Scientific notation

Scientific notation is a concise way of recording numbers, usually applied to very large or very small ones. It is most commonly used by scientists to record physical quantities to the measured precision without the need for large numbers of trailing zeros (in the case of very large quantities) or leading zeros (for the very small).

Scientific notation makes use of the fact that 10 raised to an integer positive power n is equal to 1 followed by n zeros - for instance:

10¹ = 10
10² = 100
10³ = 1000
10⁶ = 1,000,000
10²⁰ = 100,000,000,000,000,000,000

Additionally, 10 raised to a negative integer power -n is equal to 1/10ⁿ or, equivalently 0. (n-1 zeros)1:

10^-1 = 1/10 = 0.1
10^-3 = 1/1000 = 0.001
10^-9 = 1/1,000,000,000 = 0.00000001

Therefore, a large number such as 156,234,000,000,000,000,000,000,000,000 can be concisely recorded as 1.56234 × 10²⁹, and a small number such as 0.0000000000234 can be written as 2.34 × 10^-11.

Most calculators and many computer programs present very large and very small results in scientific notation; the 10 is usually omitted and the letter E for exponent is used; the above number would be represented as 1.56234 E 29. Note that this is not related to the base of the natural logarithm also commonly denoted by e.

Scientific notation is highly useful for quoting physical quantities, as they can only be measured to within certain error limits and so quoting just the digits that are certain (the "significant digits") gives all the information required without wasting space.

If a physical quantity is quoted using scientific notation, it is usually assumed to be accurate to the quoted number of digits of precision - for instance if a figure 1.2340 × 10⁶ metres is quoted, the actual figure is assumed to be between 1,233,950 metres as a lower bound and 1,234,050 metres as an upper bound. However, where precision in such measurements is crucial, much more sophisticated expressions of measurement error must be used.