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different summation of sets

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Summation

This article is about the mathematics of finite summation. For the more elementary aspects of the topic, see Addition. For infinite summation, see Series (mathematics). For other uses, see Summation (disambiguation).

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Addition (+){\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend (broad sense)}}\,+\,{\text{addend (broad sense)}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend (strict sense)}}\end{matrix}}\right\}\,=\,}{\displaystyle \scriptstyle {\text{sum}}}Subtraction (−){\displaystyle \scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\,=\,}{\displaystyle \scriptstyle {\text{difference}}}Multiplication (×){\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}{\displaystyle \scriptstyle {\text{product}}}Division (÷){\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}{\displaystyle {\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}}Exponentiation{\displaystyle \scriptstyle {\text{base}}^{\text{exponent}}\,=\,}{\displaystyle \scriptstyle {\text{power}}}nth root (√){\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}{\displaystyle \scriptstyle {\text{root}}}Logarithm (log){\displaystyle \scriptstyle \log _{\text{base}}({\text{antilogarithm}})\,=\,}{\displaystyle \scriptstyle {\text{logarithm}}}

In mathematics, summation (denoted with an enlarged capital Greek sigma symbol {\displaystyle \textstyle \sum }) is the addition of a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation.

The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group (or even monoid).

For finite sequences of such elements, summation always produces a well-defined sum. The summation of an infinite sequence of values is called a series. A value of such a series may often be defined by means of a limit (although sometimes the value may be infinite, and often no value results at all). Another notion involving limits of finite sums is integration.

The summation of the sequence [1, 2, 4, 2] is an expression whose value is the sum of each of the members of the sequence. In the example, 1 + 2 + 4 + 2 = 9. Because addition is associative, the sum does not depend on how the additions are grouped, for instance (1 + 2) + (4 + 2) and 1 + ((2 + 4) + 2) both have the value 9; therefore, parentheses are usually omitted in repeated additions. Addition is also commutative, so permuting the terms of a finite sequence does not change its sum. For infinite summations this property may fail. See Absolute convergence for conditions under which it still holds.

There is no special notation for the summation of such explicit sequences, as the corresponding repeated addition expression will do. There is only a slight difficulty if the sequence has fewer than two elements: the summation of a sequence of one term involves no plus sign (it is indistinguishable from the term itself) and the summation of the empty sequence cannot even be written down (but one can write its value "0" in its place). If, however, the terms of the sequence are given by a regular pattern, possibly of variable length, then a summation operator may be useful or even essential.

For the summation of the sequence of consecutive integers from 1 to 100, one could use an addition expression involving an ellipsis to indicate the missing terms: 1 + 2 + 3 + 4 + ... + 99 + 100. In this case, the reader can easily guess the pattern. However, for more complicated patterns, one needs to be precise about the rule used to find successive terms, which can be achieved by using the summation operator "Σ". Using this sigma notation the above summation is written as: