How this calculator works
Scientific notation writes a number as a mantissa (a value between 1 and 10, or between -10 and -1 for negative numbers) multiplied by 10 raised to some integer exponent. This calculator converts a plain decimal into that normalized form, or takes a mantissa and exponent and reconstructs the original decimal.
The format is standard in science and engineering because it makes the size of a number immediately visible from its exponent, and it avoids writing out long strings of zeros for very large or very small quantities like Avogadro's number or an atomic radius.
The formula
Decimal to scientific: value = a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integerScientific to decimal: value = mantissa × 10^exponentThe exponent is found by taking the floor of log10 of the number's absolute value; the mantissa is whatever remains once that power of 10 is divided out. A tiny floating-point correction is applied for the rare edge case where this pushes the mantissa to exactly 10 or just under 1.
Worked example: converting 123,400
- Find the exponent: 123,400 is between 10^5 (100,000) and 10^6 (1,000,000), so the exponent is 5.
- Divide out that power of 10: 123,400 ÷ 10^5 = 1.234.
- The normalized result is 1.234 × 10^5, written in e-notation as 1.234e+5.
- Check the reverse direction: 1.234 × 10^5 = 1.234 × 100,000 = 123,400, matching the original number exactly.
Frequently asked questions
Why must the mantissa be between 1 and 10?
That range is the standard normalized form of scientific notation, ensuring every number has exactly one correct representation. 12.34 × 10^4 and 1.234 × 10^5 represent the same value, but only the second is normalized.
What does the e in e-notation mean?
The letter e stands for "exponent" and is a plain-text way to write × 10^. So 1.234e+5 means 1.234 × 10^5, and 5.67e-4 means 5.67 × 10^-4. Programming languages and calculators use this format because superscripts aren't always available.
How is a negative number converted?
The sign stays on the mantissa. For example, -4,500 converts to -4.5 × 10^3, keeping the same magnitude rules as a positive number.
What is the scientific notation for zero?
Zero is a special case: it is shown as a mantissa of 0 and an exponent of 0, since there is no meaningful power of 10 to normalize a value of zero against.
Why do very large or very small decimals show extra digits?
When converting scientific notation back to a decimal, results far outside everyday magnitudes are shown in fixed 10-significant-digit scientific text rather than a long plain number, since that keeps the precision honest about what double-precision arithmetic actually supports.