Scientific Notation Calculator

How it works

Scientific notation writes a number as a mantissa times a power of ten, a × 10^n, with the mantissa between 1 and 10 (1 ≤ |a| < 10). The exponent n counts how many places the decimal point moved: positive for large numbers, negative for small ones. 149,600,000 km becomes 1.496 × 10^8 km; 0.00052 becomes 5.2 × 10^-4.

Engineering notation uses the same idea but keeps the exponent a multiple of 3 so it lines up with the SI prefixes kilo, mega, giga, milli, micro and nano. The mantissa is then between 1 and 1000: 1.496 × 10^8 is 149.6 × 10^6, or 149.6 mega. E notation is the compact form calculators and spreadsheets use: 1.496E+8 means exactly the same thing.

To multiply two numbers in scientific notation, multiply the mantissas and add the exponents; to divide, divide the mantissas and subtract the exponents. If the new mantissa falls outside 1 to 10, the calculator renormalizes it by moving the decimal point and adjusting the exponent to match. The Working line shows each of these steps.

Formula

x = a × 10^n     with 1 ≤ |a| < 10 and n = floor(log10 |x|), a = x ÷ 10^n
engineering:  n' = 3 × floor(n ÷ 3),  a' = a × 10^(n − n')   so 1 ≤ |a'| < 1000
E notation:   a × 10^n is written aE+n (or aE-n)
(a × 10^n) × (b × 10^m) = (a × b) × 10^(n + m)
(a × 10^n) ÷ (b × 10^m) = (a ÷ b) × 10^(n − m)
renormalize when |a × b| ≥ 10 or |a ÷ b| < 1

Example

The average distance from Earth to the Sun is about 149,600,000 km. The decimal point moves 8 places, so n = 8 and a = 149600000 ÷ 10^8 = 1.496. In scientific notation that is 1.496 × 10^8 km, in engineering notation 149.6 × 10^6 km, and in E notation 1.496E+8.

Going the other way, Avogadro's number 6.02214076 × 10^23 written out in full is 602214076000000000000000.

Multiplying (3 × 10^8) × (4 × 10^5): the mantissas give 3 × 4 = 12 and the exponents add to 8 + 5 = 13, so the product is 12 × 10^13. Since 12 is not between 1 and 10, it is renormalized to 1.2 × 10^14.

Assumptions and limitations

• Mantissas are kept to 12 significant digits after multiplying or dividing, which hides binary floating-point noise but also rounds a result such as 1 ÷ 3 to 3.33333333333.
• Exponents must be whole numbers between -1000 and 1000. A number entered in standard form can be anything a double-precision number can hold.
• Zero has no scientific-notation form with a mantissa between 1 and 10; it is reported simply as 0.
• The sign of the number lives in the mantissa: -0.00052 is -5.2 × 10^-4.

Frequently asked questions