Power of ten

Powers of ten , also numbered levels , are powers with the base 10 and an integer exponent. An important application of powers of ten is the exponential notation of floating point numbers .

The powers of ten

Some powers of ten have their own number names :

Surname	Number in exponential notation	Number in decimal notation
Quadrillionth	${\ displaystyle 10 ^ {- 24}}$	${\ displaystyle 0 {,} 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 001}$
Trillionths	${\ displaystyle 10 ^ {- 21}}$	${\ displaystyle 0 {,} 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 001}$
Trillionth	${\ displaystyle 10 ^ {- 18}}$	${\ displaystyle 0 {,} 000 \, 000 \, 000 \, 000 \, 000 \, 001}$
Billiardstel	${\ displaystyle 10 ^ {- 15}}$	${\ displaystyle 0 {,} 000 \, 000 \, 000 \, 000 \, 001}$
Trillionth	${\ displaystyle 10 ^ {- 12}}$	${\ displaystyle 0 {,} 000 \, 000 \, 000 \, 001}$
billionth	${\ displaystyle 10 ^ {- 9}}$	${\ displaystyle 0 {,} 000 \, 000 \, 001}$
Millionth	${\ displaystyle 10 ^ {- 6}}$	${\ displaystyle 0 {,} 000 \, 001}$
Hundred thousandths	${\ displaystyle 10 ^ {- 5}}$	${\ displaystyle 0 {,} 000 \, 01}$
Ten thousandths	${\ displaystyle 10 ^ {- 4}}$	${\ displaystyle 0 {,} 000 \, 1}$
Thousandths	${\ displaystyle 10 ^ {- 3}}$	${\ displaystyle 0 {,} 001}$
Hundredths	${\ displaystyle 10 ^ {- 2}}$	${\ displaystyle 0 {,} 01}$
tenth	${\ displaystyle 10 ^ {- 1}}$	${\ displaystyle 0 {,} 1}$
one	${\ displaystyle 10 ^ {0}}$	${\ displaystyle 1}$
ten	${\ displaystyle 10 ^ {1}}$	${\ displaystyle 10}$
Hundred	${\ displaystyle 10 ^ {2}}$	${\ displaystyle 100}$
thousand	${\ displaystyle 10 ^ {3}}$	${\ displaystyle 1 \, 000}$
Ten thousand	${\ displaystyle 10 ^ {4}}$	${\ displaystyle 10 \, 000}$
A hundred thousand	${\ displaystyle 10 ^ {5}}$	${\ displaystyle 100 \, 000}$
million	${\ displaystyle 10 ^ {6}}$	${\ displaystyle 1 \, 000 \, 000}$
billion	${\ displaystyle 10 ^ {9}}$	${\ displaystyle 1 \, 000 \, 000 \, 000}$
trillion	${\ displaystyle 10 ^ {12}}$	${\ displaystyle 1 \, 000 \, 000 \, 000 \, 000}$
Billiards	${\ displaystyle 10 ^ {15}}$	${\ displaystyle 1 \, 000 \, 000 \, 000 \, 000 \, 000}$
Trillion	${\ displaystyle 10 ^ {18}}$	${\ displaystyle 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000}$
Trillion	${\ displaystyle 10 ^ {21}}$	${\ displaystyle 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000}$
Quadrillion	${\ displaystyle 10 ^ {24}}$	${\ displaystyle 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000}$

The exponent indicates the number of zeros in the decimal notation of the power of ten. If it says 10 to the sixth (10 ⁶ ), then it's a million.

use

Hertzsprung-Russell diagram. Abscissa: color index of stars, ordinate: luminosity (sun = 1) in powers of ten

Since our usual number system , the decimal system (from Latin decimus "the tenth"), is based on powers of ten, they are suitable for writing even very large and very small numbers compactly. They form the basis of the scientific notation ( SCI for short for scientific ), which can reduce the numbers to the presented mantissa and exponent of base 10.

As examples:

{\ displaystyle 4 {,} 711 \ cdot 10 ^ {3} = 4711}

{\ displaystyle 4 {,} 711 \ cdot 10 ^ {6} = 4 \, 711 \, 000}

{\ displaystyle 0 {,} 000000004711 = 4 {,} 711 \ cdot 10 ^ {- 9}}

The axes of diagrams are also often not divided linearly, but rather into powers of ten as a logarithmic scale when very large value ranges are involved. An example from astronomy - which is associated with “astronomically large numbers” - is the Hertzsprung-Russell diagram , which in the adjacent picture shows a scale from 0.00001 to 100,000 units of the solar luminosity L ₀ on the ordinate . But then z. B. in the middle between 10 and 100 L ₀ not the value 20 or 50, but 31.63 L ₀ . This value is not the arithmetic but the geometric mean of 10 and 100, i.e. H. the root of 1000 or 10 ^1.5 .

Calculating with powers of ten

There are some simple arithmetic rules for calculating with powers of ten:

If you multiply a natural number by a step number, the zeros of the step number are appended to the natural number.
If you multiply a decimal fraction by a number of steps, the decimal point moves to the right by as many places as the number of steps has zeros.
If you divide a decimal fraction by a step number, the comma moves as many places to the left as the step number has zeros.

Powers of ten and SI prefixes

The International System of Units defines appropriate prefixes for units of measurement :

SI prefixes
Surname	Deka	Hecto	kilo	Mega	Giga	Tera	Peta	Exa	Zetta	Yotta
symbol	there	H	k	M.	G	T	P	E.	Z	Y
factor	10 ¹	10 ²	10 ³	10 ⁶	10 ⁹	10 ¹²	10 ¹⁵	10 ¹⁸	10 ²¹	10 ²⁴
Surname	Deci	Centi	Milli	Micro	Nano	Pico	Femto	Atto	Zepto	Yocto
symbol	d	c	m	µ	n	p	f	a	z	y
factor	10 ⁻¹	10 ⁻²	10 ⁻³	10 ⁻⁶	10 ⁻⁹	10 ⁻¹²	10 ^-15	10 ⁻¹⁸	10 ⁻²¹	10 ⁻²⁴

Cultural reception

Ten high

Web links

Wiktionary: power of ten - explanations of meanings, word origins, synonyms, translations