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In applied mathematics, very large and very small numbers are written in compact form by using integer powers of . For instance, the speed of light in vacuum,  meters per second (approximately), can be written more compactly as  meter per second.
More generally, a real number  is said to be expressed in scientific notation if it is written in the form ,
Where  is an integer and  is a number greater than or equal to  , but less than  . The integer  is called the characteristic of , the number  is called the mantissa of  , and factor  is called the order of magnitude of  .
To change a number from ordinary decimal form to scientific notation, move the decimal point to obtain a number between  and  (one digit in front of the decimal point) and multiply by  or by , where  is the number of places the decimal points was moved to the left or to the right, respectively. Final zeros after the decimal point can be dropped unless it is necessary to retain them to indicate the accuracy of an approximation.
Example:
Rewrite each statement so that all numbers are expressed in scientific notation.
- The volume of the earth is approximately
 cubic meters.
- The earth rotates about its axis with an angular speed of approximately
 degree per second.
Solution:
(a) We move the decimal point  places to the left 
to obtain a number between  and  and multiply by  , so that  .
Thus, the volume of the earth is approximately  cubic meters.
(b) We move the decimal point three places to the right 
to obtain a number between  and  and multiply by  , so that
Thus, the earth rotates about its axis with an angular speed of approximately  degree per second.
Example:
Rewrite the following numbers in ordinary decimal form.
(a)  (b) 
Solution:
(a) 
(b) 
(Very small numbers, written in ordinary decimal form, are easier to read if commas are used to set off zeros in groups of three.)
Names and Scientific prefixes for some integer powers of  are listed in table.
Power of  |
Names
|
Prefix
|
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Ten
|
Deka
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|
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Hundred
|
Hecto
|
|
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Thousand
|
Kilo
|
|
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Million
|
Mega
|
|
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Billion
|
Giga
|
|
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Trillion
|
Tera
|
|
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Quadrillion
|
Peta
|
|
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Tenth
|
Deci
|
|
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Hundredth
|
Centi
|
|
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Thousandth
|
Milli
|
|
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Millionth
|
Micro
|
|
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Billionth
|
Nano
|
|
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Trillionth
|
Pico
|
|
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Quadrillionth
|
Femto
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