominator === 0) {
return { num: NaN, den: NaN, error: "Denominator cannot be zero." };
}
if (numerator === 0) {
return { num: 0, den: 1, error: null };
}
var commonDivisor = gcd(numerator, denominator);
var simplifiedNum = numerator / commonDivisor;
var simplifiedDen = denominator / commonDivisor;
// Ensure the denominator is positive
if (simplifiedDen < 0) {
simplifiedNum = -simplifiedNum;
simplifiedDen = -simplifiedDen;
}
return { num: simplifiedNum, den: simplifiedDen, error: null };
}
function calculateRational(operation) {
var num1 = parseFloat(document.getElementById("numerator1").value);
var den1 = parseFloat(document.getElementById("denominator1").value);
var num2 = parseFloat(document.getElementById("numerator2").value);
var den2 = parseFloat(document.getElementById("denominator2").value);
var resultDiv = document.getElementById("result");
// Input validation
if (isNaN(num1) || isNaN(den1) || isNaN(num2) || isNaN(den2)) {
resultDiv.textContent = "Error: Please enter valid numbers for all fields.";
resultDiv.className = "error";
return;
}
if (den1 === 0 || den2 === 0) {
resultDiv.textContent = "Error: Denominator cannot be zero.";
resultDiv.className = "error";
return;
}
var resultNum, resultDen;
var error = null;
switch (operation) {
case 'add':
resultNum = (num1 * den2) + (num2 * den1);
resultDen = den1 * den2;
break;
case 'subtract':
resultNum = (num1 * den2) – (num2 * den1);
resultDen = den1 * den2;
break;
case 'multiply':
resultNum = num1 * num2;
resultDen = den1 * den2;
break;
case 'divide':
if (num2 === 0) {
resultDiv.textContent = "Error: Division by zero.";
resultDiv.className = "error";
return;
}
resultNum = num1 * den2;
resultDen = den1 * num2;
break;
default:
resultDiv.textContent = "Error: Unknown operation.";
resultDiv.className = "error";
return;
}
var simplified = simplifyFraction(resultNum, resultDen);
if (simplified.error) {
resultDiv.textContent = "Error: " + simplified.error;
resultDiv.className = "error";
} else {
if (simplified.den === 1) {
resultDiv.textContent = "Result: " + simplified.num;
} else {
resultDiv.textContent = "Result: " + simplified.num + "/" + simplified.den;
}
resultDiv.className = ""; // Remove error class if successful
}
}