How Exponential (Scientific) Notation Works
Exponential notation writes a number as a mantissa multiplied by a power of 10: a × 10ⁿ. In standard scientific notation, the mantissa a is restricted to 1 ≤ |a| < 10, so there is always exactly one nonzero digit before the decimal point. The exponent n is a positive integer for numbers 10 or larger, negative for numbers smaller than 1, and 0 for numbers between 1 and 10. This calculator converts any number you enter into that form, along with the related engineering-notation form and its fully expanded decimal value.
Formula and method
To convert a decimal number N into scientific notation, find the integer n such that N = a × 10ⁿ with 1 ≤ |a| < 10. Practically, move the decimal point in N until only one nonzero digit remains to its left, and let n equal the number of places moved (positive if you moved the point left, negative if you moved it right). For example, 6,300,000 = 6.3 × 10⁶ (decimal moved 6 places left) and 0.0000048 = 4.8 × 10⁻⁶ (decimal moved 6 places right). Engineering notation uses the same idea but forces n to be a multiple of 3, which allows the mantissa to range from 1 up to 1000 — for example 6,300,000 = 6.3 × 10⁶ in engineering notation too (since 6 is already a multiple of 3), while 630,000,000 = 630 × 10⁶ rather than 6.3 × 10⁸.
Common sources of error
- Sign of the exponent: numbers less than 1 always get a negative exponent, and numbers 10 or greater always get a positive exponent — mixing this up is the most common mistake.
- Off-by-one in counting places: count decimal-point moves carefully; 45,000 = 4.5 × 10⁴ (4 moves), not 4.5 × 10⁵.
- Confusing scientific and engineering notation: scientific notation always keeps the mantissa under 10; engineering notation allows up to 1000 but only uses exponents that are multiples of 3 (matching SI prefixes like k, M, µ, n).
Checking your result
A quick check: multiply the mantissa back out by 10 raised to the exponent shown and confirm it matches your original number (this calculator does that for you in the "Expanded Decimal Form" result). Also confirm the mantissa's magnitude matches the notation you selected — between 1 and 10 for scientific notation, or between 1 and 1000 for engineering notation.
Applications
Scientific notation is the standard way to write very large or very small measurements — Avogadro's number (6.022 × 10²³), the charge of an electron (1.602 × 10⁻¹⁹ C), or a computer's clock speed. Engineering notation is preferred in electronics and physics because its exponents (…, -6, -3, 0, 3, 6, 9, …) map directly onto SI unit prefixes such as micro, milli, kilo, mega, and giga, making unit conversion straightforward.