Order of magnitude

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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.

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 ${\displaystyle c}$.^{[citation needed]} 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

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.^{[contradictory]} For example, the number 4000000 has a logarithm (in base 10) 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 "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.

Order-of-magnitude estimate

An order-of-magnitude estimate of a variable, whose precise value is unknown, is an estimate

Order of magnitude difference

An order-of-magnitude difference between two values is a factor of 10.

Non-decimal orders of magnitude

Other orders of magnitude may be calculated using

The different

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, 10–10¹⁰, 10¹⁰–10¹⁰¹⁰, 10¹⁰¹⁰–10¹⁰¹⁰¹⁰, ... or

0–⁰10, ⁰10–¹10, ¹10–²10, ²10–³10, ³10–⁴10, ...

The "midpoints" which determine which round number is nearer are in the first case:

1.076, 2.071, 1453, 4.20×10³¹, 1.69×10³¹⁶,...

and, depending on the interpolation method, in the second case

−0.301, 0.5, 3.162, 1453, 1×10¹⁴⁵³, ${\displaystyle (10\uparrow )^{1}10^{1453}}$, ${\displaystyle (10\uparrow )^{2}10^{1453}}$,... (see notation of extremely large numbers)

For extremely small numbers (in the sense of close to zero) neither method is suitable directly, but the generalized order of magnitude of the

Similar to the

• 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 (data)
• 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