# Power of 10

A power of 10 is a number formed by multiplying 10 by itself a whole number of times, written as \( 10^{n} \), where the exponent n counts the multiplications; negative exponents give fractions, so \( 10^{-3} \) is one thousandth. Powers of 10 are the backbone of the decimal place-value system, scientific notation, the SI metric prefixes, and logarithmic scales.

| Key fact | Detail |
|---|---|
| Definition | \( 10^{n} \) means n factors of 10 multiplied together; \( 10^{0} = 1 \), and negative n gives reciprocals such as \( 10^{-2} = 0.01 \)<sup>[1](https://www.open.edu/openlearn/mod/oucontent/view.php?id=98572&section=_unit6.1.1)</sup> |
| SI prefix range | Decimal multiples and sub-multiples of SI units now run from \( 10^{30} \) (quetta, Q) to \( 10^{-30} \) (quecto, q), after four new prefixes were added in 2022<sup>[2](https://www.nist.gov/pml/special-publication-330/sp-330-section-3)</sup><sup> • </sup><sup>[3](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes)</sup>  |
| Strictly decimal | SI prefixes refer strictly to powers of 10; one kilobit is 1000 bits, not 1024 bits, and binary prefixes such as kibi (Ki, \( 2^{10} \)) are recommended for powers of 2<sup>[2](https://www.nist.gov/pml/special-publication-330/sp-330-section-3)</sup> |
| Scientific notation | A number is written as a decimal fraction, usually between 1 and 10, times a power of ten<sup>[4](https://science.nasa.gov/universe/glossary/)</sup> |
| Order of magnitude | The power of 10 in scientific notation; quantities sharing it are of the same order of magnitude, and rounding to the nearest power of ten rounds up when the mantissa is 5 or more<sup>[5](https://openstax.org/books/college-physics/pages/1-2-physical-quantities-and-units)</sup><sup> • </sup><sup>[6](https://www.open.edu/openlearn/mod/oucontent/view.php?id=98792&section=_unit3.3)</sup>  |
| Large-number names | Names for numbers \( 10^{9} \) and larger are not uniform worldwide (in many countries 1 billion = \( 10^{12} \)), so powers of 10 are the preferred way to express large numbers<sup>[7](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-7-rules-and-style-conventions-expressing-values)</sup> |
| Cultural landmark | *Powers of Ten*, a 9-minute film by Charles and Ray Eames (1968, rereleased 1977), steps through the universe one power of ten at a time and was added to the US National Film Registry in 1998<sup>[8](https://eamesfoundation.org/work/powers-of-ten-and-the-relative-size-of-things-in-the-universe/)</sup> |

## Arithmetic and notation

The rules follow directly from the meaning of exponents. Multiplying powers adds exponents, \( 10^{6} \times 10^{9} = 10^{15} \), and dividing subtracts them. This is why arithmetic with extreme numbers becomes simple once numbers are written in power form: one million times one billion is \( 10^{6} \times 10^{9} = 10^{15} \), found by adding 6 and 9.<sup>[9](https://link.springer.com/article/10.1007/s10763-025-10609-z)</sup>

**Scientific notation** expresses any number as \( x \times 10^{y} \), with x carrying the significant digits and y fixing the scale. A negative y indicates a fraction of 1: \( 4.5 \times 10^{-6} \) stands for 0.0000045.<sup>[10](https://assets.openstax.org/oscms-prodcms/media/documents/Physics-WEB_Sab7RrQ.pdf)</sup> The exponent also carries meaning for precision. In \( l = 1200 \) m it is impossible to tell whether the trailing zeroes are significant, but writing \( l = 1.200 \) km, using the kilo prefix for \( 10^{3} \), makes the two zeroes significant by construction.<sup>[7](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-7-rules-and-style-conventions-expressing-values)</sup>

## The SI prefixes

The [International System of Units](https://www.edgepedia.org/international-system-of-units) attaches named prefixes to units to produce multiples and sub-multiples at powers of ten. The full set now spans thirty orders of magnitude in each direction: deka (\( 10^{1} \)), hecto (\( 10^{2} \)), and kilo (\( 10^{3} \)) upward through mega, giga, tera, peta, exa, zetta, and yotta (\( 10^{24} \)) to ronna (\( 10^{27} \)) and quetta (\( 10^{30} \)); downward, deci, centi, and milli run through micro, nano, pico, femto, atto, zepto, and yocto (\( 10^{-24} \)) to ronto (\( 10^{-27} \)) and quecto (\( 10^{-30} \)).<sup>[2](https://www.nist.gov/pml/special-publication-330/sp-330-section-3)</sup><sup> • </sup><sup>[3](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes)</sup>  The four largest and four smallest were added in 2022 to align with version 2.01 of the BIPM SI Brochure, 9th edition.<sup>[2](https://www.nist.gov/pml/special-publication-330/sp-330-section-3)</sup>

Several conventions govern their use. A prefix symbol attaches to a unit symbol without a space and forms a new inseparable unit symbol, so \( 2.3 \ \mathrm{cm^{3}} = 2.3 \times 10^{-6} \ \mathrm{m^{3}} \); compound prefixes formed by juxtaposing two prefix symbols are not permitted, so one writes µV, not mmV.<sup>[2](https://www.nist.gov/pml/special-publication-330/sp-330-section-3)</sup><sup> • </sup><sup>[11](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication1038.pdf)</sup>  Prefix symbols cannot stand alone or attach to the symbol 1, the unit one, so particularly large or small pure numbers are expressed as powers of 10 instead.<sup>[12](https://www.nist.gov/pml/special-publication-330/sp-330-section-5)</sup> The kilogram is the only coherent SI unit whose name already contains a prefix, so multiples of mass are formed on the gram: \( 10^{-6} \) kg is written milligram, not microkilogram.<sup>[2](https://www.nist.gov/pml/special-publication-330/sp-330-section-3)</sup>

For readability, NIST recommends choosing prefixes so that numerical values fall between 0.1 and 1000 and using only prefixes whose exponents are multiples of 3, though this is not mandatory; 2703 W may be written 2.703 kW.<sup>[7](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-7-rules-and-style-conventions-expressing-values)</sup>

## Powers of 2 and the binary prefixes

Because SI prefixes mean powers of 10 strictly, they must not be used for powers of 2: kilo cannot represent \( 2^{10} = 1024 \), mega cannot represent \( 2^{20} = 1\,048\,576 \), and giga cannot represent \( 2^{30} = 1\,073\,741\,824 \).<sup>[3](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes)</sup> For binary multiples the recommended prefixes are kibi (Ki, \( 2^{10} \)), mebi (Mi, \( 2^{20} \)), gibi (Gi, \( 2^{30} \)), tebi (Ti, \( 2^{40} \)), pebi (Pi, \( 2^{50} \)), exbi (Ei, \( 2^{60} \)), zebi (Zi, \( 2^{70} \)), and yobi (Yi, \( 2^{80} \)).<sup>[2](https://www.nist.gov/pml/special-publication-330/sp-330-section-3)</sup>

## Logarithmic scales and orders of magnitude

A logarithmic scale is built directly on powers of ten: each step up or down the scale multiplies or divides the quantity by a fixed power of ten, so the intervals between divisions differ, unlike a linear scale where each step adds a fixed amount.<sup>[13](https://www.open.edu/openlearn/mod/oucontent/view.php?id=68401&section=4)</sup> On a base-10 log scale the marks label 1, 10, 100, 1000, and so on, covering a far larger range of values than the corresponding linear scale.<sup>[10](https://assets.openstax.org/oscms-prodcms/media/documents/Physics-WEB_Sab7RrQ.pdf)</sup><sup> • </sup><sup>[14](https://www.graphpad.com/support/faq/why-are-negative-values-and-zero-missing-from-graphs-using-a-logarithmic-axis/)</sup> Because the logarithm is not defined for zero or negative values, a logarithmic axis can display only positive numbers.<sup>[14](https://www.graphpad.com/support/faq/why-are-negative-values-and-zero-missing-from-graphs-using-a-logarithmic-axis/)</sup>

The order of magnitude of a quantity is the nearest power of ten to its value. Writing the quantity as \( a \times 10^{n} \), the order of magnitude is \( 10^{n} \) if a is less than 5 and \( 10^{n+1} \) if a is 5 or greater; the diameter of Mars, 6762 km, written as \( 6.762 \times 10^{3} \) km, is therefore said to be of order \( 10^{4} \) km.<sup>[6](https://www.open.edu/openlearn/mod/oucontent/view.php?id=98792&section=_unit3.3)</sup> Two quantities differing by one order of magnitude differ by a factor of ten; three orders of magnitude means a factor of one thousand.<sup>[15](https://pressbooks.cuny.edu/astrobiology/back-matter/appendix/)</sup> Logarithmic scales built on powers of ten appear wherever quantities span enormous ranges: the decibel scale for sound, in which every 10 dB increase corresponds to a tenfold increase in intensity, is a standard example.<sup>[13](https://www.open.edu/openlearn/mod/oucontent/view.php?id=68401&section=4)</sup>

## History of base ten

The decimal place-value system, in which each digit position carries a power of ten, developed in India, where a decimal scale came to be associated with the values of the places of numbers arranged left to right or right to left; the system was transmitted to the Arabs and later into Europe.<sup>[16](https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/)</sup> Early Indian fascination with large numbers produced named ranks for successive powers of ten: in the *Lalitavistara*, an account of the life of Gautama Buddha, the young Gautama, examined on mathematics, lists the powers of 10 up to \( 10^{53} \) and then continues to \( 10^{421} \).<sup>[16](https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/)</sup> The essential features of the resulting notation are only ten symbols however large the number, the use of zero to indicate the absence of a value, and the principle of local value, by which a symbol represents different numbers according to its position.<sup>[17](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Arithmetic)</sup>

## Powers of Ten in teaching and film

The 1968 documentary short *Powers of Ten*, written and directed by Ray Eames with her husband Charles Eames and rereleased in updated form in 1977, adapts Kees Boeke's 1957 book *Cosmic View*. It presents a linear view of the universe from the human scale to the farthest galaxies and then down to the nucleus of a carbon atom, showing what it means to add another zero to any number; the 1977 version runs 9 minutes, with music by [Elmer Bernstein](https://www.edgepedia.org/elmer-bernstein).<sup>[8](https://eamesfoundation.org/work/powers-of-ten-and-the-relative-size-of-things-in-the-universe/)</sup> In 1998 the Library of Congress selected it for the United States National Film Registry as culturally, historically, or aesthetically significant.<sup>[8](https://eamesfoundation.org/work/powers-of-ten-and-the-relative-size-of-things-in-the-universe/)</sup> The film has been used to teach scale: one study found that students' ability to place sizes on a number line using powers-of-ten notation improved by 20 percent in the frequency of correctly labeled items after viewing.<sup>[9](https://link.springer.com/article/10.1007/s10763-025-10609-z)</sup>

More recently, the Powers of the Universe module, a 6-hour progression of lessons trialed with students aged 7 to 13, teaches arithmetic with extreme numbers through powers-of-ten notation, using an activity book in which each page is a power of ten numbered from −37 to 90, with people on page zero, atoms on page −10, and the observable Universe on page 26; across three trials the average retention rate of the tested material was 62.5 percent.<sup>[18](https://arxiv.org/abs/2311.18192)</sup><sup> • </sup><sup>[9](https://link.springer.com/article/10.1007/s10763-025-10609-z)</sup>

## References

1. [Succeed with maths Part 2, 1.1, The Open University](https://www.open.edu/openlearn/mod/oucontent/view.php?id=98572&section=_unit6.1.1)
2. [SP 330 Section 3: Decimal multiples and sub-multiples of SI units, NIST](https://www.nist.gov/pml/special-publication-330/sp-330-section-3)
3. [NIST Guide to the SI, Chapter 4: The Two Classes of SI Units and the SI Prefixes, NIST](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes)
4. [Universe glossary, NASA Science](https://science.nasa.gov/universe/glossary/)
5. [1.2 Physical Quantities and Units, College Physics, OpenStax](https://openstax.org/books/college-physics/pages/1-2-physical-quantities-and-units)
6. [Mathematics for science and technology: 2 Scales of measurement, The Open University (OpenLearn)](https://www.open.edu/openlearn/mod/oucontent/view.php?id=98792&section=_unit3.3)
7. [NIST Guide to the SI, Chapter 7: Rules and Style Conventions for Expressing Values of Quantities, NIST](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-7-rules-and-style-conventions-expressing-values)
8. [Powers of Ten and the Relative Size of Things in the Universe, Eames Foundation](https://eamesfoundation.org/work/powers-of-ten-and-the-relative-size-of-things-in-the-universe/)
9. [Powers of the Universe: Empowering Primary School Students with the Powers of Ten Notation, International Journal of Science and Mathematics Education](https://link.springer.com/article/10.1007/s10763-025-10609-z)
10. [Chapter 1, What is Physics?, OpenStax Physics](https://assets.openstax.org/oscms-prodcms/media/documents/Physics-WEB_Sab7RrQ.pdf)
11. [NIST Special Publication 1038, The International System of Units (SI), NIST](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication1038.pdf)
12. [SP 330 Section 5: Rules and style conventions, NIST](https://www.nist.gov/pml/special-publication-330/sp-330-section-5)
13. [Mathematics for science and technology, Week 2: The logarithmic scale, The Open University](https://www.open.edu/openlearn/mod/oucontent/view.php?id=68401&section=4)
14. [Why are negative values (and zero) missing from graphs using a logarithmic axis?, GraphPad FAQ 2056](https://www.graphpad.com/support/faq/why-are-negative-values-and-zero-missing-from-graphs-using-a-logarithmic-axis/)
15. [Appendix A: Orders of Magnitude and a New Way to Count, Astrobiology, CUNY Pressbooks](https://pressbooks.cuny.edu/astrobiology/back-matter/appendix/)
16. [Indian numerals, MacTutor History of Mathematics, University of St Andrews](https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/)
17. [Arithmetic, 1911 Encyclopædia Britannica, Wikisource](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Arithmetic)
18. [Lonshakova, Blair, Treagust, Zadnik (2023). Powers of the Universe: Empowering primary school students with the powers of ten notation. arXiv.](https://arxiv.org/abs/2311.18192)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Integer sequences and partitions › Special and named integers › Powers and perfect powers*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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