1 a quantity of money; "he borrowed a large sum"; "the amount he had in cash was insufficient" [syn: sum of money, amount, amount of money]
3 the final aggregate; "the sum of all our troubles did not equal the misery they suffered" [syn: summation, sum total]
4 the choicest or most essential or most vital part of some idea or experience; "the gist of the prosecutor's argument"; "the heart and soul of the Republican Party"; "the nub of the story" [syn: kernel, substance, core, center, essence, gist, heart, heart and soul, inwardness, marrow, meat, nub, pith, nitty-gritty]
6 the basic unit of money in Uzbekistan
7 a set containing all and only the members of two or more given sets; "let C be the union of the sets A and B" [syn: union, join]
1 be a summary of; "The abstract summarizes the main ideas in the paper" [syn: summarize, summarise, sum up]
2 determine the sum of; "Add all the people in this town to those of the neighboring town" [syn: total, tot, tot up, sum up, summate, tote up, add, add together, tally, add up] [also: summing, summed]
- sŭm, /sʌm/, /sVm/
- Rhymes: -ʌm
- A quantity obtained by addition or aggregation.
- The sum of 3 and 4 is 7.
- An arithmetic computation, especially one posed to a student as an exercise (not necessarily limited to addition.)
- A quantity of money.
- A summary.
- A central idea or point.
- The utmost degree.
- An old English measure of corn equal to the quarter.
- 1882, The sum is also used for the quarter, and the strike for the bushel. — James Edwin Thorold Rogers, A History of Agriculture and Prices in England, Volume 4, p. 207.
- (quantity obtained by addition or aggregation): amount, sum total, summation, total, totality
- (arithmetic computation): calculation, computation
- (quantity of money): amount, quantity of money, sum of money
- (summary): See summary
- (central idea or point): center/centre, core, essence, gist, heart, heart and soul, inwardness, kernel, marrow, meat, nub, nitty-gritty, pith substance
- (utmost degree): See summit
- (obsolete: old English measure of corn): quarter
quantity obtained by addition or aggregation
quantity of money
- Dutch: som
- Finnish: summa
- German: Summe
- Italian: somma
- Spanish: cantidad
- Swedish: belopp, summa
- See summary
central idea or point
- Swedish: kontenta, kärna
- See summit
obsolete: old English measure of corn
to add together
to give a summary of
- See summarize
- From Uzbek.
basic unit of money in Kyrgyzstan
- Swedish: som
basic unit of money in Uzbekistan
- Swedish: som
Particlesum (relative particle)
- (joke) 숨 에ㄹ고 숨 (sum ergo sum) Breath therefore I am.
EtymologyCognates include Ancient Greek sc=polytonic, Sanskrit sc=Deva, Old English eom (English am).
Summation is the addition of a set of numbers; the result is their sum or total. The "numbers" to be summed may be natural numbers, complex numbers, matrices, or still more complicated objects. An infinite sum is a subtle procedure known as a series. Note that the term summation has a special meaning in the context of divergent series related to extrapolation.
NotationThe summation of 1, 2, and 4 is 1 + 2 + 4 = 7. The sum is 7. Since addition is associative, it does not matter whether we interpret "1 + 2 + 4" as (1 + 2) + 4 or as 1 + (2 + 4); the result is the same, so parentheses are usually omitted in a sum. Finite addition is also commutative, so the order in which the numbers are written does not affect its sum. (For issues with infinite summation, see absolute convergence.)
If a sum has too many terms to be written out individually, 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.
Capital-sigma notationMathematical notation has a special representation for compactly representing summation of many similar terms: the summation symbol, a large upright capital Sigma. This is defined thus:
- \sum_^n x_i = x_m + x_ + x_ +\cdots+ x_ + x_n.
The subscript gives the symbol for an index variable, i. Here, i represents the index of summation; m is the lower bound of summation, and n is the upper bound of summation. Here i = m under the summation symbol means that the index i starts out equal to m. Successive values of i are found by adding 1 to the previous value of i, stopping when i = n. We could as well have used k instead of i, as in
- \sum_^6 k^2 = 2^2+3^2+4^2+5^2+6^2 = 90.
Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context, as in
- \sum x_i^2
- \sum_^n x_i^2.
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:
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