Addition

Addition is one of the basic Operators of arithmetic. In its simplest form, addition combines two numbers (terms, summands), the ''augend'' and ''addend'', into a single number, the sum. Adding more numbers corresponds to repeated addition. By extension, addition of zero, one or infinitely many numbers can be defined, see below. For a definition of addition in the natural numbers, see Addition in N. See also: counting == Important properties == When adding finitely many numbers, it doesn't matter how you group the numbers and in which order you add them; you will always get the same result. (See Associativity and Commutativity.) If you add 0 (number) to any number, the quantity won't change; zero is the identity element for addition. The sum of any number and its additive inverse (in contexts where such a thing exists) is zero.: ) == Notation == If the terms are all written out individually, then addition is written using the Plus and minus signs ("+"). Thus, the sum of 1, 2, and 4 is 1 + 2 + 4 = 7. If the terms are not written out individually, then the sum may be written with an ellipsis to mark out the missing terms. Thus, the sum of all the natural numbers from 1 to 100 is 1 + 2 + … + 99 + 100. Alternatively, the sum can be represented by the summation symbol, which is the capital Sigma (letter). This is defined as: : $\backslash sum\_\{i=m\}^\{n\}\; x\_\{i\}\; =\; x\_\{m\}\; +\; x\_\{m+1\}\; +\; x\_\{m+2\}\; +\; \backslash dots\; +\; x\_\{n-1\}\; +\; x\_\{n\}.$ The subscript gives the symbol for a dummy variable, ''i''. Here, ''i'' represents the index of summation; ''m'' is the lower bound of summation, and ''n'' is the upper bound of summation. So, for example: : $\backslash sum\_\{x=2\}^\{6\}\; x^\{2\}\; =\; 2^\{2\}\; +\; 3^\{2\}\; +\; 4^\{2\}\; +\; 5^\{2\}\; +\; 6^\{2\}\; =\; 90.$ One may also consider sums of infinitely many terms; these are called infinite series. Notationally, we would replace ''n'' above by the infinity symbol (∞). The sum of such a series is defined as the limit (mathematics) of the sum of the first ''n'' terms, as ''n'' grows without bound. That is: : $\backslash sum\_\{i=m\}^\{\backslash infty\}\; x\_\{i\}\; :=\; \backslash lim\_\{n\backslash to\backslash infty\}\; \backslash sum\_\{i=m\}^\{n\}\; x\_\{i\}.$ One can similarly replace ''m'' with negative infinity, and :$\backslash sum\_\{i=-\backslash infty\}^\backslash infty\; x\_i\; :=\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash sum\_\{i=-n\}^m\; x\_i\; +\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash sum\_\{i=m+1\}^n\; x\_i,$ for some integer ''m'', provided both limits exist. One often sees generalizations of this notation in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example: : is the sum of ''f''(''x'') over all (integer) ''x'' in the specified range, :$\backslash sum\_\{x\backslash in\; S\}\; f(x)$ is the sum over all ''x'' in the set ''S'', and :$\backslash sum\_\{d|n\}\backslash mu(d)$ is the sum of μ(''d'') over all integers ''d'' dividing ''n''. == Relationships to other operations and constants == It's possible to add fewer than 2 numbers: *If you add the single term ''x'', then the sum is ''x''. *If you add zero terms, then the sum is 0 (number), because zero is the identity element for addition. This is known as the ''empty sum''. These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if ''m'' = ''n'' in the definition above, then there is only one term in the sum; if ''m'' = ''n'' + 1, then there is none. Many other operations can be thought of as generalised sums. If a single term ''x'' appears in a sum ''n'' times, then the sum is ''n''''x'', the result of a multiplication. If ''n'' is not a natural number, then the multiplication may still make sense, so that we have a sort of notion of adding a term, say, two and a half times. A special case is multiplication by -1, which leads to the concept of the additive inverse, and to subtraction, the inverse operation to addition. The most general version of these ideas is the linear combination, where any number of terms are included in the generalised sum any number of times. == Useful sums == The following are useful identities: :

: $\backslash sum\_\{i=1\}^\{n\}\; i\; =\; \backslash frac\; \{n(n+1)\}\{2\}$

(see arithmetic series);

:$\backslash sum\_\{k=i\}^nk\; =\; \backslash frac\{(n-i+1)(n+i)\}\{2\}$ : $\backslash sum\_\{i=1\}^\{n\}\; (2i\; -\; 1)\; =\; n^2;$ : $\backslash sum\_\{i=0\}^\{n\}\; i^\{2\}\; =\; \backslash frac\{n(n+1)(2n+1)\}\{6\};$ : $\backslash sum\_\{i=0\}^\{n\}\; i^\{3\}\; =\; \backslash left(\backslash frac\{n(n+1)\}\{2\}\backslash right)^\{2\};$ : $\backslash sum\_\{i=N\_1\}^\{N\_2\}\; x^\{i\}\; =\; \backslash frac\{x^\{N\_2+1\}-x^\{N\_1\}\}\{x-1\}$ (see geometric series); : $\backslash sum\_\{i=0\}^\{n\}\; x^\{i\}\; =\; \backslash frac\{x^\{n+1\}-1\}\{x-1\}$ (special case of the above where $\{N\_1\}=0$) : $\backslash sum\_\{i=0\}^\{\backslash infty\}\; x^\{i\}\; =\; \backslash frac\{1\}\{1-x\};$ (special case of the above, $\backslash lim\_\{n\backslash to\backslash infty\}$ and ); :

$\backslash sum\_\{i=0\}^\{n\}\; \{n\; \backslash choose\; i\}\; =\; 2^\{n\}$

(see binomial coefficient);

: $\backslash sum\_\{i=0\}^\{n-1\}\; \{i\; \backslash choose\; k\}\; =\; \{n\; \backslash choose\; k+1\}.$ : $\backslash left(\backslash sum\_i\; a\_i\backslash right)\backslash left(\backslash sum\_j\; b\_j\backslash right)\; =\; \backslash sum\_i\backslash sum\_j\; a\_ib\_j$ : Bernoulli number. The following are useful approximations (using big O notation): :

$\backslash sum\_\{i=1\}^\{n\}\; i^\{c\}\; =\; \backslash Theta(n^\{c+1\})$

for every real constant ''c'' greater than -1;

: $\backslash sum\_\{i=1\}^\{n\}\; \backslash frac\{1\}\{i\}\; =\; \backslash Theta(\backslash log\{n\});$ :

$\backslash sum\_\{i=1\}^\{n\}\; c^\{i\}\; =\; \backslash Theta(c^\{n\})$

for every real constant ''c'' greater than 1;

:

$\backslash sum\_\{i=1\}^\{n\}\; \backslash log(i)^\{c\}\; =\; \backslash Theta(n\; \backslash cdot\; \backslash log(n)^\{c\})$

for every nonnegative real constant ''c'';

:

$\backslash sum\_\{i=1\}^\{n\}\; \backslash log(i)^\{c\}\; \backslash cdot\; i^\{d\}\; =\; \backslash Theta(n^\{d+1\}\; \backslash cdot\; \backslash log(n)^\{c\})$

for all nonnegative real constants ''c'' and ''d'';

:

$\backslash sum\_\{i=1\}^\{n\}\; \backslash log(i)^\{c\}\; \backslash cdot\; i^\{d\}\; \backslash cdot\; b^\{i\}\; =\; \backslash Theta\; (n^\{d\}\; \backslash cdot\; \backslash log(n)^\{c\}\; \backslash cdot\; b^\{n\})$

for all nonnegative real constants ''b'' > 1, ''c'', ''d''.

== Approximation by integrals == Many such approximations can be obtained by the following connection between sums and integrals, which holds for any monotonic function function ''f'': :$\backslash int\_\{s=a-1\}^\{b\}\; f(s)\backslash ,\; ds\; \backslash le\; \backslash sum\_\{i=a\}^\{b\}\; f(i)\; \backslash le\; \backslash int\_\{s=a\}^\{b+1\}\; f(s)\backslash ,\; ds.$ For more general approximations, see the Euler-Maclaurin formula. ==In music== Sums are also used in musical set theory. George Perle provides the following example: :"C-E, D-F♯, E♭-G, are different instances of the same interval (music)… the other kind of identity… has to do with axes of symmetry. C-E belongs to a family of symmetrically related Dyad (music)s as follows:" {| |''D'' | |D♯ | |E | |F | |F♯ | |G | |''G♯'' |- |''D'' | |C♯ | |C | |B | |A♯ | |A | |''G♯'' |} ::Axis pitches italicized, the axis is pitch class determined. Thus in addition to being part of the interval-4 family, C-E is also a part of the sum-2 family (with G♯ equal to 0). The tone row to Alban Berg's ''Lyric Suite'', $\backslash \{0,11,7,4,2,9,3,8,10,1,5,6\backslash \}$, is a series of six dyads, all sum 11. If the row is rotated and retograded, so it runs $\backslash \{0,6,5,1,\backslash dots\backslash \}$, the dyads are all sum 6. {| |+'''Successive dyads from ''Lyric Suite'' tone row, all sum 11''' |- |''C'' | |G | |D | |D♯ | |A♯ | |''E♯'' |- |''B'' | |E | |A | |G♯ | |C♯ | |''F♯'' |} ::Axis pitches italicized, the axis is dyad (interval 1) determined ==See also== *Incrementation *Plus and minus signs *Equals sign *Modular arithmetic *Elementary arithmetic ==External links== * [http://www.cut-the-knot.org/do_you_know/addition.shtml Addition] * * [http://webhome.idirect.com/~totton/abacus/pages.htm#Addition1 Addition on a Japanese abacus] selected from [http://webhome.idirect.com/~totton/abacus/ Abacus: Mystery of the Bead] Arithmetic Mathematical notation simple:Addition th:การบวก

Addition

When b is replaced with the infinity (??) symbol, the sum is an infinite series. This has a countably infinite number of terms, and represents the limit of the sum of the first n terms, as n grows without bound. This needs to be explained better. User:Vera Cruz ---- The lowercase ''theta'' function used on this page needs to be replaced by uppercase ''theta'', as described at Big O notation. --User:Zero0000 06:30, 23 Dec 2003 (UTC) == Uncommon Bounds == See also Multiplication. What if the bounds are fractions? For example the series: $\backslash sum\_\{i=1\}^n\; 2i-1\; =\; n^2$
$\backslash left(\backslash frac\{a\}\{b\}\backslash right)^2\; =\; \backslash sum\_\{i=1\}^\{a/b\}\; 2i-1$
$\backslash left(\backslash frac\{a\}\{b\}\backslash right)^2\; =\; \backslash frac\{a^2\}\{b^2\}\; =\; \backslash frac\{\backslash sum\_\{i=1\}^a\; 2i-1\}\{\backslash sum\_\{i=1\}^b\; 2i-1\}$ Thus, it can be generalized that $\backslash sum\_\{i=a/b\}^\{c/d\}\; f(i)\; =\; \backslash frac\{\backslash sum\_\{i=a\}^c\; f(i)\}\{\backslash sum\_\{i=b\}^d\; f(i)\}$ Due to the commutative property of addition, $\backslash sum\_\{i=a\}^b\; f(i)\; =\; \backslash sum\_\{i=b\}^a\; f(i)$. Thus, with $b\; >\; a$, we iterate in reverse order (that is from the greater bound to the lower bound, or in reverse order) - for example:
$\backslash sum\_\{i=1\}^\{3\}\; i\; =\; 1\; +\; 2\; +\; 3\; =\; 6$
$\backslash sum\_\{i=3\}^\{1\}\; i\; =\; 3\; +\; 2\; +\; 1\; =\; 6$ (note the order) What if the bounds are negative? Also, $\backslash sum\_\{i=-1\}^\{-3\}\; i\; =\; -1\; +\; -2\; +\; -3\; =\; -1\; -\; 2\; -\; 3\; =\; -6$ and
$\backslash sum\_\{i=1\}^3\; -i\; =\; -1\; +\; -2\; +\; -3\; =\; -1\; -\; 2\; -\; 3\; =\; -6$ (note the sign at the bounds) If $f(-i)\; =\; -f(i)\backslash ,\backslash !$, then the generalization becomes
$\backslash sum\_\{i=-a\}^\{-b\}\; f(i)\; =\; \backslash sum\_\{i=a\}^b\; f(-i)\; =\; \backslash sum\_\{i=a\}^b\; -f(i)\; =\; -\backslash sum\_\{i=a\}^b\; f(i)$ What if the bounds are equal? In this case, the summation will yield the identity element for addition (that is zero or empty sum). Thus, the generalizations are: # $\backslash sum\_\{i=a/b\}^\{c/d\}\; f(i)\; =\; \backslash frac\{\backslash sum\_\{i=a\}^c\; f(i)\}\{\backslash sum\_\{i=b\}^d\; f(i)\}$ # $\backslash sum\_\{i=a\}^b\; f(i)\; =\; \backslash sum\_\{i=b\}^a\; f(i)$ # $\backslash sum\_\{i=-a\}^\{-b\}\; f(i)\; =\; \backslash sum\_\{i=a\}^b\; f(-i)$ # $\backslash sum\_\{i=a\}^\{a\}\; f(i)\; =\; 0$ # $\backslash sum\_\{i=1\}^n\; m\; =\; mn$ (see Multiplication) # $\backslash sum\_\{i=a\}^b\; m\; =\; m(b-a+1)$ (from the equation before this) # $\backslash sum\_\{i=a/b\}^\{c/d\}\; m$ is disputed because there are two possible definitions ## $\backslash frac\{m(c-a+1)\}\{m(d-b+1)\}\; =\; \backslash frac\{c-a+1\}\{d-b+1\}$ according to #1 ## $m\backslash left(\backslash frac\{c\}\{d\}\; -\; \backslash frac\{b\}\{a\}+1\backslash right)\; =\; m\backslash left(\backslash frac\{ac-bd+ad\}\{ad\}\backslash right)$ according to #6 But we prefer both definitions, i.e. $\backslash sum\_\{i=a/b\}^\{c/d\}\; m\; =\; m\backslash left(\backslash frac\{ac\; -\; bd\}\{ad\}+1\backslash right)\; \backslash mbox\{\; iff\; \}\; ad\; \backslash ne\; 0$ $\backslash sum\_\{i=a/b\}^\{c/d\}\; m\; =\; \backslash frac\{c\; -\; a\; +\; 1\}\{d\; -\; a\; +\; 1\}\; \backslash mbox\{\; iff\; \}\; d\; -\; a\; +\; 1\; \backslash ne\; 0$
$\backslash sum\_\{i=a/b\}^\{c/d\}\; m\; =\; m\backslash left(\backslash frac\{ac-bd\}\{ad\}+1\backslash right)\; \backslash or\; \backslash frac\{c-a+1\}\{d-a+1\}\; \backslash mbox\{\; both\; iff\; \}\; a,\; d\; \backslash ne\; 0\; \backslash and\; d-a\; \backslash ne\; -1$ provided that $a,\; b\; \backslash in\; \backslash mathbb\{Z\}\; \backslash and\; a,\; b\; \backslash ge\; 0\; \backslash and\; a\; \backslash ne\; b$ and that the ring is commutative over addition and that no quotient (divisor) is zero. Critics and corrections are welcome. -- User:ErikDT

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Words begining with Addition:

Addition
Addition
Additional_dialog_recording
Additional_Economical_Information_and_Dates_of_Empire_of_Japan
Additional_information_of_Japanese_industry
Additional_information_of_Japanese_industry
Additional_Members_System
Additional_Member_System
Additional_Member_System
Additional_member_system
Additional_Productions
Additiona_information_over_foreing_Commerce_and_Navigation
Additiona_information_over_foreing_Commerce_and_Navigation
Addition_(logic)
Addition_(logic)
Addition_and_subtraction_properties_of_equality
Addition_Chain
Addition_chain
Addition_chain_exponentiation
Addition_chain_exponentiation
Addition_in_N
Addition_in_N
Addition_of_HX_to_an_alkene
Addition_of_natural_numbers
Addition_of_natural_numbers
Addition_of_Velocities_Formula
Addition_of_velocities_formula
Addition_polymer
Addition_polymerization
Addition_reaction
Addition_theorem

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