Understand and apply the rules of significant figures with ease.
Significant Figures Calculator
Enter any number to see its 2 significant figure representation.
Results
—
Original Value:—
Number of Significant Figures:—
Rounded to 2 Sig Figs:—
To round to 2 significant figures, we identify the first two non-zero digits from the left. If the third digit is 5 or greater, we round up the second digit. If it's less than 5, we keep the second digit as is. All digits to the right of the rounded digit become zeros if they are before the decimal point, or are dropped if they are after the decimal point.
Comparison of Original vs. Rounded Values
Value Type
Value
Significant Figures
Original Input
—
—
Rounded to 2 Sig Figs
—
2
Summary of Significant Figures
Understanding and Using 2 Significant Figures
What are Significant Figures?
Significant figures, often abbreviated as "sig figs," are the digits in a number that carry meaning contributing to its precision. In scientific and engineering contexts, they are crucial for indicating the reliability of a measurement or calculation. They tell us how accurately a number is known. For instance, a measurement of 10.2 meters has three significant figures, implying a precision to the nearest tenth of a meter. A number like 0.0054 meters has two significant figures (5 and 4), indicating less precision.
The concept of significant figures is fundamental in scientific notation and in performing calculations where the precision of the result is limited by the least precise input. Understanding these rules ensures that results reflect the actual uncertainty in the data. Mastering the rules for identifying and rounding to a specific number of significant figures, like two, is a foundational skill in many quantitative fields.
2 Significant Figures: Formula and Mathematical Explanation
The process of determining and rounding to 2 significant figures involves a clear set of rules. First, we must correctly identify the significant figures in the original number. The general rules for identifying significant figures are:
Non-zero digits are always significant.
Zeros between non-zero digits are always significant (e.g., 102 has 3 sig figs).
Leading zeros (zeros to the left of the first non-zero digit) are never significant (e.g., 0.0045 has 2 sig figs: 4 and 5).
Trailing zeros (zeros at the end of a number) are significant only if the number contains a decimal point (e.g., 120. has 3 sig figs, but 120 has only 2).
Once the significant figures are identified, rounding to 2 significant figures involves these steps:
Locate the second significant digit from the left.
Look at the digit immediately to its right (the third significant digit).
If the third digit is 5 or greater, round up the second significant digit.
If the third digit is less than 5, keep the second significant digit as it is.
Replace all digits between the second significant digit and the decimal point with zeros.
Drop all digits to the right of the decimal point if they are no longer significant.
For example, rounding 123.45 to 2 significant figures: The first two significant digits are 1 and 2. The third digit is 3. Since 3 is less than 5, we keep 2 as it is. We replace the 3 with a 0 to maintain place value, resulting in 120. Rounding 0.00789 to 2 significant figures: The first two significant digits are 7 and 8. The third digit is 9. Since 9 is greater than or equal to 5, we round up 8 to 9. The result is 0.0079.
Practical Examples (Real-World Use Cases)
The application of 2 significant figures is widespread in various fields:
Physics Measurements: When measuring lengths, masses, or times, results are often rounded to a specific number of significant figures to reflect the precision of the measuring instrument. For instance, if a pendulum's period is measured as 1.98 seconds, rounding to 2 significant figures gives 2.0 seconds. This indicates the measurement is precise to the nearest tenth of a second.
Chemistry Calculations: In chemical reactions, concentrations, and molar masses, significant figures are vital. If a reaction involves 2.5 moles of a substance and the concentration is 0.12 M, calculations involving these values must respect their precision. A calculation might yield a result like 0.30 moles, correctly showing 2 significant figures.
Engineering Specifications: Tolerances in manufacturing and engineering designs often use significant figures. A component might be specified with a length of 5.1 cm, indicating a tolerance range around that value. If a calculation for material stress results in 15.67 MPa, it might be reported as 16 MPa to adhere to 2 significant figures, simplifying communication while maintaining reasonable precision.
Data Reporting: When presenting data in reports or scientific papers, rounding to an appropriate number of significant figures, such as 2, helps in summarizing findings clearly without overstating precision. For example, reporting a growth rate of 1.05% might be simplified to 1.1% for general audiences.
These examples highlight how 2 significant figures provide a balance between precision and simplicity in communicating quantitative information. Understanding how to achieve this rounding is essential for accurate data interpretation and reporting in fields like physics and chemistry.
How to Use This 2 Significant Figures Calculator
Using this calculator is straightforward and designed for immediate feedback:
Enter Your Number: In the "Enter a Number" field, type the numerical value you wish to analyze. This can be an integer, a decimal, or a number in scientific notation (though the input field accepts standard text for flexibility).
Click Calculate: Press the "Calculate" button. The calculator will process your input.
View Results: The calculator will display:
The original value entered.
The total number of significant figures in your original number.
The number rounded to exactly 2 significant figures. This is your primary result.
Examine Table and Chart: A table summarizes the original and rounded values along with their significant figure counts. A chart visually compares the magnitude of the original number and its 2-significant-figure representation.
Copy Results: Use the "Copy Results" button to easily transfer the key findings to another document or application.
Reset: If you need to start over or clear the fields, click the "Reset" button. It will revert the calculator to its default state.
This tool is perfect for students learning about significant figures, researchers needing to quickly round data, or anyone requiring a precise way to represent numerical precision. It's a practical aid for tasks related to data analysis.
Key Factors That Affect 2 Significant Figures Results
Several factors influence how a number is rounded to 2 significant figures and the interpretation of the result:
The Original Number's Magnitude: The size of the original number significantly impacts how rounding to 2 significant figures affects its value. Rounding 12345 to 2 sig figs gives 12000, a substantial change. Rounding 1.2345 to 2 sig figs gives 1.2, a much smaller relative change. The calculator handles these scale differences automatically.
The Third Digit: As explained in the formula, the value of the third significant digit is the sole determinant of whether the second significant digit is rounded up or stays the same. A value of 5 or greater triggers rounding up, while less than 5 does not.
Trailing Zeros: The placement and significance of trailing zeros are critical. For example, 500 has 1 significant figure (the 5), so rounding to 2 sig figs is ambiguous without context (it might become 5.0 x 10^2). However, 500. has 3 significant figures, and rounding to 2 sig figs yields 5.0 x 10^2 or 500. (if the context implies the last zero is significant). Our calculator interprets standard numerical input based on common conventions.
Leading Zeros: Leading zeros are never significant. They only serve to place the decimal point. For instance, 0.000456 rounded to 2 significant figures becomes 0.00046. The leading zeros (0.000) are preserved to show the magnitude but do not count towards the significant figures.
Context of Measurement: In real-world applications, the precision of the original measurement dictates the appropriate number of significant figures. If a measurement is inherently imprecise, forcing it to 2 significant figures might still overstate its accuracy. This calculator performs the mathematical rounding; understanding the source of the number is key to interpreting the result's true meaning.
Understanding these factors ensures that the results from the 2 significant figures calculator are used appropriately and that the precision conveyed is accurate.
Frequently Asked Questions (FAQ)
Q: What is the rule for rounding when the third digit is exactly 5?
A: The common convention is to round up if the third digit is 5. Some specific scientific or statistical contexts might use "round half to even" rules, but for general purposes, rounding up is standard.
Q: How do I handle numbers like 1000 when rounding to 2 significant figures?
A: Numbers like 1000 are ambiguous. If it means "approximately one thousand," it has 1 sig fig (1). If it means "exactly one thousand," it could have 4 sig figs (1000.). To clearly indicate 2 sig figs, it's best to use scientific notation: 1.0 x 10^3. Our calculator will interpret standard input; for ambiguous cases, consider scientific notation.
Q: Does the calculator handle negative numbers?
A: Yes, the calculator handles negative numbers. The sign does not affect the process of identifying or rounding significant figures. For example, -123.45 rounded to 2 significant figures becomes -120.
Q: Why are significant figures important in science?
A: Significant figures ensure that calculations reflect the precision of the original measurements. They prevent overstating accuracy and provide a standardized way to communicate the reliability of quantitative data.
Related Tools and Internal Resources
Scientific Notation Converter: Easily convert numbers to and from scientific notation, a format often used with significant figures.
var chartInstance = null; // Global variable to hold chart instance
function isValidNumber(value) {
return !isNaN(parseFloat(value)) && isFinite(value);
}
function countSignificantFigures(numStr) {
numStr = numStr.replace(/^\s+|\s+$/g, "); // Trim whitespace
if (numStr === ") return 0;
// Handle scientific notation
var parts = numStr.split(/[eE]/);
var mantissa = parts[0];
// Remove sign if present
if (mantissa.startsWith('-') || mantissa.startsWith('+')) {
mantissa = mantissa.substring(1);
}
// Remove leading zeros after decimal point
mantissa = mantissa.replace(/^0+/, ");
// Remove trailing zeros if no decimal point
if (!mantissa.includes('.') && mantissa.length > 1) {
mantissa = mantissa.replace(/0+$/, ");
}
// Special case: if mantissa becomes empty after removing leading zeros (e.g., "0.000")
if (mantissa === " && numStr.includes('.')) {
return 1; // Treat "0.000" as having one significant figure (the zero)
}
// Special case: if mantissa becomes empty after removing trailing zeros (e.g., "500")
if (mantissa === " && !numStr.includes('.')) {
// If original number was like "500", it has 1 sig fig. If "500.", it has 3.
// This function is called *before* rounding, so we need to be careful.
// For simplicity here, let's assume standard interpretation: 500 has 1 sig fig.
// A more robust parser would be needed for perfect ambiguity handling.
if (numStr.endsWith('.') && numStr.length > 1) return numStr.length -1; // e.g. "500." -> 3 sig figs
if (numStr.length > 0 && !isNaN(parseInt(numStr[0]))) return 1; // e.g. "500" -> 1 sig fig
return 0;
}
var sigFigs = 0;
var hasDecimal = mantissa.includes('.');
var digits = mantissa.replace('.', ").split(");
if (digits.length === 0) return 0; // Should not happen with valid numbers
// Count non-zero digits
for (var i = 0; i < digits.length; i++) {
if (digits[i] !== '0') {
sigFigs++;
}
}
// Handle zeros between non-zeros
var tempDigits = mantissa.split('');
var firstNonZeroFound = false;
for (var i = 0; i < tempDigits.length; i++) {
if (tempDigits[i] === '0') {
if (firstNonZeroFound && i < tempDigits.length – 1 && tempDigits[i+1] !== '.' && tempDigits[i+1] !== '0') {
// Zero between non-zeros (and not part of trailing zeros after decimal)
// This logic is tricky. Let's simplify based on common rules.
}
} else {
firstNonZeroFound = true;
}
}
// Simplified approach based on common rules:
// 1. Non-zero digits are significant.
// 2. Zeros between non-zeros are significant.
// 3. Leading zeros are NOT significant.
// 4. Trailing zeros ARE significant IF there is a decimal point.
var significantDigits = [];
var inSignificantPart = false;
var hasDecimalPoint = numStr.includes('.');
for (var i = 0; i = '1' && char <= '9') {
significantDigits.push(char);
inSignificantPart = true;
} else if (char === '0') {
if (inSignificantPart) {
significantDigits.push(char);
}
} else if (char === '.') {
// Decimal point itself doesn't count, but affects trailing zeros
} else if (char === '-' || char === '+') {
// Ignore sign
} else if (char === 'e' || char === 'E') {
// Stop processing mantissa for sig figs
break;
}
}
// Refined logic for trailing zeros
var finalSigFigs = 0;
var tempNumStr = numStr.split(/[eE]/)[0]; // Work with mantissa
if (tempNumStr.startsWith('-') || tempNumStr.startsWith('+')) {
tempNumStr = tempNumStr.substring(1);
}
var digitsArray = tempNumStr.split('');
var firstDigitIndex = -1;
var lastDigitIndex = -1;
for(var i = 0; i = '1' && digitsArray[i] decimalIndex; i–) {
if (tempNumStr[i] === '0') {
// This trailing zero is significant because there's a decimal
// The previous logic already counted it if it was between non-zeros or after the last non-zero.
// This part is mainly to ensure we don't *lose* sig figs if the last digit is a zero after the decimal.
} else {
break; // Stop if we hit a non-zero digit
}
}
} else {
// No decimal point. Trailing zeros are NOT significant unless specified.
// Example: 500 -> 1 sig fig. 5000 -> 1 sig fig.
// If the number was like "500." it would have 3 sig figs.
// This function is called on the raw input string.
// Let's assume standard interpretation: 500 has 1 sig fig.
// If the input was "500.0", it would have 4 sig figs.
// The logic needs to be robust.
// A simpler approach:
var num = parseFloat(numStr);
if (num === 0) return 1;
var e = numStr.toLowerCase().indexOf('e');
var mantissaStr = (e === -1) ? numStr : numStr.substring(0, e);
// Remove sign
if (mantissaStr.startsWith('-') || mantissaStr.startsWith('+')) {
mantissaStr = mantissaStr.substring(1);
}
// Remove leading zeros
mantissaStr = mantissaStr.replace(/^0+/, ");
// Handle case like "0.000" -> mantissaStr becomes ""
if (mantissaStr === "" && numStr.includes('.')) return 1; // Treat 0.000 as 1 sig fig
// Remove decimal point for counting
var digitsOnly = mantissaStr.replace('.', ");
if (digitsOnly.length === 0) return 1; // Case like "0"
// Count non-zero digits
var count = 0;
for (var i = 0; i decimalPos; i–) {
if (mantissaStr[i] === '0') {
count++; // Trailing zero after decimal is significant
} else {
break;
}
}
} else {
// No decimal point. Trailing zeros are NOT significant.
// Example: 500 -> 1 sig fig. 5000 -> 1 sig fig.
// If the original input was "500.", the logic above would handle it.
// If the input is just "500", we assume 1 sig fig.
// This requires careful parsing. Let's use a simpler rule for this function:
// Count all digits from the first non-zero digit to the last digit,
// UNLESS there's no decimal point and the last digits are zeros.
// This is complex. Let's use a common library approach mentally:
// Use regex for robust sig fig counting.
var regex = /[1-9](?:\.?[0-9]+)?(?:e[+-]?[0-9]+)?|[1-9](?:0+)?(?=\.)|[1-9](?:0+)?(?!\.)|[0](?=\.)/;
var match = numStr.match(/^-?(?:0|[1-9]\d*)(?:\.\d+)?(?:[eE][+-]?\d+)?$|^-?(?:0|[1-9]\d*)\.(?:0+)?(?:[eE][+-]?\d+)?$/);
if (!match) return 0; // Invalid number format
var numVal = parseFloat(numStr);
if (numVal === 0) return 1;
var numString = String(numVal); // Use string representation of float
// Handle scientific notation
if (numString.includes('e') || numString.includes('E')) {
var parts = numString.toLowerCase().split('e');
var mantissa = parts[0];
var exponent = parseInt(parts[1]);
// Remove sign from mantissa
if (mantissa.startsWith('-') || mantissa.startsWith('+')) {
mantissa = mantissa.substring(1);
}
// Remove leading zeros from mantissa
mantissa = mantissa.replace(/^0+/, ");
// If mantissa is empty after removing leading zeros (e.g., "0.000e…")
if (mantissa === "" && numString.includes('.')) return 1; // Treat as 1 sig fig
// Remove decimal point for counting
var digits = mantissa.replace('.', ");
if (digits.length === 0) return 1; // Case like "0"
var sigCount = 0;
for (var i = 0; i decimalIndex; i–) {
if (mantissa[i] === '0') {
sigCount++;
} else {
break;
}
}
}
return sigCount;
} else {
// Non-scientific notation
var numStrProcessed = String(numVal); // Use string representation of float
// Remove sign
if (numStrProcessed.startsWith('-') || numStrProcessed.startsWith('+')) {
numStrProcessed = numStrProcessed.substring(1);
}
// Remove leading zeros
numStrProcessed = numStrProcessed.replace(/^0+/, ");
// Handle case like "0.000"
if (numStrProcessed === "" && numString.includes('.')) return 1;
// Remove decimal point for counting
var digits = numStrProcessed.replace('.', ");
if (digits.length === 0) return 1; // Case like "0"
var sigCount = 0;
for (var i = 0; i decimalIndex; i–) {
if (numStrProcessed[i] === '0') {
sigCount++;
} else {
break;
}
}
} else {
// No decimal point. Trailing zeros are NOT significant.
// Example: 500 -> 1 sig fig.
// If the original input was "500.", parseFloat would make it 500.
// The string representation might lose the trailing dot.
// This is why parsing the original string is better.
// Let's re-parse the original input string for trailing zeros without decimal.
var originalInputNoSign = numStr.split(/[eE]/)[0];
if (originalInputNoSign.startsWith('-') || originalInputNoSign.startsWith('+')) {
originalInputNoSign = originalInputNoSign.substring(1);
}
if (!originalInputNoSign.includes('.')) {
var trailingZeros = 0;
for (var i = originalInputNoSign.length – 1; i >= 0; i–) {
if (originalInputNoSign[i] === '0') {
trailingZeros++;
} else {
break;
}
}
// If the number is purely zeros (e.g., "000"), sigCount is 0.
// If it's like "500", sigCount is 1 (for '5'). Trailing zeros are not counted.
// If it's like "5000", sigCount is 1 (for '5').
// This needs careful handling.
// Let's use a simplified rule: count digits from first non-zero to last digit.
// If no decimal, trailing zeros are NOT counted.
var firstNonZero = -1;
var lastDigit = -1;
for(var i=0; i= '1' && originalInputNoSign[i] = 0; i–) {
if (originalInputNoSign[i] === '0') {
potentialTrailingZeros++;
} else {
break;
}
}
// If the number is like "500", lastDigit points to '5'.
// If the number is like "5000", lastDigit points to '5'.
// The countWithoutTrailingZeros is correct for non-decimal numbers.
return countWithoutTrailingZeros;
}
}
return sigCount;
}
}
}
return 0; // Fallback
}
function roundToTwoSignificantFigures(numStr) {
var num = parseFloat(numStr);
if (isNaN(num) || !isFinite(num)) {
return { rounded: NaN, sigFigs: 0, original: numStr };
}
var originalNumStr = numStr; // Keep original string for display
var sigFigs = countSignificantFigures(numStr);
if (sigFigs 12. 1.2 -> 1.2. 120 -> 120. 0.12 -> 0.12.
// If sigFigs is 1, and the number is like 100, it should remain 100.
// If sigFigs is 2, and the number is like 120, it should remain 120.
// If sigFigs is 1, and the number is like 100.0, it should be 100. (3 sig figs)
// This requires careful handling of the original string's format.
// Let's re-evaluate based on the original string's structure for precision.
var numVal = parseFloat(numStr);
if (isNaN(numVal) || !isFinite(numVal)) return { rounded: NaN, sigFigs: 0, original: numStr };
var numString = String(numVal); // Use float's string representation
// Handle scientific notation conversion for consistency
if (numString.includes('e') || numString.includes('E')) {
numString = numVal.toFixed(10).replace(/0+$/, ").replace(/\.$/, "); // Convert to fixed point, remove trailing zeros
if (numString.endsWith('.')) numString = numString.slice(0, -1); // Remove trailing dot if any
}
// Re-count sig figs on the potentially modified string representation
var currentSigFigs = countSignificantFigures(numString);
if (currentSigFigs <= 2) {
// If still <= 2, return the number as is, formatted.
// Ensure trailing zeros after decimal are kept if they were significant.
var finalNumStr = numStr; // Use original input for best format preservation
var originalSigFigs = countSignificantFigures(originalNumStr);
if (originalSigFigs originalDecimalIndex; i–) {
if (originalNumStr[i] === '0') {
trailingZeros++;
} else {
break;
}
}
if (trailingZeros > 0) {
reconstructedStr += '0'.repeat(trailingZeros);
}
}
} else {
// No decimal in original. Check for trailing zeros if they were significant.
// This is complex. For simplicity, rely on parseFloat's string conversion.
}
return { rounded: reconstructedStr, sigFigs: originalSigFigs, original: originalNumStr };
}
}
}
// Proceed with rounding if sigFigs > 2
var numStrForRounding = String(num); // Use string representation of float for rounding logic
// Handle scientific notation for rounding
var exponent = 0;
if (numStrForRounding.includes('e') || numStrForRounding.includes('E')) {
var parts = numStrForRounding.toLowerCase().split('e');
numStrForRounding = parts[0];
exponent = parseInt(parts[1]);
if (numStrForRounding.startsWith('-') || numStrForRounding.startsWith('+')) {
numStrForRounding = numStrForRounding.substring(1);
}
} else {
// Remove sign for processing
if (numStrForRounding.startsWith('-') || numStrForRounding.startsWith('+')) {
numStrForRounding = numStrForRounding.substring(1);
}
}
var digits = numStrForRounding.split(");
var decimalIndex = digits.indexOf('.');
if (decimalIndex !== -1) {
digits.splice(decimalIndex, 1); // Remove decimal for easier manipulation
}
// Find the position of the second significant digit
var sigCount = 0;
var secondSigFigPos = -1;
var thirdSigFigPos = -1;
for (var i = 0; i < digits.length; i++) {
if (digits[i] !== '0') {
sigCount++;
if (sigCount === 2) {
secondSigFigPos = i;
}
if (sigCount === 3) {
thirdSigFigPos = i;
break; // Found the third significant digit
}
}
}
// If there are fewer than 2 significant figures, return original (handled above)
if (secondSigFigPos === -1) {
// This case should ideally be caught by sigFigs 2 but we couldn't find 2nd sig fig? Error.
// Or it means the number is like 1000 and sigFigs was calculated as 1.
// Let's ensure the initial sigFigs count is correct.
// If sigFigs > 2, we MUST find a second sig fig.
return { rounded: originalNumStr, sigFigs: sigFigs, original: originalNumStr };
}
var roundUp = false;
if (thirdSigFigPos !== -1) {
var thirdDigit = parseInt(digits[thirdSigFigPos]);
if (thirdDigit >= 5) {
roundUp = true;
}
}
// Perform rounding
if (roundUp) {
var j = secondSigFigPos;
while (j >= 0) {
if (digits[j] === '9') {
digits[j] = '0';
j–;
} else {
digits[j] = String(parseInt(digits[j]) + 1);
break;
}
}
// If we carried over past the beginning (e.g., rounding 99 to 100)
if (j 1.2. Exponent is 0.
// Example: rounding 99.5 to 2 sig figs -> 100. Exponent becomes 1. resultStr is "10"
// Need to reconstruct carefully.
var finalMantissa = digits.join(");
var finalResultStr = finalMantissa + 'e' + exponent;
finalRoundedValue = parseFloat(finalResultStr); // Parse it back to number
} else {
// Handle standard decimal notation
// Need to know the original position of the decimal relative to the significant digits.
var originalDecimalPos = originalNumStr.indexOf('.');
var originalSign = originalNumStr.startsWith('-') || originalNumStr.startsWith('+') ? originalNumStr[0] : ";
var originalMantissa = originalNumStr.replace(/^-|\+|e.*$/g, "); // Get mantissa without sign/exponent
var numDigitsBeforeDecimal = 0;
if (originalDecimalPos === -1) { // No decimal in original
numDigitsBeforeDecimal = originalMantissa.length;
} else {
numDigitsBeforeDecimal = originalDecimalPos;
}
// The number of digits in the result string (after rounding) is secondSigFigPos + 1
// The number of digits that should be before the decimal point in the final result
// depends on the original number's magnitude.
// Example: 12345 -> 12000. Original had 5 digits before decimal. Result should have 5 digits total.
// Example: 0.012345 -> 0.012. Original had 0 digits before decimal. Result should have 2 digits after decimal.
var numDigitsInResult = digits.length;
var numDigitsInOriginalMantissa = originalMantissa.replace('.', ").length;
var numZerosToPad = 0;
if (originalDecimalPos === -1) { // Original was integer
// If the rounded number has fewer digits than original integer part, pad with zeros.
numZerosToPad = numDigitsBeforeDecimal – numDigitsInResult;
resultStr = digits.join(");
resultStr += '0'.repeat(numZerosToPad);
} else { // Original had a decimal
// The number of digits after the decimal point in the result should correspond
// to the original number of digits after the decimal, minus the digits that were rounded off.
// Or, more simply, place the decimal such that the number of digits before it matches the original structure.
var originalDigitsAfterDecimal = originalMantissa.length – originalDecimalPos;
var digitsUsedFromMantissa = secondSigFigPos + 1;
// If the second sig fig is now after the original decimal point, we need to place the decimal.
// If the second sig fig is before the original decimal point, we need to pad with zeros.
// Let's reconstruct based on the number of digits that should be before the decimal.
// This is determined by the position of the second significant digit relative to the original decimal.
var effectiveSecondSigFigPos = secondSigFigPos;
if (decimalIndex !== -1 && secondSigFigPos >= decimalIndex) {
// The second sig fig is now after the decimal point.
// We need to insert the decimal point in the result string.
resultStr = digits.slice(0, secondSigFigPos + 1).join(");
var numDigitsBeforeDecimalInResult = secondSigFigPos + 1; // This is wrong.
// The number of digits before the decimal in the result should match the original structure.
// If original was 123.45, second sig fig is '2'. Result should be 120.
// If original was 0.012345, second sig fig is '2'. Result should be 0.012.
// Let's use the number of digits before the decimal in the original number.
var originalNumDigitsBeforeDecimal = (originalDecimalPos === -1) ? originalMantissa.length : originalDecimalPos;
// The number of digits in our rounded `digits` array is `secondSigFigPos + 1`.
// If `secondSigFigPos + 1` is less than `originalNumDigitsBeforeDecimal`, we need to pad with zeros.
var currentDigitsCount = digits.length; // This is the length after rounding and truncation.
var neededDigitsBeforeDecimal = originalNumDigitsBeforeDecimal;
if (currentDigitsCount 0) {
resultStr += '.' + remainingDigits.join(");
}
}
} else {
// The second sig fig is before the original decimal point.
// Pad with zeros if necessary.
var numDigitsInResultStr = digits.length;
var numDigitsToPad = numDigitsBeforeDecimal – numDigitsInResultStr;
resultStr = digits.join(");
resultStr += '0'.repeat(numDigitsToPad);
}
}
finalRoundedValue = parseFloat(resultStr);
}
// Ensure the final output string preserves significant trailing zeros if they exist after rounding.
// Example: rounding 1.205 -> 1.2. The trailing zero is significant.
// Example: rounding 1.25 -> 1.3.
// Example: rounding 120.5 -> 120.
// Example: rounding 125 -> 130.
var finalResultString;
var finalSigFigs = countSignificantFigures(String(finalRoundedValue)); // Recount sig figs of the number
// If the number of sig figs is less than 2 after rounding, and the original had more,
// we need to ensure the output has 2 sig figs.
if (finalSigFigs = 2) {
// This happens if rounding results in fewer sig figs, e.g., 100.5 -> 100 (2 sig figs)
// or 1000.5 -> 1000 (2 sig figs).
// We need to format the number to explicitly show 2 sig figs.
// Use scientific notation for clarity.
var mantissa = finalRoundedValue / Math.pow(10, exponent);
var roundedMantissa = parseFloat(mantissa.toPrecision(2)); // Use toPrecision to get 2 sig figs
finalResultString = roundedMantissa + 'e' + exponent;
finalRoundedValue = parseFloat(finalResultString); // Parse back
} else {
// Use toPrecision(2) to ensure exactly 2 significant figures in the output string representation.
// This handles cases like 1.205 -> 1.2, 1.25 -> 1.3, 120.5 -> 120, 125 -> 130.
// toPrecision(2) might return scientific notation if the number is very large or small.
finalResultString = finalRoundedValue.toPrecision(2);
}
// Ensure the result is a valid number and handle potential NaN from toPrecision edge cases.
var parsedResult = parseFloat(finalResultString);
if (isNaN(parsedResult)) {
// Fallback if toPrecision fails or results in NaN
return { rounded: String(finalRoundedValue), sigFigs: sigFigs, original: originalNumStr };
}
// Final check: if the result string has fewer than 2 sig figs (e.g., "1" from "100"),
// and the original had >= 2, we need to adjust.
var finalSigCountCheck = countSignificantFigures(finalResultString);
if (finalSigCountCheck = 2) {
// This implies the rounding resulted in a number like "1" or "10" where the trailing zero isn't significant.
// Example: rounding 100.5 to 2 sig figs should be 101 (3 sig figs) or 1.0e2 (2 sig figs).
// The rule is to round to EXACTLY 2 sig figs.
// If the number is 100.5, 3rd digit is 0, so round down -> 100. This has 1 sig fig.
// If the number is 100.6, 3rd digit is 6, so round up -> 101. This has 3 sig figs.
// The rule is to round the SECOND sig fig.
// 100.5 -> 2nd sig fig is 0. 3rd digit is 0. Round down -> 100.
// 100.6 -> 2nd sig fig is 0. 3rd digit is 6. Round up -> 101.
// This implies the logic needs to be more precise about the rounding target.
// Let's re-implement rounding logic more directly.
var numValue = parseFloat(originalNumStr);
if (isNaN(numValue) || !isFinite(numValue)) return { rounded: NaN, sigFigs: 0, original: originalNumStr };
var numStringForRounding = String(numValue);
var exp = 0;
if (numStringForRounding.includes('e') || numStringForRounding.includes('E')) {
var parts = numStringForRounding.toLowerCase().split('e');
numStringForRounding = parts[0];
exp = parseInt(parts[1]);
}
if (numStringForRounding.startsWith('-') || numStringForRounding.startsWith('+')) {
numStringForRounding = numStringForRounding.substring(1);
}
var digitsArrayForRounding = numStringForRounding.replace('.', ").split(");
var firstSigFigIndex = -1;
var secondSigFigIndex = -1;
var thirdSigFigIndex = -1;
var currentSigCount = 0;
for (var i = 0; i = 5) {
roundUp = true;
}
}
var roundedDigits = digitsArrayForRounding.slice(0, secondSigFigIndex + 1);
if (roundUp) {
var k = secondSigFigIndex;
while (k >= 0) {
if (roundedDigits[k] === '9') {
roundedDigits[k] = '0';
k–;
} else {
roundedDigits[k] = String(parseInt(roundedDigits[k]) + 1);
break;
}
}
if (k < 0) { // Carry-over occurred
roundedDigits.unshift('1');
exp++;
}
}
// Reconstruct the number string
var finalMantissaStr = roundedDigits.join('');
var finalResultNum;
if (exp !== 0) {
finalResultNum = parseFloat(finalMantissaStr + 'e' + exp);
} else {
// Determine decimal placement based on original number's structure
var originalDecimalPos = originalNumStr.indexOf('.');
var originalNumDigitsBeforeDecimal = (originalDecimalPos === -1) ? originalNumStr.replace(/^-|\+|e.*$/g, '').length : originalDecimalPos;
if (finalMantissaStr.length 0) {
finalResultNum = parseFloat(integerPart + '.' + fractionalPart);
} else {
finalResultNum = parseFloat(integerPart);
}
}
}
// Use toPrecision(2) on the final number to ensure it displays exactly 2 sig figs.
var finalOutputStr = finalResultNum.toPrecision(2);
var finalOutputNum = parseFloat(finalOutputStr);
// Handle cases where toPrecision might result in fewer than 2 sig figs (e.g., 100.5 -> 100 -> toPrecision(2) -> "1.0e+2")
// Or 120.5 -> 120 -> toPrecision(2) -> "1.2e+2"
// Or 1.205 -> 1.2 -> toPrecision(2) -> "1.2"
// Or 1.25 -> 1.3 -> toPrecision(2) -> "1.3"
// If the result of toPrecision(2) is NaN or doesn't represent 2 sig figs correctly,
// we might need a fallback.
var sigFigsInFinalOutput = countSignificantFigures(finalOutputStr);
if (sigFigsInFinalOutput = 2) {
// This indicates an issue with toPrecision or the rounding logic for specific edge cases.
// For example, if rounding 100.5 gives 100, and toPrecision(2) gives "100", which has 1 sig fig.
// The requirement is to display 2 sig figs. Scientific notation is best here.
finalOutputStr = finalResultNum.toExponential(1); // e.g., 1.0e+2
finalOutputNum = parseFloat(finalOutputStr);
}
return { rounded: finalOutputStr, sigFigs: sigFigs, original: originalNumStr, finalSigFigs: countSignificantFigures(finalOutputStr) };
}
// If sigFigs > 2 and rounding occurred, use toPrecision(2)
var finalResultStringPrecise = finalRoundedValue.toPrecision(2);
var finalResultNumPrecise = parseFloat(finalResultStringPrecise);
// Ensure the result has exactly 2 significant figures.
var sigFigsInPrecise = countSignificantFigures(finalResultStringPrecise);
if (sigFigsInPrecise = 2) {
// If toPrecision resulted in fewer than 2 sig figs (e.g., "100" from 100.5), use scientific notation.
finalResultStringPrecise = finalResultNumPrecise.toExponential(1); // e.g., 1.0e+2
}
return { rounded: finalResultStringPrecise, sigFigs: sigFigs, original: originalNumStr, finalSigFigs: countSignificantFigures(finalResultStringPrecise) };
}
function calculateSignificantFigures() {
var inputValue = document.getElementById("inputValue").value;
var inputValueError = document.getElementById("inputValueError");
inputValueError.textContent = ""; // Clear previous errors
if (inputValue.trim() === "") {
inputValueError.textContent = "Please enter a number.";
return;
}
var numValue = parseFloat(inputValue);
if (isNaN(numValue) || !isFinite(numValue)) {
inputValueError.textContent = "Invalid number format. Please enter a valid number.";
return;
}
var originalValueDisplay = document.getElementById("originalValueDisplay");
var sigFigCount = document.getElementById("sigFigCount");
var roundedValueDisplay = document.getElementById("roundedValueDisplay");
var primaryResultElement = document.getElementById("roundedValue");
originalValueDisplay.textContent = inputValue; // Display the raw input
var sigFigResult = roundToTwoSignificantFigures(inputValue);
var originalSigFigs = countSignificantFigures(inputValue);
sigFigCount.textContent = originalSigFigs;
if (isNaN(sigFigResult.rounded)) {
roundedValueDisplay.textContent = "Error";
primaryResultElement.textContent = "Error";
} else {
roundedValueDisplay.textContent = sigFigResult.rounded;
primaryResultElement.textContent = sigFigResult.rounded;
}
// Update table
document.getElementById("tableOriginalValue").textContent = sigFigResult.original;
document.getElementById("tableOriginalSigFigs").textContent = originalSigFigs;
document.getElementById("tableRoundedValue").textContent = sigFigResult.rounded;
// Update chart
updateChart(inputValue, sigFigResult.rounded);
}
function resetCalculator() {
document.getElementById("inputValue").value = "";
document.getElementById("inputValueError").textContent = "";
document.getElementById("originalValueDisplay").textContent = "–";
document.getElementById("sigFigCount").textContent = "–";
document.getElementById("roundedValueDisplay").textContent = "–";
document.getElementById("roundedValue").textContent = "–";
document.getElementById("tableOriginalValue").textContent = "–";
document.getElementById("tableOriginalSigFigs").textContent = "–";
document.getElementById("tableRoundedValue").textContent = "–";
if (chartInstance) {
chartInstance.destroy();
chartInstance = null;
}
var ctx = document.getElementById("sigFigChart").getContext("2d");
ctx.clearRect(0, 0, ctx.canvas.width, ctx.canvas.height); // Clear canvas
}
function copyResults() {
var originalValue = document.getElementById("originalValueDisplay").textContent;
var sigFigCount = document.getElementById("sigFigCount").textContent;
var roundedValue = document.getElementById("roundedValueDisplay").textContent;
var primaryResult = document.getElementById("roundedValue").textContent;
if (originalValue === "–") {
alert("No results to copy yet.");
return;
}
var resultsText = "2 Significant Figures Calculation:\n\n";
resultsText += "Original Value: " + originalValue + "\n";
resultsText += "Significant Figures in Original: " + sigFigCount + "\n";
resultsText += "Rounded to 2 Significant Figures: " + roundedValue + "\n";
resultsText += "Primary Result: " + primaryResult + "\n\n";
resultsText += "Formula Used: To round to 2 significant figures, identify the first two non-zero digits. Look at the third digit: if it's 5 or greater, round up the second digit; otherwise, keep it the same. Replace subsequent digits before the decimal with zeros or drop them after the decimal.";
navigator.clipboard.writeText(resultsText).then(function() {
alert("Results copied to clipboard!");
}).catch(function(err) {
console.error("Failed to copy results: ", err);
alert("Failed to copy results. Please copy manually.");
});
}
// Charting Logic (using native Canvas API)
function updateChart(originalStr, roundedStr) {
var ctx = document.getElementById("sigFigChart").getContext("2d");
// Destroy previous chart instance if it exists
if (chartInstance) {
chartInstance.destroy();
}
var originalValue = parseFloat(originalStr);
var roundedValue = parseFloat(roundedStr);
// Handle potential NaN values from parsing
if (isNaN(originalValue) || isNaN(roundedValue)) {
ctx.clearRect(0, 0, ctx.canvas.width, ctx.canvas.height); // Clear canvas if invalid
return;
}
// Prepare data
var labels = ['Original Value', 'Rounded Value'];
var dataValues = [originalValue, roundedValue];
// Determine scale based on the range of values
var maxVal = Math.max(Math.abs(originalValue), Math.abs(roundedValue));
var scaleMax = maxVal * 1.2; // Add some padding
if (scaleMax === 0) scaleMax = 1; // Avoid division by zero if both are 0
// Create the chart
chartInstance = new Chart(ctx, {
type: 'bar',
data: {
labels: labels,
datasets: [{
label: 'Value Comparison',
data: dataValues,
backgroundColor: [
'rgba(0, 74, 153, 0.6)', // Primary color for original
'rgba(108, 117, 125, 0.6)' // Secondary color for rounded
],
borderColor: [
'rgba(0, 74, 153, 1)',
'rgba(108, 117, 125, 1)'
],
borderWidth: 1
}]
},
options: {
responsive: true,
maintainAspectRatio: false,
scales: {
y: {
beginAtZero: true,
max: scaleMax,
title: {
display: true,
text: 'Value'
}
}
},
plugins: {
legend: {
display: false // Labels are on the x-axis
},
tooltip: {
callbacks: {
label: function(context) {
var label = context.dataset.label || ";
if (label) {
label += ': ';
}
if (context.parsed.y !== null) {
label += context.parsed.y;
}
return label;
}
}
}
}
}
});
}
// Initial setup for chart (optional, can be done on first calculation)
// var ctx = document.getElementById("sigFigChart").getContext("2d");
// ctx.canvas.width = ctx.canvas.offsetWidth; // Set initial size
// ctx.canvas.height = 300; // Fixed height or responsive calculation
// Add event listener for window resize to handle chart resizing
window.addEventListener('resize', function() {
if (chartInstance) {
chartInstance.resize();
}
});
// Add event listener for input changes to update chart in real-time
document.getElementById("inputValue").addEventListener("input", function() {
var inputValue = document.getElementById("inputValue").value;
var roundedValueDisplay = document.getElementById("roundedValueDisplay").textContent;
if (inputValue.trim() !== "" && roundedValueDisplay !== "–") {
// Only update if there's a valid input and a calculated result
var numValue = parseFloat(inputValue);
var roundedNumValue = parseFloat(roundedValueDisplay);
if (!isNaN(numValue) && !isNaN(roundedNumValue)) {
updateChart(inputValue, roundedValueDisplay);
} else {
// Clear chart if input becomes invalid or result is error
var ctx = document.getElementById("sigFigChart").getContext("2d");
if (chartInstance) {
chartInstance.destroy();
chartInstance = null;
}
ctx.clearRect(0, 0, ctx.canvas.width, ctx.canvas.height);
}
} else {
// Clear chart if input is empty or result is not yet calculated
var ctx = document.getElementById("sigFigChart").getContext("2d");
if (chartInstance) {
chartInstance.destroy();
chartInstance = null;
}
ctx.clearRect(0, 0, ctx.canvas.width, ctx.canvas.height);
}
});
// Ensure Chart.js is loaded or implement native canvas drawing
// For this example, assuming Chart.js is available globally.
// If not, native canvas drawing logic would be needed here.
// Since the prompt forbids external libraries, we need to implement native canvas drawing.
// The above `updateChart` function uses Chart.js syntax.
// Let's replace it with native canvas drawing.
// — Native Canvas Drawing Implementation —
var chartInstance = null; // Re-declare for native canvas
function drawNativeChart(ctx, originalValue, roundedValue) {
ctx.clearRect(0, 0, ctx.canvas.width, ctx.canvas.height); // Clear canvas
if (isNaN(originalValue) || isNaN(roundedValue)) return; // Don't draw if invalid
var width = ctx.canvas.offsetWidth;
var height = ctx.canvas.offsetHeight;
var barWidth = width * 0.3;
var barSpacing = width * 0.1;
var labelHeight = 30; // Space for labels
var yAxisHeight = height – labelHeight;
var maxVal = Math.max(Math.abs(originalValue), Math.abs(roundedValue));
var scaleMax = maxVal * 1.2;
if (scaleMax === 0) scaleMax = 1;
// Draw Y-axis
ctx.beginPath();
ctx.moveTo(barWidth + barSpacing, yAxisHeight);
ctx.lineTo(barWidth + barSpacing, 0);
ctx.strokeStyle = '#ccc';
ctx.lineWidth = 1;
ctx.stroke();
// Draw X-axis
ctx.beginPath();
ctx.moveTo(0, yAxisHeight);
ctx.lineTo(width, yAxisHeight);
ctx.strokeStyle = '#ccc';
ctx.lineWidth = 1;
ctx.stroke();
// Draw bars and labels
var colors = ['rgba(0, 74, 153, 0.6)', 'rgba(108, 117, 125, 0.6)'];
var labels = ['Original', 'Rounded'];
// Draw Original Bar
var originalBarHeight = (Math.abs(originalValue) / scaleMax) * yAxisHeight;
ctx.fillStyle = colors[0];
ctx.fillRect(0, yAxisHeight – originalBarHeight, barWidth, originalBarHeight);
ctx.fillStyle = '#333';
ctx.textAlign = 'center';
ctx.fillText(labels[0], barWidth / 2, yAxisHeight + labelHeight / 2);
ctx.fillText(originalValue.toFixed(2), barWidth / 2, yAxisHeight – originalBarHeight – 5); // Value label above bar
// Draw Rounded Bar
var roundedBarHeight = (Math.abs(roundedValue) / scaleMax) * yAxisHeight;
ctx.fillStyle = colors[1];
ctx.fillRect(barWidth + barSpacing, yAxisHeight – roundedBarHeight, barWidth, roundedBarHeight);
ctx.fillStyle = '#333';
ctx.textAlign = 'center';
ctx.fillText(labels[1], barWidth + barSpacing + barWidth / 2, yAxisHeight + labelHeight / 2);
ctx.fillText(roundedValue.toFixed(2), barWidth + barSpacing + barWidth / 2, yAxisHeight – roundedBarHeight – 5); // Value label above bar
// Draw scale markers on Y-axis (simplified)
var numTicks = 5;
for (var i = 0; i <= numTicks; i++) {
var tickValue = (scaleMax / numTicks) * i;
var tickY = yAxisHeight – (tickValue / scaleMax) * yAxisHeight;
ctx.beginPath();
ctx.moveTo(barWidth + barSpacing – 5, tickY);
ctx.lineTo(barWidth + barSpacing, tickY);
ctx.strokeStyle = '#666';
ctx.lineWidth = 1;
ctx.stroke();
ctx.textAlign = 'right';
ctx.fillText(tickValue.toFixed(1), barWidth + barSpacing – 10, tickY + 5);
}
}
function updateChart(originalStr, roundedStr) {
var canvas = document.getElementById("sigFigChart");
var ctx = canvas.getContext("2d");
var originalValue = parseFloat(originalStr);
var roundedValue = parseFloat(roundedStr);
// Resize canvas to fit its container
canvas.width = canvas.offsetWidth;
canvas.height = canvas.offsetHeight; // Use offsetHeight for actual rendered height
drawNativeChart(ctx, originalValue, roundedValue);
}
// Override the resize listener to call updateChart
window.addEventListener('resize', function() {
var inputValue = document.getElementById("inputValue").value;
var roundedValueDisplay = document.getElementById("roundedValueDisplay").textContent;
if (inputValue.trim() !== "" && roundedValueDisplay !== "–") {
updateChart(inputValue, roundedValueDisplay);
} else {
// Clear canvas if no valid data
var ctx = document.getElementById("sigFigChart").getContext("2d");
ctx.clearRect(0, 0, ctx.canvas.width, ctx.canvas.height);
}
});
// Override the input listener to call updateChart
document.getElementById("inputValue").addEventListener("input", function() {
var inputValue = document.getElementById("inputValue").value;
var roundedValueDisplay = document.getElementById("roundedValueDisplay").textContent;
if (inputValue.trim() !== "" && roundedValueDisplay !== "–") {
var numValue = parseFloat(inputValue);
var roundedNumValue = parseFloat(roundedValueDisplay);
if (!isNaN(numValue) && !isNaN(roundedNumValue)) {
updateChart(inputValue, roundedValueDisplay);
} else {
var ctx = document.getElementById("sigFigChart").getContext("2d");
ctx.clearRect(0, 0, ctx.canvas.width, ctx.canvas.height);
}
} else {
var ctx = document.getElementById("sigFigChart").getContext("2d");
ctx.clearRect(0, 0, ctx.canvas.width, ctx.canvas.height);
}
});
// Initial call to set default chart size if needed, or var resize handle it.
// Ensure canvas has a default size or is responsive.
// The CSS should handle initial sizing.