Order of magnitude is a concept used to discuss the scale of numbers in relation to one another.
Differences in order of magnitude can be measured on a base-10 logarithmic scale in "decades" (i.e., factors of ten).
Each division or multiplication by 10 is called an order of magnitude.
Below are examples of different methods of partitioning the real numbers into specific "orders of magnitude" for various purposes.
The table below enumerates the order of magnitude of some numbers using this definition: The geometric mean of
) represents a geometric halfway point within the range of possible values of
slightly: Orders of magnitude are used to make approximate comparisons.
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 growing amounts of Internet data have led to addition of new SI prefixes over time, most recently in 2022.
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.
[contradictory] For example, the number 4000000 has a logarithm (in base 10) of 6.602; its order of magnitude is 6.
In a similar example, with the phrase "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.
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 (about 3.162).
An order-of-magnitude estimate is sometimes also called a zeroth order approximation.
Similarly, if the reference value is one of some powers of 2 since computers store data in a binary format, the magnitude can be understood in terms of the amount of computer memory needed to store that value.
Other orders of magnitude may be calculated using bases other than integers.
In the field of astronomy, the nighttime brightnesses of celestial bodies are ranked by "magnitudes" in which each increasing level is brighter by a factor of
times brighter: that is, two base 10 orders of magnitude.
This series of magnitudes forms a logarithmic scale with a base of
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.
(these make sense in the long scale only), and the suffix -illion tells that the base is 1000000.
The IEC standard prefixes with base 1024 were invented for use in electronic technology.