Orders of magnitude
area
angular velocity
charge
currency
data
density
energy
frequency
length
magnetic flux density
mass
numbers
power
pressure
specific energy density
specific heat capacity
speed
temperature
time
volume
Conversion of units
physical unit
SI
SI base unit
SI derived unit
SI prefix
Planck units

In
words
DecimalPower
of ten
Order of
magnitude
ten thousandth0. 000110−4−4
thousandth0. 00110−3−3
hundredth0. 0110−2−2
tenth0. 110−1−1
one11000
ten101011
hundred1001022
thousand1,0001033
ten thousand10,0001044
million1,000,0001066
billion1,000,000,0001099
trillion1,000,000,000,000101212

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 significant figures (also called significant digits and abbreviated sig figs) of a number are those digits that carry meaning contributing to its accuracy

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. The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, truncation is the term for limiting the number of digits right of the Decimal point, by discarding the least significant ones 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 106 and 107. In a similar example, "He had a 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. Definition and base Logarithmic scales are either defined for ratios of the underlying quantity or one has to agree to measure

An order of magnitude estimate of a variable whose precise value is unknown is an estimate rounded to the nearest power of ten. For lip-rounding in phonetics see Labialisation and Roundedness. 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. Human beings, humans or man (Origin 1590–1600 L homō man OL hemō the earthly one (see Humus In Biology a population is the collection of inter-breeding organisms of a particular Species; in Sociology EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 In other words; when rounding its logarithm, a number of order of magnitude 10 is in between 109. 5 and 1010. 4. An order of magnitude estimate is sometimes also called a zeroth order approximation. Orders of approximation have been used not only in Science, Engineering, and other quantitative disciplines to make Approximations with various degrees

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. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001

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. The concept of scale is applicable if a system is represented proportionally by another system

## Non-decimal orders of magnitude

Other orders of magnitude may be calculated using bases other than 10. In mathematical numeral systems, the base or radix is usually the number of unique digits, including zero that a positional Numeral The different decimal numeral systems of the world use a larger base to better envision the size of the number, and have created names for the powers of this larger base. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. A numeral system (or system of numeration) is a Mathematical notation for representing numbers of a given set by symbols in a consistent manner Many numeral systems with base 10 use a superimposed larger base of 100 1000 10000 or 1000000 The table shows what number the order of magnitude aim at for base 10 and for base 1,000,000. It can be seen that the order of magnitude is included in the number name in this example, because bi- means 2 and tri- means 3, and the suffix -illion tells that the base is 1,000,000. But the number names billion, trillion themselves (here with other meaning than in the first chapter) are not names of the orders of magnitudes, they are names of "magnitudes", that is the numbers 1,000,000,000,000 etc. The long and short scales are two different numerical systems used throughout the world Short scale is the English translation of the French

order of magnitudeis log10 ofis log1000000 of
1101,000,000million
21001,000,000,000,000trillion
310001,000,000,000,000,000,000quintillion

SI units in the table at right are used together with SI prefixes, which were devised with mainly base 1000 magnitudes in mind. The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base Many numeral systems with base 10 use a superimposed larger base of 100 1000 10000 or 1000000 An SI prefix (also known as a metric prefix) is a name or associated symbol that precedes a unit of measure (or its symbol to form a Decimal multiple or The IEC standard prefixes with base 1024 was invented for use in context of electronic technology. In computing binary prefixes are names or associated symbols that can precede a unit of measure (such as a Byte) to indicate multiplication by a power of two

The ancient apparent magnitudes for the brightness of stars uses the base $\sqrt[5]{100} \approx 2.512$ and is reversed. The apparent magnitude ( m) of a celestial body is a measure of its Brightness as seen by an observer on Earth, normalized to the value The modernized version has however turned into a logarithmic scale with non-integer values.

### Extremely large numbers

For extremely large numbers, a generalized order of magnitude can be based on their double logarithm or super-logarithm. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce In Mathematics, the super-logarithm is one of the two inverse functions of Tetration. 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:

$0-1, 1-10, 10-10^{10}, 10^{10}-10^{10^{10}}, 10^{10^{10}}-10^{10^{10^{10}}}, \dots$, 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. In Mathematics, tetration (also known as hyper -4 076, 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. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which

Similar to the logarithmic scale one can have a double logarithmic scale (example provided here) and super-logarithmic scale. Definition and base Logarithmic scales are either defined for ratios of the underlying quantity or one has to agree to measure This is the timeline of the Universe from Big Bang to Heat Death scenario 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 Mathematics and Statistics, the quasi-arithmetic mean or generalised f -mean is one generalisation of the more familiar Means such 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). The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers