Order of magnitude

# Order of magnitude

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An order of magnitude is the class of scale or magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as 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. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...

to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the (base 10) exponent being applied to this amount (therefore, to be an order of magnitude greater is to be 10 times as large). Such differences in order of magnitude can be measured on the logarithmic scale
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

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 electronic systems, such as audio amplifiers and filters.-Calculations:The factor-of-ten in a decade can be...

" (i.e. factors of ten).

It is common among scientists and technologists to say that a parameter whose value is not accurately known, or is within a range, is "of the order of" some value. For example, standby electrical power used in a household is not accurately known and varies between households, but is typically of the order of a few tens of watts.

## Use

Orders of magnitude are generally used to make very approximate comparisons, and reflect very large differences. 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.-In mathematics:One hundred is the square of 10...

. 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 digits that carry meaning contributing to its precision. 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 formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

part of the logarithm, obtained by truncation
Truncation
In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...

. For example, the number 4,000,000 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 be 6. An order of magnitude is an approximate position on a logarithmic scale
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

.

An order-of-magnitude estimate of a variable whose precise value is unknown is an estimate rounded
Rounding
Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing \$23.4476 with \$23.45, or the fraction 312/937 with 1/3, or the expression √2 with 1.414.Rounding is often done on purpose to...

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
Humans are the only living species in the Homo genus...

population
Population
A population is all the organisms that both belong to the same group or species and live in the same geographical area. The area that is used to define a sexual population is such that inter-breeding is possible between any pair within the area and more probable than cross-breeding with individuals...

of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

) 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....

. To round a number to its nearest order of magnitude, one rounds its logarithm to the nearest integer. Thus 4,000,000, 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 1.7 × 108 is 8, whereas the nearest order of magnitude for 3.7 × 108 is 9. 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 is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...

is 95 times than of Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

, so Saturn is two orders of magnitude more massive than Earth. Order-of-magnitude differences are called decades
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 electronic systems, such as audio amplifiers and filters.-Calculations:The factor-of-ten in a decade can be...

when measured on a logarithmic scale
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

.
In words
(long scale
Long and short scales
The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...

)
In words
(short scale
Long and short scales
The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...

)
Prefix Symbol Decimal Power
of ten
Order of
magnitude
quadrillionth septillionth yocto- y 0.000.000.000.000.000.000.000.001 10−24 −24
trilliardth sextillionth zepto- z 0.000.000.000.000.000.000.001 10−21 −21
trillionth quintillionth atto- a 0.000.000.000.000.000.001 10−18 −18
billiardth quadrillionth femto- f 0.000.000.000.000.001 10−15 −15
billionth trillionth pico- p 0.000.000.000.001 10−12 −12
milliardth billionth nano- n 0.000.000.001 10−9 −9
millionth millionth micro- µ 0.000.001 10−6 −6
thousandth thousandth milli- m 0.001 10−3 −3
hundredth hundredth centi- c 0.01 10−2 −2
tenth tenth deci- d 0.1 10−1 −1
one one 1 100 0
ten ten deca- da 10 101 1
hundred hundred hecto- h 100 102 2
thousand thousand kilo- k 1,000 103 3
million million mega- M 1,000,000 106 6
milliard billion giga- G 1,000,000,000 109 9
billion trillion tera- T 1,000,000,000,000 1012 12
billiard quadrillion peta- P 1,000,000,000,000,000 1015 15
trillion quintillion exa- E 1,000,000,000,000,000,000 1018 18
trilliard sextillion zetta- Z 1,000,000,000,000,000,000,000 1021 21
quadrillion septillion yotta- Y 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
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...

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, so that the brightest level is 5 orders of magnitude brighter than the weakest, which can also be stated as a factor of 100 times brighter.

The different decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

numeral systems 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 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 (these make sense in the long scale only), 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 of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...

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 of short scale long scale
1 10 1,000,000 million million
2 100 1,000,000,000,000 trillion billion
3 1000 1,000,000,000,000,000,000 quintillion trillion

SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units in the table at right are used together with SI prefix
SI prefix
The International System of Units specifies a set of unit prefixes known as SI prefixes or metric prefixes. An SI prefix is a name that precedes a basic unit of measure to indicate a decadic multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol...

es, which were devised with mainly base 1000 magnitudes in mind. The IEC standard prefixes 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 measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the 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
This article is about large numbers in the sense of 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 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, roots and logarithms, 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, 1e10–101e10, 101e10410, 410–510, etc. (see tetration
Tetration
In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...

)

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, , ,... (see notation of extremely large numbers)

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 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 scenario. The different Eras of the Universe are shown.Usually the logarithmic scale is used for such timelines but it compresses the most interesting Stelliferous Era too much as this example shows. Therefore a double-logarithmic...

) 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, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, 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, otherwise).

• Orders of approximation
Orders of approximation
In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth order approximation, a first order...

• Big O notation
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...

• Decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

• Number sense
Number sense
In mathematics education, number sense can refer to "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations." Many other definitions exist, but are similar to the one given...

• Names of large numbers
Names of large numbers
This article lists and discusses the usage and derivation of names of large numbers, together with their possible extensions.The following table lists those names of large numbers which are found in many English dictionaries and thus have a special claim to being "real words"...

• Names of small numbers
Names of small numbers
This article lists and discusses the usage and derivation of names of small numbers.-Table of names:The following table lists English language names of small numbers used in the long and short scales, along with the power of ten, engineering notation, and International System of Units symbols and...