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

Answer

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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.

Complete step-by-step answer:

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

3) Hexadecimal 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.

Complete step-by-step answer:

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

3) Hexadecimal 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.

Last updated date: 28th Sep 2023

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