Numbers: Mathematical Operations on Natural Numbers

 In this discussion we will look at the different mathematical operations defined on the set of natural numbers. We denote the set of all natural numbers as follows

$\mathbb{N} = \{0,1,2, \cdots \}$                                                                       $(1)$

On this set we define two mathematical operations

1. Addition 
2. Multiplication

It must be noted that we are only looking at the operations that are closed. 

Let us explain this notion further. Addition as an operation takes two natural numbers and maps them to an other natural number. In mathematical notation we can write as follows.

Let $a,b,c \in \mathbb{N}$, where $\in$ means "is an element of".Then,      

$a+b =c$                                                                                                        $(2)$

or operator notation ,        $+(a,b)$                                                                                                         $(3)$

Here $(a,b)$ is called an ordered pair. This is due to the fact the order of the elements matters i.e.

                                           $(a,b) \neq (b,a)$                                                                                          $(4)$ 

If $(a,b) = (b,a)$, then $(a,b)$ is called an unordered pair.

Mathematical notation to signify an ordered pair is as follows

$(a,b) \in \mathbb{N} \times \mathbb{N}$

where $\mathbb{N} \times \mathbb{N}$ is the set of all the ordered pairs of natural numbers.

Thus, we can generalize equations $2$ and $3$ as

$+: \mathbb{N} \times \mathbb{N} \longrightarrow \mathbb{N}$                                                   $(5)$

Which states that the addition operation($+$) maps an ordered pair of natural numbers($\mathbb{N} \times \mathbb{N}$) to an other natural number ($N$).  Here, it must be noted that $\longrightarrow$ is an implication. It is a logical operation and for this to be true it must hold(must be true) for all possibilities i.e. each possible ordered pair of natural numbers in this case.

Moreover, multiplication can be written in the same way as

 $\times: \mathbb{N} \times \mathbb{N} \longrightarrow \mathbb{N}$

The set of natural numbers $\mathbb{N}$ is closed with respect to addition($+$) and multiplication($\times$) as these map elements of $\mathbb{N}$ to elements of $\mathbb{N}$. In simple terms, if we add or multiply two natural numbers the result is also a natural number. 

It can be easily seen that for the set of natural numbers the operations of subtraction and division are not closed. 

Example: $3,4$ are both natural numbers, then $3-4=-1$ that is not a natural number.

Hence, $-:\mathbb{N} \times \mathbb{N} \not\longrightarrow \mathbb{N}$.

Moreover, neither $\frac{3}{4} \in \mathbb{N}$ nor $\frac{4}{3} \in \mathbb{N}$. Thus,

$\div: \mathbb{N} \times \mathbb{N} \not\longrightarrow \mathbb{N}$ .

Note: As we have pointed out earlier that for the implication($\longrightarrow$) to be true it must hold for all possible cases. Thus, to  negate the implication($\not\longrightarrow$) i.e. to say that the implication does not hold, only one counter example is enough.

Now that we have discussed some mathematical notations and concepts let us move to the physical interpretation of addition and multiplication of natural numbers.

Physical interpretation of addition

Physical interpretation of multiplication

Using the above geometrical approach we can verify various equations we encounter in our studies e.g.

 

Thus, it can be seen that abstract arithmetic of natural numbers has physical applications in geometry. These geometric properties are also true for all extensions of natural numbers.