# 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.

$$9+0=9$$ $$0+4=4$$