Power of 10

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:

1, 10, 100, 1,000, 10,000, 100,000, 1,000,000, 10,000,000. ... (sequence A011557 in the OEIS)
Visualisation of powers of 10 from one to 1 billion.

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:

  • billion = 10[(2 + 1) × 3] = 109
  • octillion = 10[(8 + 1) × 3] = 1027
NamePowerNumberSI symbolSI prefix
one01
ten110da (D)deca
hundred2100h (H)hecto
thousand31,000k (K)kilo
ten thousand (myriad (Greek))410,000
hundred thousand (lakh (India))5100,000
million61,000,000Mmega
ten million (crore (India)) 7 10,000,000
hundred million 8 100,000,000
billion (milliard)91,000,000,000Ggiga
trillion (billion)121,000,000,000,000Ttera
quadrillion (billiard)151,000,000,000,000,000Ppeta
quintillion (trillion)181,000,000,000,000,000,000Eexa
sextillion (trilliard)211,000,000,000,000,000,000,000Zzetta
septillion (quadrillion)241,000,000,000,000,000,000,000,000Yyotta
octillion (quadrilliard)271,000,000,000,000,000,000,000,000,000Rronna
nonillion (quintillion)301,000,000,000,000,000,000,000,000,000,000Qquetta
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
googol10010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
centillion3031,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
googolplex googol 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.
googolplexian googolplex one 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:

  • billionth = 10[(2 + 1) × 3] = 109
  • quintillionth = 10[(5 + 1) × 3] = 1018
NamePowerNumberSI symbolSI prefix
one01
tenth10.1ddeci
hundredth20.01ccenti
thousandth30.001mmilli
ten-thousandth (Myriadth)40.000 1
hundred-thousandth (Lacth)50.000 01
millionth60.000 001μmicro
billionth90.000 000 001nnano
trillionth120.000 000 000 001ppico
quadrillionth150.000 000 000 000 001ffemto
quintillionth180.000 000 000 000 000 001aatto
sextillionth210.000 000 000 000 000 000 001zzepto
septillionth240.000 000 000 000 000 000 000 001yyocto
octillionth270.000 000 000 000 000 000 000 000 001rronto
nonillionth300.000 000 000 000 000 000 000 000 000 001qquecto
decillionth330.000 000 000 000 000 000 000 000 000 000 001
undecillionth360.000 000 000 000 000 000 000 000 000 000 000 001
duodecillionth390.000 000 000 000 000 000 000 000 000 000 000 000 001
tredecillionth420.000 000 000 000 000 000 000 000 000 000 000 000 000 001
quattuordecillionth450.000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
quindecillionth480.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
sexdecillionth510.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
septendecillionth540.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
octodecillionth570.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
novemdecillionth600.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
vigintillionth630.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
unvigintillionth660.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
duovigintillionth690.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
trevigintillionth720.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
quattuorvigintillionth750.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
quinvigintillionth780.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
sexvigintillionth810.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
septenvigintillionth840.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
octovigintillionth870.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
novemvigintillionth900.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
trigintillionth930.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
googolth1000.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 1
centillionth3030.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001
googolplexthgoogolten 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 -googolplex ten 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)th power

Googol

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. (Read below)

Googolplex

The number googolplex is 10googol, or 1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, and was also made by Edward Kasner's nephew. (Read above)

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
  • Powers of Ten (1977). Nine-minute film. US Public Broadcasting Service (PBS), made by Charles and Ray Eames. "An adventure in magnitudes. Starting at a picnic by the lakeside in Chicago, this film transports the viewer to the outer edges of the universe. Every ten seconds we view the starting point from ten times farther out until our own galaxy is visible only as a speck of light among many others. Returning to Earth with breathtaking speed, we move inward - into the hand of the sleeping picnicker - with ten times more magnification every two seconds. Our journey ends inside a proton of a carbon atom within a DNA molecule in a white blood cell."

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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