Use this calculator to perform basic arithmetic operations and determine the result with the correct number of significant figures or decimal places.
+
–
*
/
Results:
Raw Calculation Result:
Result with Correct Significant Figures/Decimal Places:
Explanation:
Understanding Significant Figures
Significant figures (often called "sig figs") are crucial in scientific and engineering fields because they indicate the precision of a measurement. When you record a measurement, the number of significant figures tells you how many digits are known with certainty, plus one estimated digit. Using significant figures correctly ensures that calculations do not imply a greater precision than the original measurements allow.
Rules for Counting Significant Figures:
Non-zero digits: All non-zero digits are significant. (e.g., 123.45 has 5 sig figs)
Zeros between non-zero digits: Zeros located between non-zero digits are significant. (e.g., 1002 has 4 sig figs)
Leading zeros: Zeros that precede all non-zero digits are NOT significant. They merely indicate the position of the decimal point. (e.g., 0.00123 has 3 sig figs)
Trailing zeros (with a decimal point): Trailing zeros are significant if the number contains a decimal point. (e.g., 12.00 has 4 sig figs, 120. has 3 sig figs)
Trailing zeros (without a decimal point): Trailing zeros in a number without a decimal point are generally considered NOT significant unless explicitly stated (e.g., by using scientific notation or a decimal point at the end). For example, 1200 typically has 2 sig figs, while 1200. has 4 sig figs.
Exact numbers: Numbers that are counted or defined (e.g., 12 eggs in a dozen, 100 cm in 1 meter) have an infinite number of significant figures and do not limit the precision of a calculation.
Rules for Arithmetic Operations:
1. Multiplication and Division:
When multiplying or dividing measurements, the result should be rounded to the same number of significant figures as the measurement with the fewest significant figures.
Example:
12.34 (4 sig figs) × 5.6 (2 sig figs) = 69.104
The measurement with the fewest sig figs is 5.6 (2 sig figs).
Therefore, the result should be rounded to 2 significant figures: 69.
2. Addition and Subtraction:
When adding or subtracting measurements, the result should be rounded to the same number of decimal places as the measurement with the fewest decimal places.
The measurement with the fewest decimal places is 5.6 (1 decimal place).
Therefore, the result should be rounded to 1 decimal place: 17.9.
How to Use the Calculator:
Enter your first and second numerical values into the respective input fields. Select the desired arithmetic operation (+, -, *, /) from the dropdown menu. Click "Calculate" to see the raw result, the result rounded to the correct number of significant figures or decimal places, and an explanation of the rounding rule applied.
.calculator-container {
background-color: #f9f9f9;
border: 1px solid #ddd;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
font-family: Arial, sans-serif;
}
.calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
}
.calc-input-group {
margin-bottom: 15px;
}
.calc-input-group label {
display: block;
margin-bottom: 5px;
font-weight: bold;
color: #555;
}
.calc-input-group input[type="text"],
.calc-input-group select {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 16px;
}
.calculator-container button {
background-color: #007bff;
color: white;
padding: 12px 20px;
border: none;
border-radius: 4px;
cursor: pointer;
font-size: 18px;
width: 100%;
margin-top: 10px;
}
.calculator-container button:hover {
background-color: #0056b3;
}
.calc-result {
background-color: #e9ecef;
border: 1px solid #dee2e6;
padding: 15px;
border-radius: 4px;
margin-top: 20px;
}
.calc-result h3 {
color: #333;
margin-top: 0;
border-bottom: 1px solid #ccc;
padding-bottom: 10px;
margin-bottom: 10px;
}
.calc-result p {
margin-bottom: 8px;
line-height: 1.5;
}
.calc-result strong {
color: #333;
}
.calculator-article {
max-width: 600px;
margin: 40px auto;
font-family: Arial, sans-serif;
line-height: 1.6;
color: #333;
}
.calculator-article h2, .calculator-article h3 {
color: #007bff;
margin-top: 25px;
margin-bottom: 15px;
}
.calculator-article p {
margin-bottom: 10px;
}
.calculator-article ol, .calculator-article ul {
margin-left: 20px;
margin-bottom: 10px;
}
.calculator-article li {
margin-bottom: 5px;
}
function countSigFigs(numStr) {
numStr = String(numStr).trim();
if (!numStr || !/^-?\d*\.?\d+(e[+-]?\d+)?$/i.test(numStr)) {
return 0; // Not a valid number string
}
// Handle the special case of zero
if (parseFloat(numStr) === 0) {
var decimalIndex = numStr.indexOf('.');
if (decimalIndex === -1) {
return 1; // "0" has 1 sig fig
} else {
// "0.0" has 2 sig figs, "0.00" has 3
return numStr.length – decimalIndex;
}
}
var parts = numStr.toLowerCase().split('e');
var mantissa = parts[0];
var sigFigs = 0;
var hasDecimal = mantissa.includes('.');
var nonZeroFound = false;
for (var i = 0; i < mantissa.length; i++) {
var char = mantissa[i];
if (char === '-' || char === '.') {
continue;
}
if (char !== '0') {
nonZeroFound = true;
sigFigs++;
} else { // char is '0'
if (nonZeroFound || hasDecimal) { // Zeros are significant if non-zero digit already found OR if there's a decimal point
sigFigs++;
}
}
}
return sigFigs;
}
function countDecimalPlaces(numStr) {
numStr = String(numStr).trim();
if (!numStr || !/^-?\d*\.?\d+(e[+-]?\d+)?$/i.test(numStr)) {
return 0; // Not a valid number string
}
var parts = numStr.toLowerCase().split('e');
var mantissa = parts[0];
var decimalIndex = mantissa.indexOf('.');
if (decimalIndex === -1) {
return 0;
}
return mantissa.length – 1 – decimalIndex;
}
function calculateSigFigs() {
var firstValueStr = document.getElementById('firstValue').value;
var secondValueStr = document.getElementById('secondValue').value;
var operation = document.getElementById('operation').value;
var rawResultElem = document.getElementById('rawResult');
var finalSigFigsResultElem = document.getElementById('finalSigFigsResult');
var sigFigsExplanationElem = document.getElementById('sigFigsExplanation');
rawResultElem.textContent = '';
finalSigFigsResultElem.textContent = '';
sigFigsExplanationElem.textContent = '';
var num1 = parseFloat(firstValueStr);
var num2 = parseFloat(secondValueStr);
if (isNaN(num1) || isNaN(num2)) {
alert('Please enter valid numbers for both values.');
return;
}
var rawResult;
var finalResult = '';
var explanation = '';
if (operation === 'add') {
rawResult = num1 + num2;
var dp1 = countDecimalPlaces(firstValueStr);
var dp2 = countDecimalPlaces(secondValueStr);
var finalDecimalPlaces = Math.min(dp1, dp2);
finalResult = rawResult.toFixed(finalDecimalPlaces);
explanation = 'For addition, the result is rounded to the same number of decimal places as the measurement with the fewest decimal places. (Fewest decimal places: ' + finalDecimalPlaces + ')';
} else if (operation === 'subtract') {
rawResult = num1 – num2;
var dp1 = countDecimalPlaces(firstValueStr);
var dp2 = countDecimalPlaces(secondValueStr);
var finalDecimalPlaces = Math.min(dp1, dp2);
finalResult = rawResult.toFixed(finalDecimalPlaces);
explanation = 'For subtraction, the result is rounded to the same number of decimal places as the measurement with the fewest decimal places. (Fewest decimal places: ' + finalDecimalPlaces + ')';
} else if (operation === 'multiply') {
rawResult = num1 * num2;
var sf1 = countSigFigs(firstValueStr);
var sf2 = countSigFigs(secondValueStr);
var finalSigFigs = Math.min(sf1, sf2);
if (finalSigFigs === 0) { // This can happen if one of the inputs is "0" and the other is not, or if inputs are invalid.
finalResult = rawResult.toString();
explanation = 'Cannot determine significant figures for one or both inputs. Displaying raw result.';
} else {
finalResult = rawResult.toPrecision(finalSigFigs);
explanation = 'For multiplication, the result is rounded to the same number of significant figures as the measurement with the fewest significant figures. (Fewest significant figures: ' + finalSigFigs + ')';
}
} else if (operation === 'divide') {
if (num2 === 0) {
alert('Cannot divide by zero.');
return;
}
rawResult = num1 / num2;
var sf1 = countSigFigs(firstValueStr);
var sf2 = countSigFigs(secondValueStr);
var finalSigFigs = Math.min(sf1, sf2);
if (finalSigFigs === 0) { // This can happen if one of the inputs is "0" and the other is not, or if inputs are invalid.
finalResult = rawResult.toString();
explanation = 'Cannot determine significant figures for one or both inputs. Displaying raw result.';
} else {
finalResult = rawResult.toPrecision(finalSigFigs);
explanation = 'For division, the result is rounded to the same number of significant figures as the measurement with the fewest significant figures. (Fewest significant figures: ' + finalSigFigs + ')';
}
}
rawResultElem.textContent = rawResult.toString();
finalSigFigsResultElem.textContent = finalResult;
sigFigsExplanationElem.textContent = explanation;
}