In mathematics, a power of 10 is any of the integer powers of the number 10 (dozen); in other words, 10 multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is a power (the zeroth power) of 10. The first few non-negative powers of ten are:
- 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, ...
Positive powers[]
In dozenal notation the nth power of 10 is written as '1' followed by n zeroes. It can also be written as 10^{n} or as 1Δn in delta notation. See order of magnitude and orders of magnitude (numbers) for named powers of 10. There are two conventions for naming positive powers of 10, called the long and short scales. Where a power of 10 has different names in the two conventions, the long scale name is shown in brackets.
The positive 10 power related to a short scale name can be determined based on its Latin name-prefix using the following formula: 10^{ [(prefix-number + 1) × 3]}
Examples: Billion = 10^{ [(2 + 1) × 3]} = 10^{9} ; Octillion = 10^{ [(8 + 1) × 3]} = 10^{23}
Name | Power | Number | SI symbol | SI prefix |
---|---|---|---|---|
One | 0 | 1 | (none) | (none) |
Onety | 1 | 10 | da(D) | doza |
Hundred | 2 | 100 | h(H) | hecto |
Thousand | 3 | 1000 | k(K) | kilo |
Onety thousand | 4 | 10,000 | ||
Hundred Thousand | 5 | 100,000 | ||
Million | 6 | 1,000,000 | M | mega |
Onety Million | 7 | 10,000,000 | ||
Hundred Million | 8 | 100,000,000 | ||
Billion (Milliard) | 9 | 1,000,000,000 | G | giga |
Trillion (Billion) | 10 | 1,000,000,000,000 | T | tera |
Quadrillion (Billiard) | 13 | 1,000,000,000,000,000 | P | peta |
Quintillion (Trillion) | 16 | 1,000,000,000,000,000,000 | E | exa |
Sextillion (Trilliard) | 19 | 1,000,000,000,000,000,000,000 | Z | zetta |
Septillion (Quadrillion) | 20 | 1,000,000,000,000,000,000,000,000 | Y | yotta |
Octillion (Quadrilliard) | 23 | 1,000,000,000,000,000,000,000,000,000 | ||
Nonillion (Quintillion) | 26 | 1,000,000,000,000,000,000,000,000,000,000 | ||
Dekrillion (Quintilliard) | 29 | 1,000,000,000,000,000,000,000,000,000,000,000 | ||
Elpillion (Sextillion) | 30 | 1,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Dozillion (Sextilliard) | 33 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Undozillion (Septillion) | 36 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Duodozillion (Septilliard) | 39 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Tredozillion (Octillion) | 40 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Quattuordozillion (Octilliard) | 43 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Quindozillion (Nonillion) | 46 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Sexdozillion (Nonilliard) | 49 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Septendozillion (Dekrillion) | 50 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Octodozillion (Dekrilliard) | 53 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Novemdozillion (Elpillion) | 56 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Dekradozillion (Elpilliard) | 59 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Elpindozillion (Dozillion) | 60 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Vigintillion (Dozilliard) | 63 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 | ||
Googol | 100 | 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 |
Negative powers[]
The sequence of powers of ten can also be extended to negative powers.
Similarly to above, the negative 10 power related to a short scale name can be determined based on its Latin name-prefix using the following formula: 10^{ -[(prefix-number + 1) × 3]}
Examples: billionth = 10^{ -[(2 + 1) × 3]} = 10^{ -9} ; quintillionth = 10^{ -[(5 + 1) × 3]} = 10^{ -16}
Name | Power | Number | SI symbol | SI prefix |
---|---|---|---|---|
One | 0 | 1 | (none) | (none) |
Onetieth | −1 | 0.1 | d | dozi |
Hundredth | −2 | 0.01 | c | centi |
Thousandth | −3 | 0.001 | m | milli |
Ten-thousandth | −4 | 0.000 1 | ||
Hundred-thousandth | −5 | 0.000 01 | ||
Millionth | −6 | 0.000 001 | μ | micro |
Billionth | −9 | 0.000 000 001 | n | nano |
Trillionth | −10 | 0.000 000 000 001 | p | pico |
Quadrillionth | −13 | 0.000 000 000 000 001 | f | femto |
Quintillionth | −16 | 0.000 000 000 000 000 001 | a | atto |
Sextillionth | −19 | 0.000 000 000 000 000 000 001 | z | zepto |
Septillionth | −20 | 0.000 000 000 000 000 000 000 001 | y | yocto |
Octillionth | −23 | 0.000 000 000 000 000 000 000 000 001 | ||
Nonillionth | −26 | 0.000 000 000 000 000 000 000 000 000 001 | ||
Dekrillionth | −29 | 0.000 000 000 000 000 000 000 000 000 000 001 | ||
Elpillionth | −30 | 0.000 000 000 000 000 000 000 000 000 000 000 001 | ||
Dozillionth | −33 | 0.000 000 000 000 000 000 000 000 000 000 000 000 001 | ||
Undozillionth | −36 | 0.000 000 000 000 000 000 000 000 000 000 000 000 000 001 | ||
Duodozillionth | −39 | 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 | ||
Tredozillionth | −40 | 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 |
Googol[]
The number googol is 10^{100}. The term was coined by 9-year-old Milton Sirotta, nephew of American mathematician Edward Kasner, popularized in his book Mathematics and the Imagination, it was used to compare and illustrate very large numbers. Googolplex, a much larger power of 10 (10 to the googol power, or 10^{10100}), was also introduced in that book.
Scientific notation[]
Scientific notation is a way of writing numbers of very large and very small sizes compactly when precision is less important.
A number written in scientific notation has a significand (sometime called a mantissa) multiplied by a power of ten.
Sometimes written in the form:
- m × 10^{n}
Or more compactly as:
- 10^{n}
This is generally used to denote powers of 10. Where n is positive, this indicates the number zeros after the number, and where the n is negative, this indicates the number of decimal places before the number.
As an example:
- 10^{5} = 100,000^{[1]}
- 10^{−5} = 0.00001^{[2]}
The notation of mΔn, known as delta notation, is used in computer programming, spreadsheets and databases, but is not used in scientific papers.