Power of 10

This article is about the mathematical concept. For other uses, see Power of 10 (disambiguation).

In mathematics, 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 trillion.

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

Name	Alternate names	Power	Number	SI symbol	SI prefix
one		0	1
ten		1	10	da	deca
hundred		2	100	h	hecto
thousand		3	1,000	k	kilo
ten thousand	myriad (Greek)	4	10,000
hundred thousand	lakh (India)	5	100,000
million		6	1,000,000	M	mega
ten million	crore (India)	7	10,000,000
hundred million		8	100,000,000
billion	milliard (LS)	9	1,000,000,000	G	giga
trillion	billion (LS)	12	1,000,000,000,000	T	tera
quadrillion	billiard (LS)	15	...	P	peta
quintillion	trillion (LS)	18		E	exa
sextillion	trilliard (LS)	21		Z	zetta
septillion	quadrillion (LS)	24		Y	yotta
octillion	quadrilliard (LS)	27		R	ronna
nonillion	quintillion (LS)	30		Q	quetta
decillion	quintilliard (LS)	33
googol		100
centillion		303
googolplex		10100

For further examples, see Names of large numbers. Numbers larger than about a trillion are rarely referred to by name or written out as digits, but instead are typically described with exponent notation.

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]} = 10⁻⁹
• quintillionth = 10^{−[(5 + 1) × 3]} = 10⁻¹⁸

Name	Power	Number	SI symbol	SI prefix
one	0	1
tenth	−1	0.1	d	deci
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	−12	...	p	pico
quadrillionth	−15		f	femto
quintillionth	−18		a	atto
sextillionth	−21		z	zepto
septillionth	−24		y	yocto
octillionth	−27		r	ronto
nonillionth	−30		q	quecto
decillionth	−33

Googol and googolplex

Main articles: 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

Main article: Scientific notation

Scientific notation is a way of writing numbers of very large and very small sizes compactly.

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

• Power of two
• Power of three
• SI prefix
• Cosmic View, inspiration for the film Powers of Ten
• Exponentiation
• Philip and Phylis Morrison wrote a book called "Powers of Ten: A Book About the Relative Size of Things in the Universe and the Effect of Adding Another Zero" to accompany the video of Eames.

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.