An order of magnitude is an approximation of the
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one.
Logarithmic distribution
In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion
:
-\ln(1-p) = p ...
s 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 10, 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 some powers of 2, since computers store data in a
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...
format, the magnitude can be understood in terms of the amount of computer memory needed to store that value.
Differences in order of magnitude can be
measured on a base-10
logarithmic scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
in “
decades
A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years.
Usage
Any period of ten years is a "decade". For example, the statement that "du ...
” (i.e., factors of ten). Examples of numbers of different magnitudes can be found at
Orders of magnitude (numbers)
This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantities and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as a ...
.
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
, or approximately
. Then,
represents the order of magnitude of the number. The order of magnitude can be any
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
. The table below enumerates the order of magnitude of some numbers in light of this definition:
The
geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
of
and
is
, meaning that a value of exactly
(i.e.,
) represents a geometric ''halfway point'' within the range of possible values of
.
Some use a simpler definition where