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An order of magnitude is the class of scale or magnitude
Magnitude (mathematics)

The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
 of any amount, where each class contains values of a fixed ratio
Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio....
 to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount. Differences in order of magnitude are measured on the logarithmic scale
Logarithmic scale

A logarithmic scale is a scale that uses the logarithm of a physical quantity instead of the quantity itself.Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values – the logarithm reduces this to a more manageable range....
 in "factors of ten" or decades
Decade (log scale)

One decade is a factor of 10 difference between two numbers measured on a logarithmic scale. It is especially useful when referring to frequencies and when describing frequency response of electronics, such as audio amplifiers and electronic filter....
 (meaning "power of ten", not "10 years"). The entries in the table at right lead to lists of items that are of the same order of magnitude in various units of measurement
Units of measurement

The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day....
.






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An order of magnitude is the class of scale or magnitude
Magnitude (mathematics)

The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
 of any amount, where each class contains values of a fixed ratio
Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio....
 to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount. Differences in order of magnitude are measured on the logarithmic scale
Logarithmic scale

A logarithmic scale is a scale that uses the logarithm of a physical quantity instead of the quantity itself.Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values – the logarithm reduces this to a more manageable range....
 in "factors of ten" or decades
Decade (log scale)

One decade is a factor of 10 difference between two numbers measured on a logarithmic scale. It is especially useful when referring to frequencies and when describing frequency response of electronics, such as audio amplifiers and electronic filter....
 (meaning "power of ten", not "10 years"). The entries in the table at right lead to lists of items that are of the same order of magnitude in various units of measurement
Units of measurement

The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day....
. This is useful for getting an intuitive sense of the comparative scale of familiar objects.

Orders of magnitude are generally used to make very approximate comparisons. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they differ by two orders of magnitude, they differ by a factor of about 100
100 (number)

100 is the natural number following 99 and preceding 101 ....
. Two numbers of the same order of magnitude have roughly the same scale: the larger value is less than ten times the smaller value. This is the reasoning behind significant figures
Significant figures

The significant figures of a number are those Numerical digit that carry meaning contributing to its accuracy . This includes all digits except:...
: the amount rounded by is usually a few orders of magnitude less than the total, and therefore insignificant.

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
Common logarithm

The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L ....
, usually as the integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 part of the logarithm, obtained by truncation
Truncation

In mathematics, truncation is the term for limiting the number of numerical digits right of the decimal point, by discarding the least significant ones....
. For example, 4,000,000 has a logarithm 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, "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 be 6. An order of magnitude is an approximate position on a logarithmic scale
Logarithmic scale

A logarithmic scale is a scale that uses the logarithm of a physical quantity instead of the quantity itself.Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values – the logarithm reduces this to a more manageable range....
.

An order of magnitude estimate of a variable whose precise value is unknown is an estimate rounded
Rounding

Rounding involves reducing the number of significant digits in a number. The result of rounding is a "shorter" number having fewer non-zero digits yet similar in magnitude....
 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
Human

A human being, also human or man, is a member of a species of bipedalism primates in the family Hominidae . Mitochondrial DNA evidence indicates that modern humans originated in east Africa about 200,000 years ago....
 population
Population

File:Population density.pngIn biology, a population is the collection of inter-breeding organisms of a particular species; in sociology, a collection of human beings....
 of the Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
) is 10 billion
1000000000 (number)

1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001.In scientific notation, it is written as 109....
. In other words; when rounding its logarithm, a number of order of magnitude 10 is in between 109.5 and 1010.4. An order of magnitude estimate is sometimes also called a zeroth order approximation.

An order of magnitude difference between two values is a factor of 10. For example, the mass of the planet Saturn
Saturn

Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn, along with Jupiter, Uranus and Neptune, is classified as a gas giant....
 is 95 times that of Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
, so Saturn is two orders of magnitude more massive than Earth. Order of magnitude differences are called decades
Decade (log scale)

One decade is a factor of 10 difference between two numbers measured on a logarithmic scale. It is especially useful when referring to frequencies and when describing frequency response of electronics, such as audio amplifiers and electronic filter....
 when measured on a logarithmic scale
Logarithmic scale

A logarithmic scale is a scale that uses the logarithm of a physical quantity instead of the quantity itself.Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values – the logarithm reduces this to a more manageable range....
.

In
words (long scale)		 In
words (short scale)		 Prefix Decimal Power
of ten		 Order of
magnitude
quadrillionth septillionth yocto- 0.000000000000000000000001 10-24 -24
trilliardth sextillionth zepto- 0.000000000000000000001 10-21 -21
trillionth quintillionth atto- 0.000000000000000001 10-18 -18
billiardth quadrillionth femto- 0.000000000000001 10-15 -15
billionth trillionth pico- 0.000000000001 10-12 -12
milliardth billionth nano- 0.000000001 10-9 -9
millionth millionth micro- 0.000001 10-6 -6
ten thousandth ten thousandth - 0.0001 10-4 -4
thousandth thousandth milli- 0.001 10-3 -3
hundredth hundredth centi- 0.01 10-2 -2
tenth tenth deci- 0.1 10-1 -1
one one - 1 100 0
ten ten deca- 10 101 1
hundred hundred hecto- 100 102 2
thousand thousand kilo- 1,000 103 3
ten thousand ten thousand - 10,000 104 4
million million mega- 1,000,000 106 6
milliard billion giga- 1,000,000,000 109 9
billion trillion tera- 1,000,000,000,000 1012 12
billiard quadrillion peta- 1,000,000,000,000,000 1015 15
trillion quintillion exa- 1,000,000,000,000,000,000 1018 18
trilliard sextillion zetta- 1,000,000,000,000,000,000,000 1021 21
quadrillion septillion yotta- 1,000,000,000,000,000,000,000,000 1024 24

Non-decimal orders of magnitude

Other orders of magnitude may be calculated using bases
Radix

In numeral system, the base or radix is usually the number of unique Numerical digit, including zero, that a Positional notation numeral system uses to represent numbers....
 other than 10. The ancient Greeks ranked the nighttime brightness of celestial bodies by 6 levels in which each level was twice as bright as the nearest weaker level of brightness, so that the brightest level is 5 orders of magnitude brighter than the weakest, which can also be stated as a factor of 32 times brighter.

The different decimal
Decimal

The decimal numeral system has 10 as its Base . It is the most widely used numeral system....
 numeral systems of the world use a larger base
Decimal superbase

Many numeral systems with base 10 use a superimposed larger base of 100, 1000, 10000 or 1000000. It is a power of 10 and might be called a superbase or superradix of the numeral system....
 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 1,000,000. 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, and the suffix -illion tells that the base is 1,000,000. But the number names billion, trillion themselves (here with other meaning
Long and short scales

The long and short scales are two different numerical systems used throughout the world:Note that the difference between the two scales grows as numbers get larger....
 than in the first chapter) are not names of the orders of magnitudes, they are names of "magnitudes", that is the numbers 1,000,000,000,000 etc.

order of magnitude is log10
Common logarithm

The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L ....
 of 		is log1000000
Decimal superbase

Many numeral systems with base 10 use a superimposed larger base of 100, 1000, 10000 or 1000000. It is a power of 10 and might be called a superbase or superradix of the numeral system....
 of
1 10 1,000,000 million
2 100 1,000,000,000,000 trillion
3 1000 1,000,000,000,000,000,000 quintillion


SI
Si

Si, si, or SI may refer to :...
 units in the table at right are used together with SI prefix
SI prefix

An SI prefix is a name or associated symbol that precedes a basic unit of measure to form a decimal multiple . The abbreviation SI is from the French language name Syst?me International d?Unit?s ....
es, which were devised with mainly base 1000 magnitudes in mind. The IEC standard prefixes
Binary prefix

In computing, a binary prefix is a set of letters that precede a unit of measure to indicate multiplication by a power of two. In certain contexts in computing, such as computer memory sizes, units of information storage and communication traffic have traditionally been reported in multiples of powers of two....
 with base 1024 was invented for use in context of electronic technology.

The ancient apparent magnitude
Apparent magnitude

The apparent magnitude of a celestial body is a measurement of its brightness as seen by an observer on Earth, normalized to the value it would have in the absence of the Earth's atmosphere....
s for the brightness of stars uses the base and is reversed. The modernized version has however turned into a logarithmic scale with non-integer values.

Extremely large numbers

For extremely large numbers
Large numbers

Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions....
, a generalized order of magnitude can be based on their double logarithm
Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
 or super-logarithm
Super-logarithm

In mathematics, the super-logarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions: Nth root and logarithms, likewise tetration has two inverse functions: super-roots and super-logarithms....
. 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, 10100–101000, ...
(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:
, or


negative numbers, 0–1, 1–10, 10–1e10, 1e10–10^1e10, 10^1e10–10^^4, 10^^4–10^^5, etc. (see tetration
Tetration

In mathematics, tetration is an iterated function exponential function, the first hyper operator after exponentiation. The portmanteau tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration....
)


The "midpoints" which determine which round number is nearer are in the first case:
1.076, 2.071, 1453, 4.20e31, 1.69e316,...
and, depending on the interpolation method, in the second case
-.301, .5, 3.162, 1453, 1e1453, 10^1e1453, 10^^2@1e1453,... (see notation of extremely large numbers
Large numbers

Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions....
)


For extremely small numbers (in the sense of close to zero) neither method is suitable directly, but of course the generalized order of magnitude of the reciprocal can be considered.

Similar to the logarithmic scale
Logarithmic scale

A logarithmic scale is a scale that uses the logarithm of a physical quantity instead of the quantity itself.Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values – the logarithm reduces this to a more manageable range....
 one can have a double logarithmic scale (example provided here
Graphical timeline from Big Bang to Heat Death

This is the timeline of the Universe from Big Bang to Heat death of the universe scenario. The Five Ages of the Universe are shown.Usually the Logarithmic timeline is used for such timelines but it compresses the most interesting Stelliferous Era too much as Future of an expanding universe#Graphical timeline example shows....
) 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
Geometric mean

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average," except that instead of adding the set of numbers and then dividing the sum by the count of numbers in the...
, 2 and 8 giving 4), but unlike in the case of log log log x (4 and 65536 giving 16 if the base is 2, but different otherwise).

See also


  • Orders of approximation
    Orders of approximation

    Orders of approximation have been used not only in science, engineering, and other quantitative disciplines to make approximations with various degrees of precision but also more generally, and more loosely, to indicate relative precision outside these disciplines in the form of "first level", "second level" and so on, "approximations"....
  • Powers of Ten
    Powers of Ten

    Powers of Ten is a 1977 short documentary film written and directed by Ray Eames and her husband, Charles Eames. The film depicts the relative Scale of the Universe in factors of ten ....
  • Orders of magnitude (length)
    Orders of magnitude (length)

    To help compare different orders of magnitude, the following list describes various lengths between 1.6 m and 1.3 m.|}Detailed List...
  • Orders of magnitude (mass)
    Orders of magnitude (mass)

    To help compare different Order of magnitude, the following list describes various mass levels between 10−36 kilogram and 1053 kg....
  • Orders of magnitude (numbers)
    Orders of magnitude (numbers)

    This list compares various sizes of positive numbers, including counts of things, dimensionless quantity and probability. Each number is given a name in the so called Long and short scales which is used in English speaking countries, as well as a name in the Long and short scales which is used in a series of countries that do not have English as th...
  • Big O notation
    Big O notation

    In mathematics, big O notation describes the asymptotic analysis of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions....
  • Decibel
    Decibel

    The decibel is a logarithmic units of measurement that expresses the magnitude of a physical quantity relative to a specified or implied reference level....
  • Logarithmic scale
    Logarithmic scale

    A logarithmic scale is a scale that uses the logarithm of a physical quantity instead of the quantity itself.Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values – the logarithm reduces this to a more manageable range....


External links

  • , a graphic animated illustration that starts with a view of the Milky Way
    Milky Way

    The Milky Way, sometimes called simply the Galaxy, is the galaxy in which the Solar System is located. It is a barred spiral galaxy that is part of the Local Group of galaxies....
     at 1023 meters and ends with subatomic particle
    Subatomic particle

    A subatomic particle is an elementary particle or composite particle particle smaller than an atom. Particle physics and nuclear physics are concerned with the study of these particles, their interactions, and non-atomic QCD matter....
    s at 10-16 meters.