Izvorni kôd stranice Sabiranje

{{Nedostaju_izvori}}
[[Datoteka:Addition01.svg|desno|mini|120px|3 + 2 = 5 sa [[jabuka]]ma, popularnim primjerom u školama<ref>From Enderton (p.138): "...select two sets ''K'' and ''L'' with card ''K'' = 2 and card ''L'' = 3. Sets of fingers are handy; sets of apples are preferred by textbooks."</ref>]]
'''Sabiranje''' je osnovna [[Aritmetika|aritmetička]] operacija, kojom saznajemo informaciju kad dvije ili više veličina (brojeva) skupimo zajedno, koliko ih ukupno ima. Sabirati možemo jabuke, kruške, lubenice, ovce u snu (sve su to cijeli brojevi), no i tekućine utočene i istočene iz spremnika, težine razne hrane i neprehrambenih artikala (decimalni brojevi).

[[Matematika|Matematički]] sabiranje je predstavljamo znakom '''plus +''', npr. :<math>1 + 2 = 3</math>. Brojeve koje sabiremo nazivamo ''pribrojnici''.

Sabiranje je [[Komutativnost|komutativno]], što znači da je :<math>1 + 2 = 2 + 1</math> , tj. možemo slobodno zamijeniti mjesta pribrojnika, a rezultat sabiranja se neće promijeniti.

Sabiranje je i [[Asocijativnost|asocijativno]], jer vrijedi :<math> ( 1 + 2 ) + 3 = 1 + ( 2 + 3 )</math>

Kod sabiranja članova nekog niza koristi se veliko grčko slovo sigma:

: <math> \sum_{i=1}^{n} x_{i} = x_{1} + x_{2} + x_{3} + \dots + x_{n-1} + x_{n}, </math>

što znači da sabiramo prvih ''n'' članova [[niz]]a, od x<sub>1</sub> do x<sub>n</sub>. Zbir članova '''Broj''' je apstraktni pojam koji koristimo za opis količina, bez brojeva ne bi bilo [[Matematika|matematike]].

== Notacija i terminologija ==

[[Datoteka:PlusCM128.svg|desno|100px|mini|Znak '''plus''']]
Sabiranje se zapisuje korištenjem [[znakovi plus i minus|znakom plus]] "+" koji se stavlja između dva člana koji se sabiru, to jest, u [[infiksna notacija|infiksnoj notaciji]]. Rezultat se izražava sa [[znak jednakosti|znakom jednakosti]]. Na primjer,

:<math>1 + 1 = 2 </math> (verbalno, "jedan plus jedan jednako je dva")
:<math>2 + 2 = 4 </math> (verbalno, "dva plus dva jednako je četiri")
:<math>5 + 4 + 2 = 11 </math> (pogledajte "asocijativnost" [[#Asocijativnost|ispod]])
:<math>3 + 3 + 3 + 3 = 12</math> (pogledajte "množenje" [[#Slične operacije|ispod]])

== Osobine ==

=== Komutativnost ===

[[Datoteka:AdditionComm01.svg|desno|113px|mini|4 + 2 = 2 + 4 sa blokovima]]
Sabiranje je [[komutativnost|komutativno]], što znači da članovi, koji se sabiru, mogu, međusobno, zamijeniti mjesta, a da rezultat ostane nepromijenjen. Simbolički, ako su ''a'' i ''b'' dva broja, tada vrijedi

:<math> a  + b = b + a</math>

=== Asocijativnost ===

[[Datoteka:AdditionAsc.svg|lijevo|100px|mini|2+(1+3) = (2+1)+3 sa segnetntovanim štapovima]]

Još jedna osobina sabiranja je [[asocijativnost]], koju dobijamo kada pokušamo definisati uzastopno sabiranje više članova sume. Da li bi izraz

:<math>a + b +c</math>  
trebao biti definisan kao 
:<math>(a + b)+ c</math>  ili :<math>a + (b + c)</math>

Činjenica da je sabiranje asocijativno govori nam da je odabir definicije nebitan. Za bilo koja tri broja ''a'', ''b'' i ''c'', važi da je

:<math>(a + b)+ c = a + (b+c)</math>

Na primjer

:<math>(1 + 2 )+ 3) = 2+ 3 = 6= 1+5 = 1+ (2+3)</math> .

Nisu svi operatori asocijativni, tako da u izrazima sa ostalim operatorima, kao što je dijeljenje, važno je naznačiti [[redoslijed operacija]].

=== Neutralni elemenat===

Postoji jedan realan broj koji ako se sabere sa bilo kojim realnim brojem daje taj isti realan broj, tj. njegovo dodavanje na neki broj ne utiče na taj broj; taj realni broj se naziva [[neutral]], i kod sabiranja realnih brojeva se obično predstavlja simbolom <math>0</math> i zove „[[0 (broj)|nula]]“:
:<math>\exists 0 \in R, \forall x \in R, x+0=x</math>

=== Inverzni elemenat===
Za svaki uzeti realni broj, postoji njemu suprotan, označen sa znakom [[minus]], koji kad se sabere sa tim brojem daje nulu; takav „suprotni“ broj nekog broja  naziva se njegovim [[inverzni elemenat|inverznim elementom]]:
:<math>\forall x \in R, \exists {-x}, x+(-x)=0</math>

Uopćeno govoreći, sabiranje ne mora zadovoljavati sve navedene osobine za sve skupove nad kojim je definirano.

Naprimjer
:sabiranje nad skupom cijelih brojeva ne zadovoljava uslove 3. i 4., sabiranje nad skupom [[ordinal]]a ne zadovoljava uslove 1. i 4., itd.
==Tablica sabiranja==
{| class="wikitable"
|-
! + !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
|-
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11
|-
| 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 
|-
| 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12  || 13 
|-
| 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12  || 13  || 14 
|-
|  5 || 6 || 7 || 8 || 9 || 10 || 11 || 12  || 13  || 14 || 15
|-
| 6 || 7 || 8 || 9 || 10 || 11 || 12  || 13  || 14 || 15 || 16 
|-
| 7 || 8 || 9 || 10 || 11 || 12  || 13  || 14 || 15 || 16  || 17
|-
| 8 || 9 || 10 || 11 || 12  || 13  || 14 || 15 || 16  || 17 || 18
|-
| 9 || 10 || 11 || 12  || 13  || 14 || 15 || 16  || 17 || 18 || 19
|-
| 10 || 11 || 12  || 13  || 14 || 15 || 16  || 17 || 18 || 19 || 20

|}

== Napomene ==
{{refspisak}}

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</div>
{{Commonscat|Addition}}

{{Elementarna aritmetika}}

[[Kategorija:Elementarna aritmetika]]
[[Kategorija:Binarne operacije]]