Power of 10

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Visualisation of powers of 10 from one to 1 billion. Visualisation 1 billion.svg
Visualisation of powers of 10 from one to 1 billion.

A power of 10 is any of the integer powers of the number ten; in other words, ten 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 ten. The first few non-negative powers of ten are:

Contents

1, 10, 100, 1,000, 10,000, 100,000, 1,000,000, 10,000,000... (sequence A011557 in the OEIS )

Positive powers

In decimal notation the nth power of ten is written as '1' followed by n zeroes. It can also be written as 10n or as 1En in E notation. See order of magnitude and orders of magnitude (numbers) for named powers of ten. There are two conventions for naming positive powers of ten, beginning with 109, called the long and short scales. Where a power of ten has different names in the two conventions, the long scale name is shown in parentheses.

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:

NamePowerNumber SI symbol SI prefix
one 01
ten 110da (D) deca
hundred 2100h (H) hecto
thousand 31,000k (K) kilo
ten thousand (myriad (Greek))410,000
hundred thousand (lakh (India))5100,000
million 61,000,000M mega
ten million (crore (India))710,000,000
hundred million8100,000,000
billion (milliard)91,000,000,000G giga
trillion (billion)121,000,000,000,000T tera
quadrillion (billiard)151,000,000,000,000,000P peta
quintillion (trillion)181,000,000,000,000,000,000E exa
sextillion (trilliard)211,000,000,000,000,000,000,000Z zetta
septillion (quadrillion)241,000,000,000,000,000,000,000,000Y yotta
octillion (quadrilliard)271,000,000,000,000,000,000,000,000,000R ronna
nonillion (quintillion)301,000,000,000,000,000,000,000,000,000,000Q quetta
decillion (quintilliard)331,000,000,000,000,000,000,000,000,000,000,000
undecillion (sextillion)361,000,000,000,000,000,000,000,000,000,000,000,000
duodecillion (sextilliard)391,000,000,000,000,000,000,000,000,000,000,000,000,000
tredecillion (septillion)421,000,000,000,000,000,000,000,000,000,000,000,000,000,000
quattuordecillion (septilliard)451,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
quindecillion (octillion)481,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
sexdecillion (octilliard)511,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
septendecillion (nonillion)541,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
octodecillion (nonilliard)571,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
novemdecillion (decillion)601,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
vigintillion (decilliard)631,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
unvigintillion (undecillion)661,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
duovigintillion (undecilliard)691,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
trevigintillion (duodecillion)721,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
quattuorvigintillion (duodecilliard)751,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
quinvigintillion (tredecillion)781,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
sexvigintillion (tredecilliard)811,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
septenvigintillion (quattuordecillion)841,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
octovigintillion (quattuordecilliard)871,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
novemvigintillion (quindecillion)901,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
trigintillion (quindecilliard)931,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
googol 10010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
centillion 3031,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
googolplex googolone then 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 zeros.
googolplexiangoogolplexone then (one then 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 zeros) zeros

Negative powers

The sequence of powers of ten can also be extended to negative powers.

Similar to the positive powers, the negative power of 10 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:

NamePowerNumber SI symbol SI prefix
one01
tenth−10.1d deci
hundredth −20.01c centi
thousandth −30.001m milli
ten-thousandth (Myriadth)−40.000 1
hundred-thousandth (Lacth)−50.000 01
millionth −60.000 001μ micro
billionth −90.000 000 001n nano
trillionth −120.000 000 000 001p pico
quadrillionth −150.000 000 000 000 001f femto
quintillionth −180.000 000 000 000 000 001a atto
sextillionth−210.000 000 000 000 000 000 001z zepto
septillionth−240.000 000 000 000 000 000 000 001y yocto
octillionth−270.000 000 000 000 000 000 000 000 001r ronto
nonillionth−300.000 000 000 000 000 000 000 000 000 001q quecto
decillionth−330.000 000 000 000 000 000 000 000 000 000 001
undecillionth−360.000 000 000 000 000 000 000 000 000 000 000 001
duodecillionth−390.000 000 000 000 000 000 000 000 000 000 000 000 001
tredecillionth−420.000 000 000 000 000 000 000 000 000 000 000 000 000 001
quattuordecillionth−450.000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
quindecillionth−480.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
sexdecillionth−510.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
septendecillionth−540.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
octodecillionth−570.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
novemdecillionth−600.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
vigintillionth−630.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
unvigintillionth−660.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
duovigintillionth−690.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
trevigintillionth−720.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
quattuorvigintillionth−750.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
quinvigintillionth−780.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
sexvigintillionth−810.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
septenvigintillionth−840.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
octovigintillionth−870.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
novemvigintillionth−900.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
trigintillionth−930.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
googolth−1000.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 1
centillionth−3030.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
googolplexth−googolten to the negative 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000th power
googolplexianth−googolplexten to the negative (ten to the 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000th power) power

Googol and googolplex

The number googol is 10100. The term was coined by 9-year-old Milton Sirotta, nephew of American mathematician Edward Kasner. It was popularized in Kasner's 1940 book Mathematics and the Imagination , where it was used to compare and illustrate very large numbers. Googolplex, a much larger power of ten (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 of 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 mEn, known as E notation , is used in computer programming, spreadsheets and databases, but is not used in scientific papers.

See also

Further reading

Video

Related Research Articles

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A googol is the large number 10100. In decimal notation, it is written as the digit 1 followed by one hundred zeroes: 10,​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. Its systematic name is 10 duotrigintillion. (The short scale names are standard in the English-speaking world.) Its prime factorization is

References

  1. ↑ "Powers of 10". www.mathsteacher.com.au. Retrieved 2020-03-17.
  2. ↑ "Powers of Ten". hesperia.gsfc.nasa.gov. Retrieved 2020-03-17.
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