Scientific notation is a way of expressing very large or very small numbers in a compact form: a × 10ⁿ, where a (the coefficient) is a decimal number greater than or equal to 1 and less than 10, and n (the exponent) is an integer indicating how many places the decimal point has been moved.
Why Use Scientific Notation?
In fields such as physics, chemistry, engineering, and finance, numbers can span many orders of magnitude. Scientific notation makes calculations easier, reduces transcription errors, and improves readability.
How the Calculator Works
Decimal → Scientific Notation
Identify the absolute value of the number.
Determine the exponent n by counting how many places the decimal point must move to place the first non‑zero digit immediately to the left of the decimal point. This is Math.floor(Math.log10(|number|)).
Calculate the coefficient a as number / 10ⁿ. The coefficient is then rounded to a reasonable number of decimal places (default 5).
Combine them as a × 10ⁿ.
Scientific Notation → Decimal
Enter the coefficient a and exponent n.
Multiply: decimal = a × 10ⁿ.
The result is displayed in standard decimal form.
Example Conversions
Decimal to Scientific: 12345.67 → 1.23457 × 10⁴
Scientific to Decimal: 3.2 × 10⁻³ → 0.0032
Practical Use Cases
Representing astronomical distances (e.g., 9.46 × 10¹² km for a light‑year).
Expressing atomic scales (e.g., 1.67 × 10⁻²⁷ kg for a proton mass).
Financial modeling of extremely large market caps or tiny interest rates.
Tips for Accurate Results
Ensure the coefficient is between 1 (inclusive) and 10 (exclusive) for a proper scientific notation.
When converting back, keep enough decimal places to avoid rounding errors in subsequent calculations.
function convertToScientific(){
var input=document.getElementById('decimalNumber').value.trim();
var num=parseFloat(input);
var resultDiv=document.getElementById('result');
var stepsDiv=document.getElementById('steps');
if(isNaN(num)){
resultDiv.innerHTML='Please enter a valid number.';
stepsDiv.innerHTML=";
return;
}
if(num===0){
resultDiv.innerHTML='0 × 10⁰';
stepsDiv.innerHTML='Zero is already in scientific notation.';
return;
}
var sign=num<0?-1:1;
var absNum=Math.abs(num);
var exponent=Math.floor(Math.log10(absNum));
var coefficient=absNum/Math.pow(10,exponent);
coefficient=coefficient*sign;
// round coefficient to 5 decimal places
coefficient=Math.round(coefficient*100000)/100000;
resultDiv.innerHTML=coefficient+' × 10⁽'+exponent+'⁾';
var steps='Step 1: Identify absolute value = '+absNum+'\\n';
steps+='Step 2: Determine exponent n = floor(log10('+absNum+')) = '+exponent+'\\n';
steps+='Step 3: Compute coefficient a = '+absNum+' / 10ⁿ = '+absNum+'/10⁽'+exponent+'⁾ = '+(absNum/Math.pow(10,exponent)).toFixed(5)+'\\n';
if(sign===-1){steps+='Step 4: Apply original sign → a = -'+Math.abs(coefficient)+'\\n';}
steps+='Result: '+coefficient+' × 10⁽'+exponent+'⁾';
stepsDiv.textContent=steps;
}
function convertToDecimal(){
var coeffInput=document.getElementById('coeff').value.trim();
var expInput=document.getElementById('exponent').value.trim();
var coeff=parseFloat(coeffInput);
var exponent=parseInt(expInput,10);
var resultDiv=document.getElementById('result');
var stepsDiv=document.getElementById('steps');
if(isNaN(coeff)||isNaN(exponent)){
resultDiv.innerHTML='Please enter valid coefficient and exponent.';
stepsDiv.innerHTML='';
return;
}
var decimal=coeff*Math.pow(10,exponent);
// limit to 12 significant digits to avoid long tails
var displayDecimal=Number(decimal.toPrecision(12));
resultDiv.innerHTML=displayDecimal;
var steps='Step 1: Multiply coefficient by 10 raised to exponent.\\n';
steps+='Step 2: '+coeff+' × 10⁽'+exponent+'⁾ = '+coeff+' × '+Math.pow(10,exponent)+'\\n';
steps+='Step 3: Result = '+displayDecimal;
stepsDiv.textContent=steps;
}