Rationalize Denominator Calculator
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Rationalize Denominator Calculator
Enter the fraction where the denominator needs rationalization.
Numerator:
Denominator:
Rationalize
Your rationalized fraction will appear here.
What is Rationalizing the Denominator?
Rationalizing the denominator is a mathematical process used to eliminate any radical (like square roots) or complex numbers (involving 'i') from the denominator of a fraction. While it might seem like an arbitrary rule, it's a standard practice in mathematics for simplifying expressions and making them easier to work with, especially when performing further calculations or comparisons.
Why Rationalize?
Simplification: Expressions with rational denominators are generally considered simpler and easier to evaluate.Standardization: It's a conventional way to present mathematical answers.Approximation: It simplifies finding decimal approximations. For example, 1/√2 is harder to approximate than √2/2.Calculations: It's crucial for certain algebraic manipulations and higher-level mathematical procedures.
How it Works (Different Cases):
Case 1: Denominator is a simple radical (e.g., √b)
To rationalize a denominator like √b, you multiply both the numerator and the denominator by √b. This is because (√b) * (√b) = b, which is rational.
Example: 1 / √2
Multiply by √2 / √2: (1 * √2) / (√2 * √2) = √2 / 2
Case 2: Denominator is a binomial radical (e.g., a + √b or a – √b)
To rationalize a denominator of the form a + √b or a – √b, you use the "conjugate." The conjugate of a + √b is a – √b, and vice versa. Multiplying a binomial by its conjugate results in a difference of squares (a² – b), eliminating the radical.
Example: 1 / (2 + √3)
The conjugate is (2 - √3). Multiply by (2 - √3) / (2 - √3):
(1 * (2 – √3)) / ((2 + √3) * (2 – √3)) = (2 – √3) / (2² – (√3)²) = (2 – √3) / (4 – 3) = 2 - √3
Case 3: Denominator is a complex number (e.g., a + bi)
Similar to binomial radicals, you multiply by the complex conjugate. The conjugate of a + bi is a – bi. Multiplying a complex number by its conjugate results in a real number (a² + b²).
Example: 3 / (1 + 2i)
The conjugate is (1 - 2i). Multiply by (1 - 2i) / (1 - 2i):
(3 * (1 – 2i)) / ((1 + 2i) * (1 – 2i)) = (3 – 6i) / (1² – (2i)²) = (3 – 6i) / (1 – 4i²) = (3 – 6i) / (1 – 4(-1)) = (3 – 6i) / (1 + 4) = (3 - 6i) / 5
Using the Calculator
Enter the numerator and denominator of your fraction. The calculator will attempt to rationalize the denominator and display the simplified result.
function rationalizeDenominator() {
var numStr = document.getElementById("numerator").value.trim();
var denStr = document.getElementById("denominator").value.trim();
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = "Calculating…";
if (numStr === "" || denStr === "") {
resultDiv.innerHTML = "Please enter both numerator and denominator.";
return;
}
try {
// Attempt to parse and rationalize
var rationalizedFraction = performRationalization(numStr, denStr);
resultDiv.innerHTML = "Rationalized Fraction: " + rationalizedFraction;
} catch (error) {
resultDiv.innerHTML = "Error: " + error.message + " Please check your input format.";
}
}
// — Core Rationalization Logic —
// This simplified JS handles basic cases. For a truly robust calculator,
// a symbolic math library would be needed. This version focuses on
// common patterns like sqrt(x), and a+sqrt(b), and a+bi.
function performRationalization(numerator, denominator) {
// Case 1: Simple Radical in Denominator (e.g., sqrt(2))
if (denominator.startsWith("sqrt(") && denominator.endsWith(")")) {
var radicalTerm = denominator.substring(5, denominator.length – 1);
var multiplyBy = "sqrt(" + radicalTerm + ")";
return simplifyFraction(numerator + " * " + multiplyBy, denominator + " * " + multiplyBy);
}
// Case 2: Binomial Radical in Denominator (e.g., 2 + sqrt(3))
var radicalMatch = denominator.match(/(\+|\-)\s*sqrt\((.+?)\)/);
if (radicalMatch) {
var sign = radicalMatch[1];
var radicalValue = "sqrt(" + radicalMatch[2] + ")";
var baseTerm = denominator.replace(radicalMatch[0], "").trim();
if (baseTerm === "") baseTerm = "0"; // Handle cases like sqrt(3) where base is implied 0
var conjugate;
if (sign === "+") {
conjugate = baseTerm + " – " + radicalValue;
} else {
conjugate = baseTerm + " + " + radicalValue;
}
return simplifyFraction(numerator + " * (" + conjugate + ")", "(" + denominator + ") * (" + conjugate + ")");
}
// Case 3: Complex Number in Denominator (e.g., 1 + 2i)
var complexMatch = denominator.match(/(\-?\d*\.?\d*)\s*([\+\-])\s*(\d*\.?\d*)i/);
if (complexMatch || denominator.endsWith("i")) {
var realPart = 0;
var imagPart = 1;
var conjugateParts = [];
var parts = denominator.replace(/\s/g, ").split(/([\+\-])/).filter(Boolean);
var currentSign = 1;
var tempNum = "";
for(var i = 0; i < parts.length; i++) {
var part = parts[i];
if (part === '+') {
currentSign = 1;
} else if (part === '-') {
currentSign = -1;
} else if (part.endsWith('i')) {
var coeffStr = part.substring(0, part.length – 1);
if (coeffStr === "" || coeffStr === "+") imagPart = currentSign * 1;
else if (coeffStr === "-") imagPart = currentSign * -1;
else imagPart = currentSign * parseFloat(coeffStr);
} else {
tempNum = parseFloat(part);
// Check if the next part is 'i' to determine if this is real or imaginary
if (i + 1 1) imagPart *= -1;
}
var conjugate = realPart + (imagPart > 0 ? " – " : " + ") + Math.abs(imagPart) + "i";
return simplifyFraction(numerator + " * (" + conjugate + ")", "(" + denominator + ") * (" + conjugate + ")");
}
// Case 4: Denominator is a rational number (or variable) – no rationalization needed
// Or if the input is complex and not matching the above patterns
if (!isNaN(parseFloat(denominator)) || denominator.match(/^[a-zA-Z0-9]+$/)) {
return numerator + " / " + denominator;
}
throw new Error("Unsupported denominator format or already rationalized.");
}
// Basic simplification – attempts to resolve multiplication and powers of sqrt
function simplifyFraction(numExpression, denExpression) {
// This is a placeholder for symbolic simplification.
// A real implementation would involve parsing expressions, applying rules, etc.
// For this calculator, we'll just return the expression as is after multiplying.
// We can attempt some very basic sqrt simplifications.
var simplifiedNum = evaluateBasicExpression(numExpression);
var simplifiedDen = evaluateBasicExpression(denExpression);
return simplifiedNum + " / " + simplifiedDen;
}
function evaluateBasicExpression(expression) {
expression = expression.replace(/\s/g, "); // Remove whitespace
// Resolve multiplications like sqrt(a)*sqrt(b) = sqrt(a*b)
expression = expression.replace(/sqrt\((.+?)\)\*sqrt\((.+?)\)/g, function(match, p1, p2) {
// This part requires careful parsing if p1 or p2 are themselves complex expressions
// For simplicity, assume p1 and p2 are simple terms or numbers
try {
var val1 = parseFloat(p1);
var val2 = parseFloat(p2);
if (!isNaN(val1) && !isNaN(val2)) {
return "sqrt(" + (val1 * val2) + ")";
}
} catch(e) { /* ignore */ }
return match; // Return original if simplification fails
});
// Resolve sqrt(a) * sqrt(a) = a
expression = expression.replace(/sqrt\((.+?)\)\*sqrt\((.+?)\)/g, function(match, p1, p2) {
if (p1 === p2) {
return p1;
}
return match;
});
// Resolve simple multiplication e.g., 2 * sqrt(3)
expression = expression.replace(/(\d+)\s*\*\s*(sqrt\(.*?\)|[a-zA-Z0-9]+)/g, '$1 $2');
expression = expression.replace(/(sqrt\(.*?\)|[a-zA-Z0-9]+)\s*\*\s*(\d+)/g, '$1 $2');
// Resolve powers like (a+b)*(a-b) = a^2 – b^2 (limited scope)
var binomialMatch = expression.match(/\((.+?)\)\*\((.+?)\)/);
if (binomialMatch) {
var term1 = binomialMatch[1];
var term2 = binomialMatch[2];
// Very basic check for conjugates
if (term1.includes('-') && term2.includes('+') && term1.replace('-', ") === term2.replace('+', ")) {
var parts = term1.split(/([\+\-])/).filter(Boolean);
var base = parts[0];
var radical = parts[1] ? parts[1].substring(1) : "; // remove sign and get radical term
if (radical.startsWith("sqrt(")) {
var radicalContent = radical.substring(5, radical.length -1);
return base + "^2 – (" + radicalContent + ")";
}
} else if (term1.includes('+') && term2.includes('-') && term1.replace('+', ") === term2.replace('-', ")) {
var parts = term1.split(/([\+\-])/).filter(Boolean);
var base = parts[0];
var radical = parts[1] ? parts[1].substring(1) : ";
if (radical.startsWith("sqrt(")) {
var radicalContent = radical.substring(5, radical.length -1);
return base + "^2 – (" + radicalContent + ")";
}
}
}
// Resolve i*i = -1
expression = expression.replace(/i\*i/g, '-1');
expression = expression.replace(/i\^2/g, '-1');
// Attempt to evaluate numeric parts after multiplications and powers
// This is highly complex for symbolic math.
// For now, let's simplify simple cases like (a*b) / c
// Or numerical evaluation for specific cases like (2-sqrt(3))/(4-3)
// Resolve a^2 – b for specific conjugate cases
expression = expression.replace(/(\d+)\^2\s*\-\s*(\d+)/g, function(match, base, sq) {
return (parseInt(base) * parseInt(base)) – parseInt(sq);
});
expression = expression.replace(/(\d+)\^2\s*\-\s*sqrt\((\d+)\)/g, function(match, base, sq) {
return (parseInt(base) * parseInt(base)) – "sqrt(" + sq + ")";
});
expression = expression.replace(/(\d+)\^2\s*\+\s*(\d+)/g, function(match, base, sq) {
return (parseInt(base) * parseInt(base)) + parseInt(sq);
});
// Resolve simple divisions like (a) / b where b is a number
var divisionMatch = expression.match(/([^\/]+)\/(\d+)/);
if (divisionMatch) {
var numeratorPart = divisionMatch[1];
var denominatorPart = parseInt(divisionMatch[2]);
if (!isNaN(denominatorPart) && denominatorPart !== 0) {
// Attempt to distribute division if numerator is binomial
var numParts = numeratorPart.split(/([\+\-])/).filter(Boolean);
if (numParts.length > 1) { // Binomial numerator
var resultParts = [];
for (var i = 0; i < numParts.length; i++) {
var part = numParts[i];
if (part === '+' || part === '-') {
resultParts.push(part);
} else {
try {
var termValue = parseFloat(part);
if (!isNaN(termValue)) {
resultParts.push(termValue / denominatorPart);
} else {
resultParts.push(part + " / " + denominatorPart);
}
} catch (e) {
resultParts.push(part + " / " + denominatorPart);
}
}
}
return resultParts.join('');
} else { // Monomial numerator
try {
var termValue = parseFloat(numeratorPart);
if (!isNaN(termValue)) {
return (termValue / denominatorPart).toString();
}
} catch (e) { /* ignore */ }
}
}
}
// Final cleanup for expressions like "sqrt(4)" to "2"
expression = expression.replace(/sqrt\((\d+)\)/g, function(match, num) {
var root = Math.sqrt(parseInt(num));
return Number.isInteger(root) ? root.toString() : match;
});
return expression;
}