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Encyclopedia > Orders of magnitude
Orders of magnitude
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Usually, orders of magnitude refers to a series of powers of ten; this article discusses the decimal scale. In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... It has been suggested that this article or section be merged with decimal representation. ...

 Powers of ten Order of magnitude 0.001 0.01 0.1 1 10 100 1,000 10,000 -3 -2 -1 0 1 2 3 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. 100 (one hundred) (the Roman numeral is C for centum) is the natural number following 99 and preceding 101. ...

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. Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1. ... The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ... A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. ...

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. [[{{{diversity_link}}}|Diversity]] {{{diversity}}} Binomial name Homo sapiens Linnaeus, 1758 Trinomial name {{{trinomial}}} Type Species {{{type_species}}} Subspecies Homo sapiens idaltu (extinct) Homo sapiens sapiens [[Image:{{{range_map}}}|{{{range_map_width}}}|]] Synonyms {{{synonyms}}} Homo (genus). ... Earth, also known as Terra, and Tellus mostly in the 19th century, is the third-closest planet to the Sun. ... The word billion and its equivalents in other languages refer to one of two different numbers, depending on whether the writer is using the long or short scale. ... Often in science, engineering, or other quantitative disciplines, it is necessary to make approximations with various degrees of precision. ...

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. // Introduction The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. ... A scale is either a device used for measurement of weights, or a series of ratios against which different measurements can be compared. ... The International System of Units (abbreviated SI from the French language name SystÃ¨me International dUnitÃ©s) is the modern form of the metric system. ... An SI prefix is a prefix that can be applied to an SI unit to form a decimal multiple or submultiple. ...

Extremely large numbers GA_googleFillSlot("encyclopedia_square");

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. Large numbers are numbers that are large compared with the numbers used in everyday life. ... Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1. ... Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ...

The first gives rise to the categories

..., 1.023-1.26, 1.26-10, 101 − 1010, 1010 − 10100, 10100 − 101000, 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). Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ...

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.) Large numbers are numbers that are large compared with the numbers used in everyday life. ...

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). A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. ... In mathematics and statistics, the generalised f-mean is the natural generalisation of the more familar means such as the arithmetic mean and the geometric mean, using a function f(x). ... The geometric mean of a set of positive data is defined as the product of all the members of the set, raised to a power equal to the reciprocal of the number of members. ...

Often in science, engineering, or other quantitative disciplines, it is necessary to make approximations with various degrees of precision. ... Powers of Ten is a 1977 short documentary film which depicts the relative scale of the Universe in factors of ten. ... This list compares various sizes of positive numbers, including counts of things, dimensionless numbers and probabilities. ... Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ... Results from FactBites:

 Orders of magnitude - Free Encyclopedia (296 words) An order of magnitude is a factor of ten. Orders of magnitude are quite easily and commonly described through the use of scientific notation and powers of ten. An order of magnitude estimate is not concerned with the precise value, but instead with the number of decimal digits after the first.
More results at FactBites »

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