An order of magnitude is an approximation of the logarithm 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 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 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 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 in "decades" (i.e., factors of ten).^[1] Examples of numbers of different magnitudes can be found at Orders of magnitude (numbers).

Definition

Generally, the order of magnitude of a number is the smallest power of 10 used to represent that number.^[2] To work out the order of magnitude of a number ${\displaystyle N}$, the number is first expressed in the following form:

${\displaystyle N=a\times 10^{b}}$

where ${\displaystyle {\frac {1}{\sqrt {10}}}\leq a<{\sqrt {10}}}$, or approximately ${\displaystyle 0.316\lesssim a\lesssim 3.16}$. Then, ${\displaystyle b}$ 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 ${\displaystyle N}$	Expression in ${\displaystyle N=a\times 10^{b}}$	Order of magnitude ${\displaystyle b}$
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 ${\displaystyle 10^{b-1/2}}$ and ${\displaystyle 10^{b+1/2}}$ is ${\displaystyle 10^{b}}$, meaning that a value of exactly ${\displaystyle 10^{b}}$ (i.e., ${\displaystyle a=1}$) represents a geometric halfway point within the range of possible values of ${\displaystyle a}$.

Some use a simpler definition where ${\displaystyle 0.5\leq a<5}$,^[3] perhaps because the arithmetic mean of ${\displaystyle 10^{b}}$ and ${\displaystyle 10^{b+c}}$ approaches ${\displaystyle 5\times 10^{b+c-1}}$ for increasing ${\displaystyle c}$. This definition has the effect of lowering the values of ${\displaystyle b}$ slightly:

Number ${\displaystyle N}$	Expression in ${\displaystyle N=a\times 10^{b}}$	Order of magnitude ${\displaystyle b}$
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	3.2 × 101	1
999	0.999 × 103	3
1000	1 × 103	3

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. The growing amounts of Internet data have led to addition of new SI prefixes over time, most recently in 2022.^[4]

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 (about 2.512) 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 (100^1/5)⁵ or a factor of 100 times brighter. The modernized version has however turned into a logarithmic scale with non-integer values.

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 1000000. 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 (these make sense in the long scale only), and the suffix -illion tells that the base is 1000000. 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 1000000000000 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 were invented for use in electronic technology.

See also

• Big O notation
• Decibel
• Mathematical operators and symbols in Unicode
• Names of large numbers
• Names of small numbers
• Number sense
• Orders of magnitude (acceleration)
• Orders of magnitude (area)
• Orders of magnitude (bit rate)
• Orders of magnitude (current)
• Orders of magnitude (energy)
• Orders of magnitude (force)
• Orders of magnitude (frequency)
• Orders of magnitude (illuminance)
• Orders of magnitude (length)
• Orders of magnitude (mass)
• Orders of magnitude (numbers)
• Orders of magnitude (power)
• Orders of magnitude (pressure)
• Orders of magnitude (radiation)
• Orders of magnitude (speed)
• Orders of magnitude (temperature)
• Orders of magnitude (time)
• Orders of magnitude (voltage)
• Orders of magnitude (volume)
• Powers of Ten
• Scientific notation
• Unicode symbols for CJK Compatibility includes SI Unit symbols
• Valuation (algebra), an algebraic generalization of "order of magnitude"
• Scale (analytical tool)

References

Further reading

• Asimov, Isaac, The Measure of the Universe (1983).

External links

• The Scale of the Universe 2 Interactive tool from Planck length 10⁻³⁵ meters to universe size 10²⁷
• Cosmos – an Illustrated Dimensional Journey from microcosmos to macrocosmos – from Digital Nature Agency
• Powers of 10, a graphic animated illustration that starts with a view of the Milky Way at 10²³ meters and ends with subatomic particles at 10⁻¹⁶ meters.
• What is Order of Magnitude?