Scientific notation

Scientific notation (standard index notation) is floating-point notation with radix (base) 10. It is a concise way of recording numbers using integer powers of ten, that is used to record numbers which are notably large or small.

Nonzero numbers are written in the form <math>a \times 10^b<math> where b is an integer; a is called the significand. The same number can be written in different ways: adding one to b reduces a by a factor 10. Usually a is chosen in the range 1-10. In that case b is the integer part of the common logarithm. Such a fixed range allows easy comparison of two numbers: the one with the larger exponent is larger.

Another term used for a is mantissa, but this may give confusion with its alternative meaning of fractional part of the common logarithm.

For very small numbers the advantage is that leading zeros are not needed. Large numbers are often (rounded to) a multiple of a power of 10. In that case an advantage of scientific notation is that trailing zeros which are the result of rounding are not needed. An additional advantage is that the rounding accuracy is shown: if one or more trailing zeros are not the result of rounding they are written (unless it is clear from the context that an exact number is referred to). For example, when the speed of light is expressed as 3.00 × 10⁸ m/s or 3.00 × 10⁵ km/s then it is clear that it is between 299 500 and 300 500 km/s. (See also below and significant figures)

• 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 (in plain text 1.56234e29 and 2.34e-11, or with a capital E). 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.

See also

• orders of magnitude
• floating-point
• engineering notation
• large number

ca:Notació científica de:Wissenschaftliche Notation es:Notación científica fr:Notation scientifique it:Notazione scientifica pl:Postać wykładnicza pt:Notação científica sv:Grundpotens zh:科学记数法