Gourt Home::Gourt :: 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 bank balance that is probably orders of magnitude larger than mine.
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 | −3 |
| 0.01 | −2 |
| 0.1 | −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.
Extremely large numbers
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
- ..., 1.023–1.26, 1.26–10, 101–1010, 1010–10100, 10100–101000, etc.
The second gives rise to the categories
- negative numbers, 0–1, 1–10, 10
See also
External links
- 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.
- Orders of Magnitude - Distance
- What is Order of Magnitude?
Größenordnung | Orden de magnitud | Ordre de grandeur | 규모의 비교 | Ordini di grandezza | סדר גודל | Nagyságrend | 数量の比較 | Rząd wielkości | Red velikosti | Suuruusluokka | 数量级
"Order of magnitude"