Order 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 ratio most commonly used is 10.

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. 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 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.

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. 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 is 95 times that of Earth, so Saturn is two orders of magnitude more massive than Earth. Order of magnitude differences are called decades when measured on a logarithmic scale).

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.

Non-decimal orders of magnitude
Other orders of magnitude may be calculated using bases 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 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, and the suffix -illion tells that the base is 1,000,000. But the number names billion, trillion themselves (here with other meaning 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.

SI units in the table at right are used together with SI prefixes, 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 magnitudes for the brightness of stars uses the base $$\sqrt[5]{100} \approx 2.512$$ 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 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 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 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) 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).