Order of magnitude
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 ratios most commonly used are
10,
2,
1000,
1024 or
e (
Euler's number, a
transcendental number approximately equal to 2.71828182846 that is used as the base for
natural logarithms).
Colloquially, the phrase
orders of magnitude is usually used to describe a value that is many times larger than the value to which it is being compared. For example:
Bill Gates has a
net worth that is
orders of magnitude larger than the average person.
Usually, orders of magnitude refers to a series of
powers of ten; this article discusses the
decimal scale.
Powers
of ten | Order of
magnitude | | 0.001 | âˆ'3 |
| 0.01 | âˆ'2 |
| 0.1 | âˆ'1 |
| 1 | 0 |
| 10 | 1 |
| 100 | 2 |
| 1,000 | 3 |
| 10,000 | 4 |
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.
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 decimal
logarithm, usually as the
integer part of the logarithm. For example, 4,000,000 has a logarithm of 6.602; its order of magnitude is 6. Thus, an order of magnitude is an approximate position on a
logarithmic scale.
An
order of magnitude estimate of a variable whose precise value is unknown is an estimate rounded to the nearest power of ten. For example, an order of magnitude estimate for a variable between about 3 billion and 30 billion (such as the
human population of the
Earth) is 10
billion. An order of magnitude estimate is sometimes also called a
zeroth order approximation.
An
order of magnitude difference between two values is a factor of 10. For example, the mass of the planet
Saturn is 95 times that of
Earth, so Saturn is
two orders of magnitude more massive than Earth.
The pages in the table at right contain lists of items that are of the same order of magnitude in various
units of measurement. This is useful for getting an intuitive sense of the comparative
scale of familiar objects.
SI units are used together with
SI prefixes, which were devised with orders of magnitude in mind.
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 first gives rise to the categories:..., 1.023â€"1.26, 1.26â€"10, 10
1â€"10
10, 10
10â€"10
100, 10
100â€"10
1000, etc.(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 second gives rise to the categories:negative numbers, 0â€"1, 1â€"10, 10
*
Orders of approximation*
Powers of Ten*
Orders of magnitude (numbers)*
Big O notation*
Powers of 10, a graphic animated illustration that starts with a view of the
Milky Way at 10
23 meters and ends with
subatomic particles at 10
-16 meters.
*
Orders of Magnitude - Distance*
What is Order of Magnitude?
