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An order of magnitude is an approximation of the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one.
Logarithmic distribution In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion : -\ln(1-p) = ...
s are common in nature and considering the order of magnitude of values sampled from such a distribution can be more intuitive. When the reference value is 10, the order of magnitude can be understood as the number of digits in the base-10 representation of the value. Similarly, if the reference value is one of some powers of 2, since computers store data in a binary format, the magnitude can be understood in terms of the amount of computer memory needed to store that value. Differences in order of magnitude can be measured on a base-10
logarithmic scale A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
in “ decades” (i.e., factors of ten). Examples of numbers of different magnitudes can be found at
Orders of magnitude (numbers) This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantities and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as a ...
.


Definition

Generally, the order of magnitude of a number is the smallest power of 10 used to represent that number. To work out the order of magnitude of a number N, the number is first expressed in the following form: :N =a\times10^b where \frac\leq a<\sqrt, or approximately 0.316\lesssim a \lesssim 3.16. Then, b represents the order of magnitude of the number. The order of magnitude can be any
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. The table below enumerates the order of magnitude of some numbers in light of this definition: The
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
of 10^ and 10^ is 10^b, meaning that a value of exactly 10^b (i.e., a=1) represents a geometric ''halfway point'' within the range of possible values of a. Some use a simpler definition where 0.5, perhaps because the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
of 10^b and 10^ approaches 5\times10^ for increasing c. This definition has the effect of lowering the values of b slightly: Yet others restrict a to values where 1\leq a<10, making the order of magnitude of a number exactly equal to its exponent part in
scientific notation Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
.


Uses

Orders of magnitude are used to make approximate comparisons. If numbers differ by one order of magnitude, ''x'' is ''about'' ten times different in quantity than ''y''. If values differ by two orders of magnitude, they differ by a factor of about 100. Two numbers of the same order of magnitude have roughly the same scale: the larger value is less than ten times the smaller value. The growing amounts of Internet data have led to addition of new prefixes over time, most recently in 2022.


Calculating the order of magnitude

The order of magnitude of a number is, intuitively speaking, the number of powers of 10 contained in the number. More precisely, the order of magnitude of a number can be defined in terms of the
common logarithm In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered ...
, usually as the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
part of the logarithm, obtained by
truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathb ...
. For example, the number has a logarithm (in base 10) of 6.602; its order of magnitude is 6. When truncating, a number of this order of magnitude is between 106 and 107. In a similar example, with the phrase "He had a seven-figure income", the order of magnitude is the number of figures minus one, so it is very easily determined without a calculator to 6. An order of magnitude is an approximate position on a
logarithmic scale A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
.


Order-of-magnitude estimate

An order-of-magnitude estimate of a variable, whose precise value is unknown, is an estimate rounded to the nearest power of ten. For example, an order-of-magnitude estimate for a variable between about 3 billion and 30 billion (such as the
human Humans (''Homo sapiens'') are the most abundant and widespread species of primate, characterized by bipedalism and exceptional cognitive skills due to a large and complex brain. This has enabled the development of advanced tools, cultu ...
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction usi ...
of the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
) is 10
billion Billion is a word for a large number, and it has two distinct definitions: *1,000,000,000, i.e. one thousand million, or (ten to the ninth power), as defined on the short scale. This is its only current meaning in English. * 1,000,000,000,000, i. ...
. To round a number to its nearest order of magnitude, one rounds its logarithm to the nearest integer. Thus , which has a logarithm (in base 10) of 6.602, has 7 as its nearest order of magnitude, because "nearest" implies rounding rather than truncation. For a number written in scientific notation, this logarithmic rounding scale requires rounding up to the next power of ten when the multiplier is greater than the square root of ten (about 3.162). For example, the nearest order of magnitude for is 8, whereas the nearest order of magnitude for is 9. An order-of-magnitude estimate is sometimes also called a
zeroth order approximation In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is. Usage in science and engineering In formal expressions, the ordinal number used b ...
.


Order of magnitude difference

An order-of-magnitude difference between two values is a factor of 10. For example, the mass of the planet
Saturn Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius of about nine and a half times that of Earth. It has only one-eighth the average density of Earth; h ...
is 95 times that of
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
, so Saturn is ''two orders of magnitude'' more massive than Earth. Order-of-magnitude differences are called decades when measured on a
logarithmic scale A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
.


Non-decimal orders of magnitude

Other orders of magnitude may be calculated using bases other than 10. The ancient Greeks ranked the nighttime brightness of celestial bodies by 6 levels in which each level was the fifth root of one hundred (about 2.512) as bright as the nearest weaker level of brightness, and thus the brightest level being 5 orders of magnitude brighter than the weakest indicates that it is (1001/5)5 or a factor of 100 times brighter. The different
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
numeral systems A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbo ...
of the world use a larger base to better envision the size of the number, and have created names for the powers of this larger base. The table shows what number the order of magnitude aim at for base 10 and for base . It can be seen that the order of magnitude is included in the number name in this example, because bi- means 2 and tri- means 3 (these make sense in the long scale only), and the suffix -illion tells that the base is . But the number names billion, trillion themselves (here with other meaning than in the first chapter) are not names of the ''orders of'' magnitudes, they are names of "magnitudes", that is the ''numbers'' etc. SI units in the table at right are used together with
SI prefix The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
es, which were devised with mainly base 1000 magnitudes in mind. The IEC standard prefixes with base 1024 were invented for use in electronic technology. The ancient
apparent magnitude Apparent magnitude () is a measure of the brightness of a star or other astronomical object observed from Earth. An object's apparent magnitude depends on its intrinsic luminosity, its distance from Earth, and any extinction of the object's ...
s for the brightness of stars uses the base \sqrt \approx 2.512 and is reversed. The modernized version has however turned into a logarithmic scale with non-integer values.


Extremely large numbers

For extremely large numbers, a generalized order of magnitude can be based on their double logarithm or super-logarithm. Rounding these downward to an integer gives categories between very "round numbers", rounding them to the nearest integer and applying the inverse function gives the "nearest" round number. The double logarithm yields the categories: : ..., 1.0023–1.023, 1.023–1.26, 1.26–10, 10–1010, 1010–10100, 10100–10, ... (the first two mentioned, and the extension to the left, may not be very useful, they merely demonstrate how the sequence mathematically continues to the left). The super-logarithm yields the categories: :0–1, 1–10, 10–1010, 1010–101010, 101010–10101010, ... or :0–010, 010–110, 110–210, 210–310, 310–410, ... The "midpoints" which determine which round number is nearer are in the first case: :1.076, 2.071, 1453, , ,... and, depending on the interpolation method, in the second case :−0.301, 0.5, 3.162, , , (10 \uparrow)^1 10^, (10 \uparrow)^2 10^,... (see notation of extremely large numbers) For extremely small numbers (in the sense of close to zero) neither method is suitable directly, but the generalized order of magnitude of the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
can be considered. Similar to the
logarithmic scale A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
one can have a double logarithmic scale (example provided
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a ...
) and super-logarithmic scale. The intervals above all have the same length on them, with the "midpoints" actually midway. More generally, a point midway between two points corresponds to the generalised ''f''-mean with ''f''(''x'') the corresponding function log log ''x'' or slog ''x''. In the case of log log ''x'', this mean of two numbers (e.g. 2 and 16 giving 4) does not depend on the base of the logarithm, just like in the case of log ''x'' (
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
, 2 and 8 giving 4), but unlike in the case of log log log ''x'' (4 and giving 16 if the base is 2, but not otherwise).


See also

*
Big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund L ...
*
Decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a ...
*
Mathematical operators and symbols in Unicode The Unicode Standard encodes almost all standard characters used in mathematics. Unicode Technical Report #25 provides comprehensive information about the character repertoire, their properties, and guidelines for implementation. Mathematical op ...
* Names of large numbers * Names of small numbers *
Number sense In psychology, number sense is the term used for the hypothesis that some animals, particularly humans, have a biologically determined ability that allows them to represent and manipulate large numerical quantities. The term was popularized by Sta ...
*
Orders of magnitude (acceleration) This page lists examples of the acceleration occurring in various situations. They are grouped by orders of magnitude. See also *G-force *Gravitational acceleration *Mechanical shock *Standard gravity * International System of Units (SI) *SI pref ...
* Orders of magnitude (area) * Orders of magnitude (current) *
Orders of magnitude (energy) This list compares various energies in joules (J), organized by order of magnitude. Below 1 J 1 to 105 J 106 to 1011 J 1012 to 1017 J 1018 to 1023 J Over 1023 J } , - , 1050 , , ≳1050 J , Upper limit of 'apparent'/isotropic energy ...
* Orders of magnitude (force) *
Orders of magnitude (frequency) The following list illustrates various frequencies, measured in hertz, according to decade in the order of their magnitudes, with the negative decades illustrated by events and positive decades by acoustic or electromagnetic uses. See also *Hert ...
*
Orders of magnitude (length) The following are examples of orders of magnitude for different lengths. __TOC__ Overview Detailed list To help compare different orders of magnitude, the following list describes various lengths between 1.6 \times 10^ metres and 10 ...
*
Orders of magnitude (mass) To help compare different orders of magnitude, the following lists describe various mass levels between 10−59  kg and 1052 kg. The least massive thing listed here is a graviton, and the most massive thing is the observable universe. ...
*
Orders of magnitude (numbers) This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantities and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as a ...
* Orders of magnitude (pressure) *
Orders of magnitude (radiation) Recognized effects of higher acute radiation doses are described in more detail in the article on radiation poisoning. Although the International System of Units (SI) defines the sievert (Sv) as the unit of radiation dose equivalent, chronic radi ...
*
Orders of magnitude (speed) To help compare different orders of magnitude, the following list describes various speed levels between approximately 2.2  m/s and 3.0 m/s (the speed of light). Values in bold are exact. List of orders of magnitude for speed See als ...
*
Orders of magnitude (temperature) List of orders of magnitude for temperature Detailed list for 100 K to 1000 K Most ordinary human activity takes place at temperatures of this order of magnitude. Circumstances where water naturally occurs in liquid form are shown in light gr ...
* Orders of magnitude (time) *
Orders of magnitude (voltage) To help compare different orders of magnitude, the following list describes various voltage levels. SI multiple Notes External links {{Orders of magnitude Voltage Voltage, also known as electric pressure, electric tension, or (electr ...
*
Orders of magnitude (volume) The table lists various objects and units by the order of magnitude of their volume. Sub-microscopic Microscopic Human measures Terrestrial Astronomical References {{DEFAULTSORT:Orders Of Magnitude (Volume) Vo ...
* ''
Powers of Ten 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 fi ...
'' *
Scientific notation Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
* Unicode symbols for CJK Compatibility includes SI Unit symbols * Valuation (algebra), an algebraic generalization of "order of magnitude" * Scale (analytical tool)


References


Further reading

* Asimov, Isaac, ''The Measure of the Universe'' (1983).


External links


The Scale of the Universe 2
Interactive tool from Planck length 10−35 meters to universe size 1027
Cosmos – an Illustrated Dimensional Journey from microcosmos to macrocosmos
– from Digital Nature Agency

a graphic animated illustration that starts with a view of the
Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye. ...
at 1023 meters and ends with
subatomic particle In physical sciences, a subatomic particle is a particle that composes an atom. According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a p ...
s at 10−16 meters.
What is Order of Magnitude?
{{Authority control Orders of magnitude Elementary mathematics Logarithmic scales of measurement