Order of magnitude

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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 10, 2, 1000, 1024 or e (Euler's number, a transcendental number approximately equal to 2.71828182846 that 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	Order of magnitude
0.001	−3
0.01	−2
0.1	−1
1	0
10	1
100	2
1,000	3
10,000	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, 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 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.

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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, 10¹–10¹⁰, 10¹⁰–10¹⁰⁰, 10¹⁰⁰–10¹⁰⁰⁰, etc.

(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 second gives rise to the categories

negative numbers, 0–1, 1–10, 10

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