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Orders of magnitude

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. The ratios most commonly used are 1000, 10, 2, 1024 or e (Euler's number, a transcendental number (approximately equal to 2.71828182846) which is used as the base for natural logarithms).

Usually, orders of magnitude refers to a series of powers of ten; this article discusses the decimal scale.

Powers of ten 0.001 0.01 0.1 1 10 100 1,000 10,000
Order of magnitude -3 -2 -1 0 1 2 3 4

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. 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 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 decimal logarithm, usually as the integer part of the logarithm. For example, 4,000,000 has a logarithm of 6.602; its order of magnitude is 6. Thus, the order of magnitude is the approximate position on a logarithmic scale.

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 population of the Earth) is 10 billion. An order of magnitude estimate is sometimes also called a zeroth order approximation.

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 of familiar objects. SI units are used together with SI prefixes, which were devised with orders of magnitude in mind.

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 first gives rise to the categories

..., 1.023-1.26, 1.26-10, 10-1e10, 1e10-1e100, 1e100-1e1000, etc.

(the first two mentioned, and the extension to the left, may not be very useful, the two just demonstrate how the sequence mathematically continues to the left).

The second gives rise to the categories

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 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 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

External links

Last updated: 08-18-2005 09:52:03