• Swedish: som

• Swedish: som

Faroese

Conjunction

sum

1. like, as
2. when, as

Particle

sum (relative particle)

1. that, who, which

Synonyms

• ið

Icelandic

Adverb

sum (comparative fleiri, superlative flest)

1. some

   Ég þekkti sum barnanna.

   I knew some of the children.

Korean

Noun

sum

1. breath

   (joke) 숨 에ㄹ고 숨 (sum ergo sum) Breath therefore I am.

Latin

Etymology

Cognates include Ancient Greek sc=polytonic, Sanskrit sc=Deva, Old English eom (English am).

Pronunciation

/sʊ̃m/

Verb

present active

1. I am, I exist
   • René Descartes

     Cogito, ergo sum.

     I think, therefore I am.

   Sum sine regno

   I am without a kingdom.

   Sic sum ut vides.

   Thus I am as you see.

   Dixit duas res ei rubori fuisse.

   He said that two things had abashed him.

Conjugation

Irregular conjugation.

Participles

Infinitives

Derived terms

• absum
• adsum
• desum
• insum
• intersum
• obsum
• possum
• praesum
• prodsum
• subsum
• supersum

Descendants

• English: essence, future, in esse, quintessence
• Italian: essere
• French: être (estre > essere > esse)
• Spanish: ser

Old English

Etymology

Common Germanic *sumaz, whence also Old High German sum, Old Norse sumr

Pronoun

sum

1. some

Polish

Pronunciation

• /sum/

Noun

1. catfish

Shabo

Verb

sum

1. say

For evaluation of sums in closed form see evaluating sums.

Summation is also a term used to describe a process in synapse biology.

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.

Notation

The 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 notation

Mathematical 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

which is informally equivalent to

\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:

\sum_

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