# Scientific notation

Scientific notation is a convenient way to write very large and very small numbers. [itex]1 \times 10^9[itex], for example, means one billion (a 1 followed by nine zeros: 1,000,000,000). [itex]1 \times 10^{-9}[itex] means one billionth, or 0.0000000001. Writing [itex]10^9[itex] instead of nine zeros saves the reader the effort and hazard of counting a long string of zeros to see how large the number is.

Adding a 0 onto the end of a number multiplies it by ten: 100 is ten times 10. In scientific notation, however, multiplying a number by 10 increases the exponent by one, for example from [itex]10^1[itex] to [itex]10^2[itex]. Remember then, when reading numbers in scientific notation, that small changes in the exponent equate to large changes in the number itself: [itex]2.5 \times 10^5[itex] dollars ($250,000) is a common price for new homes in the U.S., while [itex]2.5 \times 10^{10}[itex] dollars ($25 billion) would make you one of the world's richest people.

Formally, scientific notation (also known as standard index notation) is floating-point notation with radix (base) 10. Nonzero numbers are written in the form [itex]\pm \, a \times 10^b[itex] 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 (excluding 10 itself). In that case b is the integer part of the common logarithm of the absolute value of the number. 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 can be 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 Template:Sn m/s then it is clear that it is between 299,500,000 and 300,500,000 m/s, whereas 300,000,000 m/s suggests that the number has been rounded to the nearest one and is exact to 9 places. (See also below.)

• 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 about 4.6 × 1026 m and the mass of a proton is about 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 which is commonly denoted by e.

Scientific notation is useful for describing physical quantities which can only be measured within certain error limits. Giving just the digits that are known to be reliable (the "significant figures") conveys an indication of the error in the measurement. 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" (see SI prefixes), thus avoiding misunderstanding.

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