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| :∑<sub>''i''=''m''</sub><sup>''n''</sup> ''x''<sub>''i''</sub> = ''x''<sub>''m''</sub> + ''x''<sub>''m''+1</sub> + ''x''<sub>''m''+2</sub> + ... + ''x''<sub>''n''-1</sub> + ''x''<sub>''n''</sub> | :∑<sub>''i''=''m''</sub><sup>''n''</sup> ''x''<sub>''i''</sub> = ''x''<sub>''m''</sub> + ''x''<sub>''m''+1</sub> + ''x''<sub>''m''+2</sub> + ... + ''x''<sub>''n''-1</sub> + ''x''<sub>''n''</sub> | ||
| The subscript gives the symbol for a ] and its lower value; the superscript gives its upper value (some texts will repest "i=" in the superscript). So for instance | |||
| So for instance | |||
| :∑<sub>''i''=2</sub><sup>6</sup> ''i''<sup>2</sup> = 2<sup>2</sup> + 3<sup>2</sup> + 4<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> = 90. | :∑<sub>''i''=2</sub><sup>6</sup> ''i''<sup>2</sup> = 2<sup>2</sup> + 3<sup>2</sup> + 4<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> = 90. | ||
Revision as of 23:02, 4 January 2003
Addition is one of the basic operations of arithmetic. Addition combines two or more numbers, the summands, into a single number, the sum. (If there are only two terms, these are the augend and addend respectively.) For a definition of addition in the natural numbers, see Addition in N.
Notation
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. Alternatively, the sum can be with the summation symbol, which is a capital Sigma from the Greek alphabet. This is defined as:
- ∑i=m xi = xm + xm+1 + xm+2 + ... + xn-1 + xn
The subscript gives the symbol for a dummy variable and its lower value; the superscript gives its upper value (some texts will repest "i=" in the superscript). So for instance
- ∑i=2 i = 2 + 3 + 4 + 5 + 6 = 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 of the sum of the first n terms, as n grows without bound.
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 zero, because zero is the identity for addition. These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if a = b in the definition above, then there is only one term in the sum.
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 nx, 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:
- ∑i=1 i = n(n+1)/2
- ∑i=0 i = (2n+3n+n)/6
- ∑i=0 x = (x -1) / (x-1)
- ∑i=0 x = 1 / (1-x)
- ∑i=0 C(i, k) = C(n, k+1) (see binomial coefficient)
The following are useful approximations (using theta notation):
- ∑i=1 i = Θ(n) for every real constant c ≠ -1.
- ∑i=1 1/i = Θ(log(n))
- ∑i=1 c = Θ(c) for every real constant c.
- ∑i=1 log(i) = Θ(n log(n)) for every real constant c ≥ 0.
- ∑i=1 log(i) i = Θ(n log(n)) for all real constants c ≥ 0 and d ≥ 0.
- ∑i=1 log(i) i b = Θ(n log(n) b) for all real constants c ≥ 0, d ≥ 0 and b > 1.
See also: Infinite series