In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.

The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.

While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers.

Notably, for skewed distributions, the arithmetic mean may not accord with one's notion of " middle ", and robust statistics such as the median may be a better description of central tendency.

Then the arithmetic mean is defined via the equation

The arithmetic mean of a variable is often denoted by a bar, for example ( read " x bar ") would be the mean of some sample space.

The arithmetic mean has several properties that make it useful, especially as a measure of central tendency.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

* For a normal distribution, the arithmetic mean is equal to both the median and the mode, other measures of central tendency.

In mathematics, the arithmetic – geometric mean ( AGM ) of two positive real numbers and is defined as follows:

First compute the arithmetic mean of and and call it.

These two sequences converge to the same number, which is the arithmetic – geometric mean of and ; it is denoted by, or sometimes by.

To find the arithmetic – geometric mean of and, first calculate their arithmetic mean and geometric mean, thus:

An " elementary " proof can be given using the fact that geometric mean of positive numbers is less than arithmetic mean

A key problem in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values.

The prehistory of arithmetic is limited to a small number of artifacts which may indicate conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20, 000 and 18, 000 BC although its interpretation is disputed.

If the addition operation produces a result too large for the CPU to handle, an arithmetic overflow flag in a flags register may also be set.

As a general rule of thumb, if the condition number, then you may lose up to digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods.

Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations ( addition, subtraction, multiplication, division and exponentiation ).

Sometimes " long double " is used for this in the C language family ( the C99 and C11 standards " IEC 60559 floating-point arithmetic extension-Annex F " recommend the 80-bit extended format to be provided as " long double " when available ), though " long double " may be a synonym for " double " or may stand for quadruple precision.

As it tends strongly toward the least elements of the list, it may ( compared to the arithmetic mean ) mitigate the influence of large outliers and increase the influence of small values.

Logic circuits include such devices as multiplexers, registers, arithmetic logic units ( ALUs ), and computer memory, all the way up through complete microprocessors, which may contain more than 100 million gates.

There are three main classes of instruction: arithmetic, logical, and move ; conditional jump ; conditional skip ( which may have side effects ).

Wittgenstein did, however, concede that Principia may nonetheless make some aspects of everyday arithmetic clearer.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements ( so an analogue of the fundamental theorem of arithmetic holds ); any two elements of a PID have a greatest common divisor ( although it may not be possible to find it using the Euclidean algorithm ).

In modern applications of the liberal arts as curriculum in colleges or universities, the quadrivium may be considered as the study of number and its relationship to physical space or time: arithmetic was pure number, geometry was number in space, music number in time, and astronomy number in space and time.

Note that arithmetic left shift may cause an overflow ; this is the only way it differs from logical shift | logical left shift.

In the RGB color space, brightness can be thought of as the arithmetic mean μ of the red, green, and blue color coordinates ( although some of the three components make the light seem brighter than others, which, again, may be compensated by some display systems automatically ):

Some structures are better for fixed-point arithmetic and others may be better for floating-point arithmetic.

Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic.

It provides arithmetic and logic operations on 64-bit integer numbers ( the software may choose to instead perform two 32-bit, four 16-bit or eight 8-bit operations in a single instruction ).

For all of the above reasons, the following theorem, referred to as the Lasker – Noether theorem, may be seen as a certain generalization of the fundamental theorem of arithmetic:

The most common statistic is the arithmetic mean, but depending on the nature of the data other types of central tendency may be more appropriate.

: You may have been looking for arithmetic, a branch of mathematics.

The proof of the independence result also shows that a wide class of mathematical statements, including all statements that can be phrased in the language of Peano arithmetic, are provable in ZF if and only if they are provable in ZFC.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

Because and can be very similar numbers, the precision of the result can be much less than the inherent precision of the floating-point arithmetic used to perform the computation.

These include all quadratic surds, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots.

* Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of nth roots ( such as the roots of ).

The Analytical Engine incorporated an arithmetic logic unit, control flow in the form of conditional branching and loops, and integrated memory, making it the first design for a general-purpose computer that could be described in modern terms as Turing-complete.

An arithmetical unit ( the " mill ") would be able to perform all four arithmetic operations, plus comparisons and optionally square roots.

The characteristic which distinguishes one register as being the accumulator of a computer architecture is that the accumulator ( if the architecture were to have one ) would be used as an implicit operand for arithmetic instructions.

:" The first part of our arithmetic organ ... should be a parallel storage organ which can receive a number and add it to the one already in it, which is also able to clear its contents and which can store what it contains.

Professional mathematicians sometimes use the term ( higher ) arithmetic when referring to more advanced results related to number theory, but this should not be confused with elementary arithmetic.

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > k </ sub > are primes and the a < sub > j </ sub > are positive integers.

It is not feasible to carry out such a computation using the above recursive formulae, since at least ( a constant multiple of ) p < sup > 2 </ sup > arithmetic operations would be required.

If, for instance, an addition operation was requested, the arithmetic logic unit ( ALU ) will be connected to a set of inputs and a set of outputs.

Rather than totally removing the clock signal, some CPU designs allow certain portions of the device to be asynchronous, such as using asynchronous ALUs in conjunction with superscalar pipelining to achieve some arithmetic performance gains.

Since computers can make arithmetic calculations much faster and more accurately than humans, it was thought to be only a short matter of time before the technical details could be taken care of that would allow them the same remarkable capacity to process language.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae ( Latin, Arithmetical Investigations ), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon ( 17-sided polygon ) can be constructed with straightedge and compass.

For each specific consistent effectively represented axiomatic system for the natural numbers, such as Peano arithmetic, there exists a constant N such that no bit of Ω after the Nth can be proven to be one or zero within that system.