Orders of magnitude

An order of magnitude is the class of bigness, the class of size, the class of magnitude, of any amount, where each separate class contains ten times larger amounts than the one before. It is generally used to make very approximate comparisons. If two numbers differ by one order of magnitude, one is about 10 times larger than the other. In a sense, it is one class bigger. If number A is two orders of magnitude smaller than B, it is about 100 times smaller. Three orders of magnitude would be 1000, four 10,000 etc. If A and B are of the same order of magnitude, their difference is less than ten times. The word order is used in an unusual sense, though very common in various sciences: a class or group, of similar things, here similar amounts.

The order of magnitude of a number is, intuitively speaking, the number of powers of 10 contained in the number, or a close approximation to it. More precisely, the order of magnitude of a number can be defined in terms of the logarithm of the number to the base of 10, usually as the integer part of the logarithm. Thus the order of magnitude of 4,000,000 with a logarithm of 6.602 is 6. Equivalently, this is the exponent of the power of 10 when the number is represented using scientific notation: 4.0E+06.

Alternatively, the logarithm is rounded to the nearest integer, and e.g. 500, is in the same category as 1000.

Thus, the order of magnitude is the approximate position on a logarithmic scale.

An order of magnitude estimate of a variable whose precise value is unknown is an estimate rounded in some way to the nearest power of 10. For example, an accurate order of magnitude estimate for the human population of the Earth in the year 2000 is 10 billion. An order of magnitude estimate is sometimes also called a zeroth order approximation.

One way of categorising things in the physical world is by their size. The pages below contain lists of items that are of the same order of magnitude in time, length, area, volume, mass, energy or temperature. This is useful for getting an intuitive sense of the comparative size of things and the overall scale of the universe. SI units are used together with SI prefixes: these were devised with orders of magnitude in mind. Each individual page also gives other units; see also conversion of units.

Orders of magnitude of various quantities

In the following table the different quantities are lined up so that the following are in the same row:

• length and the approximate time taken by light to cross that length
• area of a square and the length of one side
• volume of a cube and the area of one face
• mass of some water and its volume at 4 degrees Celsius or 277.16 K

See also the separate tables for time, length, area, volume, mass, energy, power, temperature and dimensionless numbers.

* Each time shown is linked to that time. However, the time taken for light to cross the corresponding length is 3 times the time shown.

** These are the standard units but this table uses a variety of units, which can make it harder to read.

Units used in the table

The table uses units and prefixes that are commonly recognized:

• Time:
  • femtosecond (fs)
  • nanosecond (ns)
  • microsecond (μs)
  • millisecond (ms)
  • second (s)
  • hour (h)
  • day (d)
  • year (yr)
• Length:
  • attometre (am)
  • femtometre (fm)
  • picometre (pm)
  • nanometre (nm)
  • micrometre (µm)
  • millimetre (mm)
  • centimetre (cm)
  • metre (m)
  • kilometre (km)
  • astronomical unit (AU)
  • light year (LY)
• Area:
  • square metre (m²)
  • hectare (ha)
  • square kilometre (km²)
• Mass:
  • gram (g)
  • kilogram (kg)
  • tonne (t)
• Volume:
  • millilitre (ml)
  • litre (l)
  • cubic metre (m³)
• Energy:
  • millielectronvolt (meV),
  • electronvolt (eV)
  • megaelectronvolt (MeV)
  • gigaelectronvolt (GeV)
  • teraelectronvolt (TeV)
  • erg
  • joule (J)
  • kilowatthour kWh
  • megawatthour MWh
  • gigawatthour GWh
• Temperature:
  • nanokelvin (nK)
  • microkelvin (µK)
  • millikelvin (mK)
  • kelvin (K)

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, 10-1e10, 1e10-1e100, 1e100-1e1000, etc.

(the first two mentioned, and the extension to the left, may not be very useful, the two just demonstrate how the sequence mathematically continues to the left).

The second gives rise to the categories

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 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 (e.g. 2 and 16 giving 4) does not 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).

See also

• Large numbers
• Orders of approximation
• Orders of magnitude (area)
• Orders of magnitude (data)
• Orders of magnitude (density)
• Orders of magnitude (energy)
• Orders of magnitude (length)
• Orders of magnitude (mass)
• Orders of magnitude (numbers)
• Orders of magnitude (power)
• Orders of magnitude (speed)
• Orders of magnitude (temperature)
• Orders of magnitude (time)
• Orders of magnitude (U.S. money)
• Orders of magnitude (volume)
• Powers of Ten
• SI base units
• SI derived units
• Complete list of SI prefixes
• Timeline of the Big Bang
• Timeline of the Universe

External links

• Powers of 10 (http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html), 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.
• Calculation: Orders of Magnitude (http://www.sengpielaudio.com/ConvPref.htm)
• Orders of Magnitude - Distance (http://www.alcyone.com/max/physics/orders/metre.html)

de:Größenordnung es:Orden de magnitud fr:Ordre de grandeur ja:数量の比較 sl:Red velikosti zh:数量级