Order of magnitude

An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually ten, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic distributions are common in nature and
considering the order of magnitude of values sampled from such a distribution can be more intuitive. When the reference value is ten, the order of magnitude can be understood as the number of digits in the base-10 representation of the value. Similarly, if the reference value is one of certain powers of two, the magnitude can be understood as the amount of computer memory needed to store the exact integer value.
Differences in order of magnitude can be measured on a base-10 logarithmic scale in “decades”. Examples of numbers of different magnitudes can be found at Orders of magnitude.

Definition

Generally, the order of magnitude of a number is the smallest power of 10 used to represent that number. To work out the order of magnitude of a number, the number is first expressed in the following form:
where. Then, represents the order of magnitude of the number. The order of magnitude can be any integer. The table below enumerates the order of magnitude of some numbers in light of this definition:
 Number Expression in Order of magnitude 0.2 2 × 10−1 −1 1 1 × 100 0 5 0.5 × 101 1 6 0.6 × 101 1 31 3.1 × 101 1 32 0.32 × 102 2 999 0.999 × 103 3 1000 1 × 103 3

The geometric mean of and is, meaning that a value of exactly represents a geometric "halfway point" within the range of possible values of.
Some use a simpler definition where, perhaps because the arithmetic mean of and approaches for increasing. This definition has the effect of lowering the values of slightly:
 Number Expression in Order of magnitude 0.2 2 × 10−1 −1 1 1 × 100 0 5 5 × 100 0 6 0.6 × 101 1 31 3.1 × 101 1 32 3.2 × 101 1 999 0.999 × 103 3 1000 1 × 103 3

Yet others restrict to values where, making the order of magnitude of a number exactly equal to its exponent part in scientific notation.

Uses

Orders of magnitude are used to make approximate comparisons. If numbers differ by one order of magnitude, x is about ten times different in quantity than y. If values 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.
 In words In words Prefix Decimal Powerof ten Order ofmagnitude quadrillionth septillionth yocto- 10−24 −24 trilliardth sextillionth zepto- 10−21 −21 trillionth quintillionth atto- 10−18 −18 billiardth quadrillionth femto- 10−15 −15 billionth trillionth pico- 10−12 −12 milliardth billionth nano- 10−9 −9 millionth millionth micro- 10−6 −6 thousandth thousandth milli- 0.001 10−3 −3 hundredth hundredth centi- 0.01 10−2 −2 tenth tenth deci- 0.1 10−1 −1 one one 1 100 0 ten ten deca- U+3372 10 101 1 hundred hundred hecto- 100 102 2 thousand thousand kilo- 103 3 million million mega- 106 6 milliard billion giga- 109 9 billion trillion tera- 1012 12 billiard quadrillion peta- 1015 15 trillion quintillion exa- 1018 18 trilliard sextillion zetta- 1021 21 quadrillion septillion yotta- 1024 24 In words In words Prefix Decimal Powerof ten Order ofmagnitude

Calculating the order of magnitude

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, the number 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, 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 6. An order of magnitude is an approximate position on a logarithmic scale.

Order-of-magnitude estimate

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 is 10 billion. To round a number to its nearest order of magnitude, one rounds its logarithm to the nearest integer. Thus, which has a logarithm 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. For example, the nearest order of magnitude for is 8, whereas the nearest order of magnitude for is 9. An order-of-magnitude estimate is sometimes also called a zeroth order approximation.

Order of magnitude difference

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.

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 the fifth root of one hundred as bright as the nearest weaker level of brightness, and thus the brightest level being 5 orders of magnitude brighter than the weakest indicates that it is 5 or a factor of 100 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. 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. But the number names billion, trillion themselves are not names of the orders of magnitudes, they are names of "magnitudes", that is the numbers etc.
 Order of magnitude Is log10 of Is log of Short scale Long scale 1 million million 2 trillion billion 3 quintillion trillion

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 were invented for use in electronic technology.
The ancient apparent magnitudes 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, 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:
.
The super-logarithm yields the categories:
The "midpoints" which determine which round number is nearer are in the first case:
and, depending on the interpolation method, in the second case
For extremely small numbers neither method is suitable directly, but the generalized order of magnitude of the reciprocal can be considered.
Similar to the logarithmic scale one can have a double logarithmic scale 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 the corresponding function log log x or slog x. In the case of log log x, this mean of two numbers does not depend on the base of the logarithm, just like in the case of log x, but unlike in the case of log log log x.