Scientific notation (standard index notation) is a concise way of recording numbers by integer powers of ten, that is used to record numbers which are notably large or small. Such notation is used to record physical quantities without including trailing, or leading, zeros.
- 101 = 10
- 102 = 100
- 103 = 1000
- 106 = 1,000,000
- 109 = 1,000,000,000
- 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 = 0.000000001
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. For example, the distance to the edge of the observable universe is ~4.6 × 1026 m and the mass of a proton is ~1.67 x 10−27 kg. 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; for example, 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 useful for describing physical quantities, as they can only be measured within certain error limits, and so giving just the digits that are known to be correct (the "significant digits") conveys the information that can safely be used.
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, more sophisticated expressions of measurement error must be used.
Scientific notation also avoids regional differences in certain quantifiers, such as "billion", where the use of scientific notation avoids misunderstanding.