Hums

Table of contents 1 word historyand - meanings 2 sum as result of an addition 3 sum of a consequence, row 4 notation with the sigma sign 5 infinite sums of 6 related terms

word history and -

in the medium high German by latin summa one took to meanings the word sum. Summa was common into the 19te century beside sum. Latin summa goes to that on the Superlativ summus ( too superus , superior), the highest, highest, largest designation, back.

In the broader sense sum marks a whole oran epitome.

In the everyday life language sum marks a money unhängig of it whether it came by addition.

the numbers are called sum as

result of an addition in

the mathematical term 2+3 2 and 3 addends. Thatentire term is called the “sum of 2 and 3”.

One can form a sum of more than two addends, so for example

for 4+7+1.

Due to the Assoziativität of the addition does not have to be indicated for it, in which order the additionsto implement are. Thus it applies that (4+7) +1 = 4+ (7+1) is and can the sum also without clips be written.

Due to the commutative law of the addition is also the sequence of the addends no matter, i.e. it is for example

4+7+1 = 7+4+1.

Becomes <math>n </math> - times the same number <math> A </math> added, then the sum can also as product <math> n \ cdot A </math> are written.

sum of a consequence, row

if a sum a great many addends has, is appropriate it,to agree a shortened way of writing. The sum of the first 100 natural numbers can be indicated for example

as 1+2+…

+100, because it is easy to guessed, which addends were replaced by the points of omission.

As one in the elementary arithmetic ofZahlenrechnungen such as 2+3=5 to algebraic expressions as <math> 2+x=y </math> turns into, then one can e.g. the sum from one hundred completely determined numbers to the sum of any number of arbitrary numbers generalize. In addition first a variable is selected, for example <math> n </math>,those the number of addends designates. In the above case, the sum of the first one hundred natural numbers, would be <math> n =100 </math>. There of any size <math> n </math> to be certified are, are not not possible, all <math> n </math> Addends through<math> n </math> to designate different letters. Instead e.g. becomes individual letters. <math> A </math> selected and around an index supplements. This index takes successively the values 1, 2,… on. The addends are called corresponding <math> a_1, \ a_2, \ dots</math>. The addends form thus a sequence of numbers (see to consequence (mathematics)).

We can now for arbitrary natural numbers <math> n </math> the sum first <math> n </math> Members of the sequence of numbers as

< math> s_n = a_1 +a_2 + \ dots +a_n </math>

write. If one for <math> n </math> different values 1, 2,… begins, form <math> s_1, \ s_2, \ dots </math> for their part likewise a consequence. Such a consequence of partial sums over the first links of a consequence becomes as Row marks.

Example: For the consequence of the Quadratzahlen is <math> a_1 =1 </math>, <math> a_2 =4 </math>, <math> a_3=9 </math>. Math a_n=n^2 /math

< completely> generally< applies>.

The number of the partial sums of this consequence begins with <math> s_1=1 </math> , <math> s_2=5 </math>,<math> s_3=14 </math>. A summation formula means now for arbitrary <math> n </math>:

<math> s_n= \ frac {n (n+1) (2n+1)}{6}. </math>

Further summation formulas like for example the small Gauss

< math> 1+2+… +n = \ frac {n (n+1)}{2}< /math>

are in the collection of formulae algebra. The proof of such formulas is made by induction principle over natural number.

notation with the sigma sign

sums over finite or infinite rows can be noted instead of with points of omission also with the sigma sign:

<math> \ sum_ {k=m} ^ {n} a_k = \ sum_ {m \ leq k \ leq n} a_k = a_m + a_ {m+1} + \ dots + a_n< /math>

The sigma signsigma, followed from a consequence member , consists that by before a not used index of the large Greek letter (here <math> k </math> is designated). This index often becomes as run index or summation variable and/or. Run or count variable designation. WhichRate the control variable to assume can, becomes at the lower surface, if necessary also the top side <math> \ Sigma </math> indicated. There are for it two possibilities: Either down a starting and above are indicated a final value (here: <math> m </math> and <math> n</math>), or it are placed down one or more conditions for the counting variable (here: <math> m \ leq k \ leq n </math>). These data can be reduced or omitted, if it can be accepted to supplement that the reader it from the contextis able.

For <math> m=n </math> the sum of only one addends exists <math> a_n </math>. It proved as useful, for <math> n=m-1 </math> to introduce the following convention:

<math> \ sum_ {k=m} ^ {m-1} a_k: = 0< /math> (sum empties).

One notes that this the only casewith <math> n< m< /math> is, which meaningfully be defined can, contrary to the integral notation.

In the tensor calculation one agrees upon frequently the Einstein summation convention, according to which the summation character can be omitted, since it is clear from the context that over all doubles occurringIndices to sum up is.

To double sums in mathematical physics the convention applies that an apostrophe at the sigma sign

< math> \ sum_ {ij} {'} \ dots = \ sum_ {i \ ne j} \ dots \, </math>

mentioned that with the summation addends are to be omitted, for which the two control variables agree. ToDesignation of counting variables the letters mostly become <math> i </math>, <math> j </math> and <math> k </math> used. If not clearly comes out, which variable is the counting variable, this must in the text is marked.

When programming the sigma sign corresponds to a For loopwith summation of the result from each loop passage.

infinite one sums

if infinitely many expressions to be summed up, thus for example

< math> \ sum_ {j=1} ^ {\ infty} a_j = \ sum _ {j \ geq 1} a_j = a_1 + a_2 + a_3 + \ dots< /math>

with infinitely many Addends not equal zero, must be used methods of the analysis, in order to find the appropriate limit value. Such a sum is called infinite row. As upper limit one writes the symbol for infinity (<math> \ infty </math>). See in addition the article row (mathematics).

It is to be marked however that each sum, which possesses ∞ as upper limit does not have to be an infinite sum. For example the sum

< left> math \ sum_ {>k 0} \ left [\ frac {n} {p^k} \ right] = \ left [\ frac {n} {p} \ right] + \ left [\ frac {n} {p^2} \ right] + \ [\ frac {n} {p^3} \ right] + \ dots< /math>

for prime numbers< math> p </math> andthe integer function <math> [x] </math>, infinitely many addends, but only finally many are not equal zero. (This sum indicates, how often the factor <math> p </math> in the prime factorization of n! .) [

terms those

occurs disjunkte combination of quantities has a certain formal similarity with the addition of numbers; are for example <math> X< /math> and <math> Y< /math> finite quantities, then is the number of elements of <math> X \ sqcup Y< /math> equal the sum number of elements of <math> X< /math> and <math> Y< /math>. That cartesian product is distributiv over this accumulation:

<math> X \ times (Y \ sqcup Z) \ cong (X \ times Y) \ sqcup (X \ times Z). </math>

Out categoryal view the similar construction for vector spaces or abelsche groups direct sum one calls; generally one speaks of a Koprodukt.