# Sum of natural numbers and its properties

In the sum: $$2+42+9$$, the addends are: $$2, 42$$ and $$9$$.

The sum of natural numbers fulfills two basic properties:

• Commutative property $$a+b=b+a$$

When we do a sum, we can change the order of the the addends and the result will be the same

$$4+6=6+4$$$$$2+42+9= 42+2+9=9+2+42$$$

• Associative property $$(a+b)+c=a+(b+c)$$

When you add up three or more numbers at once, we can add them up in pairs and finally add up the results.

In the addition $$15+2+7$$

can be calculated first $$15 + 2 = 17$$ and then $$17+7 = 24$$.

Or if you prefer, you can start adding up the last numbers: $$2+7=9$$, and then $$15+9 = 24$$.

• In mathematics, brackets are used to show which operation is done first (in this case, the sum).

$$(15+2)+7$$ means that firstly we do $$15+2=17$$ and then $$17+7 = 24$$.

Looking at this example, the associative property can be applied as follows: $$(15+2)+7=15+(2+7)$$$• Neutral element $$a+0=0+a=a$$ The $$0$$ is the neutral element of addition, since any natural number added to $$0$$ results in the same number. $$9+0=9$$$ $$0+4=4$$\$