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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 10n 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] = 109 ; Octillion = 10 [(8 + 1) × 3] = 1023

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 10100. 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 1010100), 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 × 10n

Or more compactly as:

10n

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:

105 = 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.

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