Scientific Notation Calculator

Scientific notation writes a number as a mantissa between 1 and 10 times a power of ten. 47,000 becomes 4.7 x 10^4; 0.0000472 becomes 4.72 x 10^-5. The exponent counts how many places the decimal point moved, positive for large numbers and negative for small ones.

The requirement that the mantissa sit in [1, 10) is what makes the form canonical. 47 x 10^3 and 0.47 x 10^5 are both arithmetically equal to 47,000, and neither is scientific notation. Normalizing them is the step most often skipped after a multiplication, because multiplying two valid mantissas frequently produces one that is not: 4 x 10^3 times 5 x 10^6 gives 20 x 10^9, which has to be rewritten as 2 x 10^10.

Arithmetic in the notation is easier than in the numbers. To multiply, multiply the mantissas and add the exponents. To divide, divide the mantissas and subtract the exponents. (4.72 x 10^-5)(3 x 10^8) = 14.16 x 10^3 = 1.416 x 10^4. Addition and subtraction are the awkward ones, because the exponents have to match first: 3 x 10^5 plus 4 x 10^4 means rewriting the second as 0.4 x 10^5 to get 3.4 x 10^5.

The real purpose of the notation is significant figures, not brevity. Writing 47,000 is ambiguous: it might be measured to the nearest thousand or to the nearest unit, and the zeros do not say which. 4.7 x 10^4 claims two significant figures. 4.700 x 10^4 claims four. That distinction is unavailable in ordinary decimal writing and is the reason scientific disciplines use the notation even for numbers that are not especially large or small. The rules follow from it: a product or quotient carries the significant figures of its least precise input, so multiplying a 3-figure measurement by a 5-figure one gives a 3-figure answer, and reporting more is a false claim about precision.

Engineering notation is the same idea with the exponent restricted to multiples of three. 47,000 becomes 47 x 10^3 rather than 4.7 x 10^4, which breaks the mantissa rule on purpose, because multiples of three are where the SI prefixes sit. 47 x 10^3 ohms is 47 kilohms, 10^-6 is micro, 10^-9 is nano, 10^9 is giga. Electrical and mechanical work uses this form almost exclusively, since every value then reads directly as a prefixed unit.

E notation is the typing-friendly version: 4.7e4 means 4.7 x 10^4. It is what calculator displays and programming languages produce, and the "e" has nothing to do with Euler's number. Anything beyond about 10^15 in a typical double-precision float loses exact integer representation, which is why very large counts are often held as strings rather than numbers.

A final practical note. The notation makes orders of magnitude comparable at a glance, which is its everyday value. A proton's mass is 1.67 x 10^-27 kg and the Sun's is 1.99 x 10^30 kg: a ratio of about 10^57, which is legible written this way and nearly impossible written out.