Enter two mathematical expressions to check if they are equivalent.
Results will appear here.
Understanding Expression Equivalence
In mathematics, two expressions are considered equivalent if they produce the same output for all possible valid inputs. This means that no matter what value you substitute for the variables within the expressions, the result will always be identical. This concept is fundamental in algebra and is crucial for simplifying equations, verifying mathematical identities, and understanding the behavior of functions.
How Equivalence is Determined
Determining if two expressions are equivalent typically involves a combination of algebraic manipulation and, for computational verification, testing with various inputs:
Algebraic Simplification: The most rigorous way to prove equivalence is to simplify both expressions independently using algebraic rules (like the distributive property, combining like terms, factorization, etc.) until they reach their simplest forms. If the simplest forms are identical, the original expressions are equivalent.
Substitution (for verification): While not a formal proof on its own, substituting specific values for the variables and checking if the results match can provide strong evidence for equivalence. If the results differ for even one set of input values, the expressions are definitely not equivalent. This is the approach our calculator uses for practical verification.
Key Algebraic Properties Used:
Commutative Property: The order of operands does not change the result (e.g., a + b = b + a, a * b = b * a).
Associative Property: The way operands are grouped does not change the result (e.g., (a + b) + c = a + (b + c), (a * b) * c = a * (b * c)).
Distributive Property: Multiplication distributes over addition/subtraction (e.g., a * (b + c) = a * b + a * c).
Combining Like Terms: Terms with the same variable(s) raised to the same power can be added or subtracted (e.g., 3x + 2x = 5x).
Use Cases for Equivalence Checking:
Solving Equations: Simplifying complex equations by replacing parts with equivalent, simpler forms.
Simplifying Formulas: Making mathematical or scientific formulas easier to understand and compute.
Programming: Compilers often optimize code by replacing expressions with equivalent, more efficient ones.
Mathematical Proofs: Verifying identities and theorems.
Educational Tools: Helping students grasp algebraic manipulation concepts.
This calculator provides a quick way to check for probable equivalence by evaluating the expressions at a few sample points. For absolute mathematical certainty, manual algebraic simplification is recommended.
function evaluateExpression(expression, valueMap) {
// Basic sanitization and variable extraction
var variables = {};
var expr = expression.replace(/[^a-zA-Z0-9+\-*/().^ ]/g, "); // Allow basic math ops, vars, numbers, spaces
// Identify variables in the expression
var foundVars = expr.match(/[a-zA-Z]+/g);
if (foundVars) {
foundVars.forEach(function(v) {
if (!(v in variables)) {
variables[v] = 0; // Placeholder for substitution
}
});
}
// Substitute values from valueMap if provided
for (var varName in valueMap) {
if (variables.hasOwnProperty(varName)) {
variables[varName] = valueMap[varName];
}
}
// Prepare the expression for evaluation
var evalString = expr;
for (var varName in variables) {
// Use a regex to replace whole words only, avoiding partial matches
var regex = new RegExp('\\b' + varName + '\\b', 'g');
evalString = evalString.replace(regex, variables[varName]);
}
// Replace '^' with '**' for exponentiation
evalString = evalString.replace(/\^/g, '**');
try {
// Use a restricted Function constructor for safer evaluation
// We are providing a limited scope for evaluation
var evaluator = new Function('return ' + evalString);
return evaluator();
} catch (e) {
console.error("Evaluation error for expression:", expression, "with error:", e);
return NaN; // Return Not a Number on error
}
}
function checkEquivalence() {
var expr1 = document.getElementById("expression1").value.trim();
var expr2 = document.getElementById("expression2").value.trim();
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = 'Results will appear here.'; // Clear previous results
if (expr1 === "" || expr2 === "") {
resultDiv.innerHTML = "Please enter both expressions.";
return;
}
// Define sample values for variables
// We'll try to identify common variables and test them
var sampleValues = [
{ x: 2, y: 3, z: 4 },
{ x: -1, y: 5, z: 0.5 },
{ x: 0, y: -2, z: 10 },
{ x: 10, y: 1, z: -3 }
];
var allResultsMatch = true;
var testedVariables = {}; // To keep track of variables we've seen
// Attempt to parse variables from both expressions
var allVars = new Set();
var vars1 = expr1.match(/[a-zA-Z]+/g);
var vars2 = expr2.match(/[a-zA-Z]+/g);
if (vars1) vars1.forEach(v => allVars.add(v));
if (vars2) vars2.forEach(v => allVars.add(v));
var variableList = Array.from(allVars);
// If no variables, just evaluate directly
if (variableList.length === 0) {
var val1 = evaluateExpression(expr1, {});
var val2 = evaluateExpression(expr2, {});
if (isNaN(val1) || isNaN(val2)) {
resultDiv.innerHTML = "Error evaluating expressions. Please check syntax.";
return;
}
if (Math.abs(val1 – val2) < 1e-9) { // Use tolerance for floating point comparison
resultDiv.innerHTML = "The expressions are likely equivalent.";
} else {
resultDiv.innerHTML = "The expressions are NOT equivalent.";
}
return;
}
// Test with sample values for identified variables
for (var i = 0; i < sampleValues.length; i++) {
var currentValues = {};
var missingVars = false;
for (var j = 0; j 1e-9) { // Use tolerance for floating point comparison
allResultsMatch = false;
break; // If any test fails, they are not equivalent
}
}
if (allResultsMatch) {
resultDiv.innerHTML = "The expressions are likely equivalent.";
} else {
resultDiv.innerHTML = "The expressions are NOT equivalent.";
}
}