Orders Of Magnitude

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An order of magnitude is the class of scale (measurement) or magnitude (mathematics) of any amount, where each class contains values of a fixed Geometric progression to the class preceding it. The ratio most commonly used is 10.

{| align="center" cellpadding="3"!style="border-bottom: black solid thin;"| In
words!style="border-bottom: black solid thin;"| Decimal!style="border-bottom: black solid thin;"| Exponent#Powers of ten
of ten!style="border-bottom: black solid thin;"| Order of
magnitude|-|style="text-align: center;"| ten thousandths
(these terms may be confusive)|style="text-align: center;"| 0.0001|style="text-align: center;"| 10-4|style="text-align: center;"| −4|-|style="text-align: center;"| thousandth|style="text-align: center;"| 0.001|style="text-align: center;"| 10-3|style="text-align: center;"| −3|-|style="text-align: center;"| hundredth|style="text-align: center;"| 0.01|style="text-align: center;"| 10-2|style="text-align: center;"| −2|-|style="text-align: center;"| tenth|style="text-align: center;"| 0.1|style="text-align: center;"| 10-1|style="text-align: center;"| −1|-|style="text-align: center;"| one|style="text-align: center;"| 1|style="text-align: center;"| 100|style="text-align: center;"| 0|-|style="text-align: center;"| ten|style="text-align: center;"| 10|style="text-align: center;"| 101|style="text-align: center;"| 1|-|style="text-align: center;"| hundred|style="text-align: center;"| 100|style="text-align: center;"| 10²|style="text-align: center;"| 2|-|style="text-align: center;"| thousand|style="text-align: center;"| 1,000|style="text-align: center;"| 10³|style="text-align: center;"| 3|-|style="text-align: center;"| ten thousand|style="text-align: center;"| 10,000|style="text-align: center;"| 104|style="text-align: center;"| 4|-|style="text-align: center;"| million|style="text-align: center;"| 1,000,000|style="text-align: center;"| 106|style="text-align: center;"| 6|-|style="text-align: center;"| billion|style="text-align: center;"| 1,000,000,000|style="text-align: center;"| 109|style="text-align: center;"| 9|}

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 (number). 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: 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, usually as the integer part of the logarithm, obtained by truncation. 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 10^{6} and 10^{7}. 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.

An order of magnitude estimate of a variable whose precise value is unknown is an estimate Rounding 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 population of the Earth) is 10 1000000000 (number). In other words; when rounding its logarithm, a number of order of magnitude 10 is in between 10^{9.5} and 10^{10.4}. 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 is 95 times that of Earth, so Saturn is two orders of magnitude more massive than Earth.

The pages in the table at right contain lists of items that are of the same order of magnitude in various units of measurement. This is useful for getting an intuitive sense of the comparative scale (measurement) of familiar objects.

Non-decimal orders of magnitude Other orders of magnitude may be calculated using Radix other than 10. The different decimal numeral systems of the world use a Decimal superbase 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 1000000. 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 1000000. But the number names billion, trillion themselves (here with Long and short scales than in the first chapter) are not names of the orders of magnitudes, they are names of "magnitudes", that is the numbers 1 000000 000000 etc.

{| border="1"! order of magnitude !! is Common logarithm of !! colspan=2 | is Decimal superbase#Mathematical description of|-| 1 || 10 || 1 000000 || million|-| 2 || 100 || 1 000000 000000 || trillion|-| 3 || 1000 || 1 000000 000000 000000 || quintillion|}

SI units in the table at right are used together with SI prefixes, which were devised with mainly base 1000 magnitudes in mind. Binary prefix#IEC standard prefixes with base 1024 was invented for use in context of electronic technology.

The ancient apparent magnitudes for the brightness of stars uses the base \sqrt{100} \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 Logarithm#Generalizations or Tetration#Extension to real numbers. 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: 0-1, 1-10, 10-10^{10}, 10^{10}-10^{10^{10-->, 10^{10^{10-->-10^{10^{10^{10-->}, \dots, or

negative numbers, 0-1, 1-10, 10-1e10, 1e10-10^1e10, 10^1e10-10^^4, 10^^4-10^^5, etc. (see tetration)

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 Large_numbers#Standardized_system_of_writing_very_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 (mathematics) can be considered.

Similar to the logarithmic scale#Graphic representation one can have a double logarithmic scale (example provided Graphical timeline from Big Bang to Heat Death) 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, 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
Powers of Ten
Orders of magnitude (numbers)
Big O notation

External links

Powers of 10, a graphic animated illustration that starts with a view of the Milky Way at 1023 meters and ends with subatomic particles at 10-16 meters.
Orders of Magnitude - Distance
What is Order of Magnitude?

External links

Powers of 10, a graphic animated illustration that starts with a view of the Milky Way at 1023 meters and ends with subatomic particles at 10-16 meters.
Orders of Magnitude - Distance
What is Order of Magnitude?

Order of magnitude - Wikipedia, the free encyclopedia
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it.

Orders of magnitude (numbers) - Wikipedia, the free encyclopedia
This list compares various sizes of positive numbers, including counts of things, dimensionless numbers and probabilities.

EisenBlog - Marc Eisenstadt's Home Page Blog at The Open University's ...
The URL you should use to send a TrackBack ping to this entry is: http://kmi.open.ac.uk/people/marc/2005/06/01/two-orders-of-magnitude-overload-conjecture/trackback/

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