Everything about Orders Of Magnitude totally explained
An
order of magnitude is the class of
scale or
magnitude of any amount, where each class contains values of a fixed
ratio to the class preceding it. The ratio most commonly used is 10.
In
words |
Decimal |
Power
of ten |
Order of
magnitude |
| ten thousandth |
0.0001 |
10-4 |
−4 |
| thousandth |
0.001 |
10-3 |
−3 |
| hundredth |
0.01 |
10-2 |
−2 |
| tenth |
0.1 |
10-1 |
−1 |
| one |
1 |
100 |
0 |
| ten |
10 |
101 |
1 |
| hundred |
100 |
102 |
2 |
| thousand |
1,000 |
103 |
3 |
| ten thousand |
10,000 |
104 |
4 |
| million |
1,000,000 |
106 |
6 |
| billion |
1,000,000,000 |
109 |
9 |
| trillion |
1,000,000,000,000 |
1012 |
12 |
Orders of magnitude are generally used to make very approximate comparisons. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they 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. This is the reasoning behind
significant figures: the amount rounded by is usually a few orders of magnitude less than the total, and therefore insignificant.
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, 4,000,000 has a logarithm of 6.602; its order of magnitude is 6. When truncating, a number of this order of magnitude is between
, or
» negative numbers, 0-1, 1-10, 10-1e10, 1e10-10^1e10, 10^1e10-10^^4, 10^^4-10^^5, etc. (see
tetration)
The "midpoints" which determine which round number is nearer are in the first case:
» 1.076, 2.071, 1453, 4.20e31, 1.69e316,...
and, depending on the interpolation method, in the second case
» -.301, .5, 3.162, 1453, 1e1453, 10^1e1453, 10^^2@1e1453,... (see
notation of extremely large numbers)
For extremely small numbers (in the sense of close to zero) neither method is suitable directly, but of course the generalized order of magnitude of the
reciprocal can be considered.
Similar to the
logarithmic scale one can have a double logarithmic scale (example provided
here) 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(x) the corresponding function log log x or slog x. In the case of log log x, this mean of two numbers (for example 2 and 16 giving 4) doesn't depend on the base of the logarithm, just like in the case of log x (
geometric mean, 2 and 8 giving 4), but unlike in the case of log log log x (4 and 65536 giving 16 if the base is 2, but different otherwise).
Further Information
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