Question

# Is there any natural number which has no successor? Is there a last number?

Hint: In this question, we need to first look at the definition of successor; it gives us a clear picture of how to answer. We need to know about natural numbers and where they start and end.

Let us look at some of the basic definitions.
NUMBER: A number tells us how many times a unit is contained in a given quantity.
Types of number system:
1) Binary number system
2) Octal number system
4) Roman number system
5) Decimal number system
DECIMAL NUMBER SYSTEM: Numeric values are represented by using digits from 0 to 9.
Again numbers in the decimal system are classified as real numbers.
Real numbers are classified into rational numbers and irrational numbers.
Rational numbers are again classified into integer and non-integer rational numbers.
Again integers are classified into:
1) Positive integer
2) Negative integer
3) Whole number
4) Natural number
NATURAL NUMBERS: Numbers starting from 1, having no fraction part, which we use in counting the objects, denoted by N.
N = {1, 2, 3, …..}
Successor: Successor refers to the number directly after a given number, respectively.
To find the successor of a given number we need to add one to the given number.
Let us assume that the given number is a.
Hence, Successor of a will be a+1.
The successor function is denoted by S. So,
$S\left( n \right)=n+1$
As we know that natural numbers start from 1 and there is no certain end to that series.
N = {1, 2, 3, …..}
S for natural numbers will be:
\begin{align} & \Rightarrow S=\left\{ 1+1,2+1,3+1, ….. \right\} \\ & \therefore S=\left\{ 2,3,4, ….. \right\} \\ \end{align}
Hence, as there is no specific end to natural numbers we can say that every natural number has a successor.

Note: The successor function is one of the basic components used to build a primitive recursive function.
Let us consider an example to find the successor:
Successor of 4 is 5 where 4 is the predecessor of 5.
Predecessor refers to the number directly before a given number. When we consider natural numbers we can say that not all natural numbers have a predecessor.
$P\left( n \right)=n-1$
If we calculate the predecessor of 1 we get 0. But, 0 is not a natural number.
So, 1 has no predecessor in natural numbers but all natural numbers have successors.