Addition

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Addition (or summation) is one of the basic operations of arithmetic. In its simplest form, addition combines two numbers, the augend and addend, into a single number, the sum. Adding more numbers can be viewed as repeated addition. In the case of repeated addition the augend is the first addend. (Repeated addition of the number one is the most basic form of counting.) By extension, the addition of zero numbers, one number, or infinitely many numbers can be defined.

Addition can also be defined for mathematical objects other than numbers — for example, matrices or polynomials. Regardless of the nature and number of objects being added, the individual constituents of a sum typically are called summands or terms. (This is to be distinguished from factors, which are multiplied.)

Contents

Important properties

Addition is used to model two physical processes. First, when two or more collections are combined into a single collection, the number of objects in the single collection is the sum of the number of objects in the original collections. Second, when an original measure is extended or amplified by a given amount, the final measure is the sum of the original measure and the measure of the extension or amplification.

The most fundamental property of addition is that you can only add like quantities. In order to add quantities, you must given them a common name, or common denominator. For example, two apples plus three oranges are five fruit, since "fruit" is a common name for apples and oranges. If a measure of five feet is extended by two inches, the sum is 62 inches, since 60 inches is another name for five feet. One special case of this general rule is the use of a common denominator when adding fractions.

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 zero 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.

For a more formal treatment in the context of natural numbers, please see Addition of natural numbers.

Notation

Image:Plus sdkfdkfd.gif

If the terms are all written out individually, then addition is written using the plus sign ("+").

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 = 5050. There are two situations where addition is "understood" even though no symbol appears. The first is that a column of numbers, with the last number in the column underlined, usually (but not always) indicates that the numbers in the column are to be added, with the sum written below the underlined number. The second is in the notation for mixed numbers, where a whole number followed by a fraction indicates the sum of the whole number and the fraction. This notation often causes confusion for those not throughly familiar with arithmetic, because in most other cases an "understood" operation is multiplication.

Summation notation

Alternatively, the sum can be represented by the summation symbol, which is the capital Sigma. This is defined as:

<math> \sum_{i=m}^{n} x_{i} = x_{m} + x_{m+1} + x_{m+2} + \dots + x_{n-1} + x_{n}. </math>

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:

<math> \sum_{x=2}^{6} x^{2} = 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} = 90. </math>

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 of the sum of the first n terms, as n grows without bound. That is:

<math> \sum_{i=m}^{\infty} x_{i} := \lim_{n\to\infty} \sum_{i=m}^{n} x_{i}. </math>

One can similarly replace m with negative infinity, and

<math>\sum_{i=-\infty}^\infty x_i := \lim_{n\to\infty}\sum_{i=-n}^m x_i + \lim_{n\to\infty}\sum_{i=m+1}^n x_i,</math>

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:

<math>\sum_{0\le x< 100} f(x)</math>

is the sum of f(x) over all (integer) x in the specified range,

<math>\sum_{x\in S} f(x)</math>

is the sum of f(x) over all integers x in the set S, and

<math>\sum_{d|n}\;\mu(d)</math>

is the sum of μ(d) over all integers d dividing n.

There are also ways to generalize the use of many sigma signs. For example,

<math>\sum_{ll^{'}}</math>

is the same as

<math>\sum_l\sum_{l^{'}}.</math>

Computerized notation

The summation can also be represented in a programming language.

<math> \sum_{i=m}^{n} x_{i}</math>

is equivalent to the following JavaScript program: 

sum=0;
for(i=m; i<=n; i++)
  sum += x[i];
<math> \sum_{x=2}^{6} x^{2} = 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} = 90. </math> 
sum=0;
for(x=2; x<=6; x++)
  sum += x*x;

Special cases

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 zero, because zero is the identity 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.

Relationships to other operations

Subtraction can be thought of as a kind of addition, that is, addition of an additive inverse.

Multiplication can be thought of as a generalization of addition. If a single term x appears in a sum n times, then the sum is n x, the product of n and x. If n is not a natural number, then the multiplication may still make sense, so that we have a notion for a sum in which a term appears, say, two and a half times.

A special case is multiplication of a given number by -1, where the product is the additive inverse of the given number.

Another generalization of addition is a linear combination, where each term in the sum has a multiplier from some given set, usually either the set of real numbers or the set of complex numbers.

Vectors can also be added, see linear algebra.

The most general form of addition occurs in abstract algebra, where addition may be any well defined binary operation.

Useful sums

The following are useful identities:

<math> \sum_{i=1}^{n} i = \frac {n(n+1)}{2} </math>
 (see arithmetic series);
<math>\sum_{i=1}^n m = mn</math>
<math>\sum_{k=i}^nk = \frac{(n-i+1)(n+i)}{2}</math>
<math> \sum_{i=1}^{n} (2i - 1) = n^2; </math>
<math> \sum_{i=0}^{n} i^{2} = \frac{n(n+1)(2n+1)}{6}; </math>
<math> \sum_{i=0}^{n} i^{3} = \left(\frac{n(n+1)}{2}\right)^{2}; </math>
<math> \sum_{i=N_1}^{N_2} x^{i} = \frac{x^{N_2+1}-x^{N_1}}{x-1}</math> (see geometric series);
<math> \sum_{i=0}^{n} x^{i} = \frac{1-x^{n+1}}{1-x} </math> (special case of the above where <math>{N_1}=0</math>)
<math> \sum_{i=0}^{\infty} x^{i} = \frac{1}{1-x};</math> (special case of the above, <math>\lim_{n\to\infty}</math> and <math>|x|<1</math>);
<math> \sum_{i=0}^\infty ix^{i-1}=\frac{1}{(1-x)^2}</math> (only for <math>|x|<1</math>);
<math> \sum_{i=0}^{n} {n \choose i} = 2^{n} </math> (see binomial coefficient);
<math> \sum_{i=0}^{n-1} {i \choose k} = {n \choose k+1}. </math>
<math>\left(\sum_i a_i\right)\left(\sum_j b_j\right) = \sum_i\sum_j a_ib_j</math>
<math>{\left(\sum_i a_i\right)}^2 = 2\sum_i\sum_{j<i} a_ia_j + \sum_i a_i^2</math>

In general, the sum of the first n mth powers is

<math>\sum_{i=0}^n i^m = \frac{(n+1)^{m+1}}{m+1} + \sum_{k=1}^m\frac{B_k}{m-k+1}{m\choose k}(n+1)^{m-k+1},</math>

where <math>B_k</math> is the kth Bernoulli number.

The following are useful approximations (using theta notation):

<math> \sum_{i=1}^{n} i^{c} = \Theta(n^{c+1}) </math> for every real constant c greater than -1;
<math> \sum_{i=1}^{n} \frac{1}{i} = \Theta(\log{n}); </math>
<math> \sum_{i=1}^{n} c^{i} = \Theta(c^{n}) </math> for every real constant c greater than 1;
<math> \sum_{i=1}^{n} \log(i)^{c} = \Theta(n \cdot \log(n)^{c}) </math> for every nonnegative real constant c;
<math> \sum_{i=1}^{n} \log(i)^{c} \cdot i^{d} = \Theta(n^{d+1} \cdot \log(n)^{c}) </math> for all nonnegative real constants c and d;
<math> \sum_{i=1}^{n} \log(i)^{c} \cdot i^{d} \cdot b^{i} = \Theta (n^{d} \cdot \log(n)^{c} \cdot b^{n}) </math> 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 increasing function f:

<math> \int_{s=a-1}^{b} f(s)\, ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\, ds.</math>

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… the other kind of identity… has to do with axes of symmetry. C-E belongs to a family of symmetrically related dyads 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, <math>\{0,11,7,4,2,9,3,8,10,1,5,6\}</math>, is a series of six dyads, all sum 11. If the row is rotated and retrograded, so it runs <math>\{0,6,5,1,\dots\}</math>, 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

External links

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