Question

Write down, if possible, the largest natural number.

Answer 1

Hint: We will first understand the set of Natural Numbers and which elements does it consist of. After that, we will assume that it has some largest element and then prove by contradiction that it is
not possible.

Complete step-by-step answer:
We know that the set of Natural Numbers is given by {1, 2, 3, 4, ……………….. }.
We can clearly observe that 1 < 2 < 3 < 4 < ……….
\[\therefore \] 1 is the least element.
But, we need the greatest of Natural Numbers.
Let us assume that it has a largest number which is say a.
Now, since a is the largest among all the Natural Numbers.
\[\therefore n < a,\forall n \in \mathbb{N}\] ……………..(1)
Now, we know that the successor of a natural number is also a natural number.
\[\therefore \] a + 1 is also a natural number.
So, by using (1), we will get:
$ \Rightarrow a + 1 < a$
This implies that:
$ \Rightarrow 1 < 0$, which is absolutely absurd.
Therefore, our assumption is wrong.

Hence, the set of Natural numbers do not have any largest number.

Note: The students must note that “This property of Natural Numbers that it does not have any maximum is known as the “Archimedean Property” which is going to be extremely handy to you in future. Archimedean property states that “the set of Natural Numbers are unbounded above”. But the students must also note that the set of Natural Numbers have the “Well Ordering Property” which states that it always has a least element. Like in natural numbers 1 is the least element.
The students must note that we used the fact: successor of natural numbers is also a natural number. But we could also have used the fact that “The set of Natural Numbers are closed under addition”.