Scientific notation is a standardized way of expressing very large or very small numbers concisely. It is widely used in science, engineering, and mathematics. A number in scientific notation is expressed as a product of two parts: a coefficient (or mantissa) and a power of 10.
The general form is: a × 10^b
a is the coefficient (or mantissa), a number greater than or equal to 1 and less than 10 (1 ≤ |a| < 10). For negative numbers, the coefficient can be negative.
b is the exponent, an integer indicating how many places the decimal point has been moved. A positive exponent means the number is large, and a negative exponent means the number is small (close to zero).
How to Perform Calculations in Scientific Notation:
1. Addition and Subtraction:
To add or subtract numbers in scientific notation, the exponents of both numbers must be the same. If they are not, you must adjust one or both numbers until their exponents match. Then, add or subtract the coefficients and keep the common exponent.
Example: (3.14 × 106) + (2.71 × 105)
Adjust the second number to have an exponent of 6: 2.71 × 105 = 0.271 × 106
Add the coefficients: 3.14 + 0.271 = 3.411
Keep the common exponent: 3.411 × 106
2. Multiplication:
To multiply numbers in scientific notation, multiply the coefficients and add the exponents.
Example: (3.14 × 106) × (2.71 × 103)
Multiply the coefficients: 3.14 × 2.71 = 8.5094
Add the exponents: 6 + 3 = 9
Combine: 8.5094 × 109
If the resulting coefficient is not between 1 and 10, you may need to adjust it (and the exponent accordingly).
3. Division:
To divide numbers in scientific notation, divide the coefficients and subtract the exponent of the divisor from the exponent of the dividend.
Example: (8.5094 × 109) / (3.14 × 106)
Divide the coefficients: 8.5094 / 3.14 ≈ 2.71
Subtract the exponents: 9 – 6 = 3
Combine: 2.71 × 103
Again, adjust the coefficient if it falls outside the 1 to 10 range.
Use Cases:
Astronomy: Distances to stars, size of planets.
Chemistry: Number of atoms or molecules (Avogadro's number).
Physics: Mass of subatomic particles, speed of light.
Computer Science: Handling very large or small floating-point numbers.
Biology: Population sizes, cell dimensions.
function formatResult(mantissa, exponent) {
if (isNaN(mantissa) || isNaN(exponent)) {
return "–";
}
// Normalize: Ensure mantissa is between 1 and 10 (or -1 and -10)
while (mantissa >= 10) {
mantissa /= 10;
exponent++;
}
while (mantissa -1) {
mantissa *= 10;
exponent–;
}
// Handle cases where original input might lead to 0 mantissa after normalization
if (mantissa === 0) {
return "0";
}
// Format to a reasonable precision, e.g., 5 decimal places
var formattedMantissa = mantissa.toFixed(5);
// Remove trailing zeros after the decimal point
formattedMantissa = formattedMantissa.replace(/\.0+$/, ").replace(/(\.\d*?)0+$/, '$1');
if (formattedMantissa === ".") formattedMantissa = "0"; // Handle case like 0.00000
return formattedMantissa + " x 10" + exponent + "";
}
function parseScientificInput(mantissaStr, exponentStr) {
var mantissa = parseFloat(mantissaStr);
var exponent = parseInt(exponentStr, 10);
if (isNaN(mantissa) || isNaN(exponent)) {
return { valid: false, error: "Invalid input. Please enter valid numbers for mantissa and exponent." };
}
return { valid: true, mantissa: mantissa, exponent: exponent };
}
function calculateScientific(operation) {
var num1Data = parseScientificInput(document.getElementById('number1_mantissa').value, document.getElementById('number1_exponent').value);
var num2Data = parseScientificInput(document.getElementById('number2_mantissa').value, document.getElementById('number2_exponent').value);
var errorMessageDiv = document.getElementById('error-message');
var resultValueSpan = document.getElementById('result-value');
errorMessageDiv.textContent = ""; // Clear previous errors
resultValueSpan.innerHTML = "–"; // Reset result
if (!num1Data.valid) {
errorMessageDiv.textContent = num1Data.error;
return;
}
if (!num2Data.valid) {
errorMessageDiv.textContent = num2Data.error;
return;
}
var m1 = num1Data.mantissa;
var e1 = num1Data.exponent;
var m2 = num2Data.mantissa;
var e2 = num2Data.exponent;
var resultMantissa = 0;
var resultExponent = 0;
if (operation === 'add' || operation === 'subtract') {
if (e1 !== e2) {
if (e1 > e2) {
m2 = m2 * Math.pow(10, e2 – e1);
e2 = e1;
} else {
m1 = m1 * Math.pow(10, e1 – e2);
e1 = e2;
}
}
if (operation === 'add') {
resultMantissa = m1 + m2;
} else { // subtract
resultMantissa = m1 – m2;
}
resultExponent = e1; // The common exponent
// Normalize result for addition/subtraction
while (resultMantissa >= 10 || resultMantissa <= -10) {
resultMantissa /= 10;
resultExponent++;
}
while (resultMantissa -1 && resultMantissa !== 0) {
resultMantissa *= 10;
resultExponent–;
}
} else if (operation === 'multiply') {
resultMantissa = m1 * m2;
resultExponent = e1 + e2;
// Normalize result for multiplication
while (resultMantissa >= 10 || resultMantissa <= -10) {
resultMantissa /= 10;
resultExponent++;
}
while (resultMantissa -1 && resultMantissa !== 0) {
resultMantissa *= 10;
resultExponent–;
}
} else if (operation === 'divide') {
if (m2 === 0) {
errorMessageDiv.textContent = "Error: Division by zero.";
return;
}
resultMantissa = m1 / m2;
resultExponent = e1 – e2;
// Normalize result for division
while (resultMantissa >= 10 || resultMantissa <= -10) {
resultMantissa /= 10;
resultExponent++;
}
while (resultMantissa -1 && resultMantissa !== 0) {
resultMantissa *= 10;
resultExponent–;
}
}
// Handle cases where result mantissa becomes exactly 0 after operations
if (resultMantissa === 0) {
resultValueSpan.innerHTML = "0";
} else {
resultValueSpan.innerHTML = formatResult(resultMantissa, resultExponent);
}
}