The question is, are all natural numbers part of rational numbers ?
To understand the explanation, you should have basic understanding of the concept of natural and rational numbers.
Let us revise both the concept.
What are natural numbers ?
The positive integers starting from 1 are called natural numbers.
Numbers like 1, 2, 3, 4, 5, 6 . . .etc. belongs to natural number group.
If you want to learn about natural numbers in detail, click the red link.
What are rational numbers?
The numbers that can be written in the form of P / Q are called rational numbers.
Where P & Q are integer values.
The numbers \mathtt{\frac{3}{5} ,\ \frac{7}{4} ,\ \frac{1}{6}} are all examples of rational numbers.
To learn about rational number in detail, click the red link.
Are all natural numbers part of rational number ?
Yes !!!
Because all the natural numbers can be represented in the form of P / Q by inserting 1 as denominator.
For example;
(i) Natural number 5
Insert 1 in the denominator, the number 5 can be written as \mathtt{\ \frac{5}{1}}
Since the number \mathtt{\ \frac{5}{1}} is in the form of P / Q, it is a rational number.
(ii) Natural Number 106
Insert 1 in the denominator we get \mathtt{\ \frac{106}{1}} .
Since the number \mathtt{\ \frac{106}{1}} is in form of P / Q, it is a rational number.
Conclusion
All natural numbers are also part of rational numbers.
All rational numbers are natural numbers?
NO ! !
Rational numbers may or may not be the part of natural numbers.
For example;
(i) Rational number \mathtt{\frac{2}{5}}
On converting fraction into decimal we get;
\mathtt{\frac{2}{5} \Longrightarrow 0.4}
Here the number 0.4 is a decimal and not a natural number.
(ii) Rational number \mathtt{\frac{12}{6}}
On simplifying the fraction we get;
\mathtt{\frac{12}{6} \Longrightarrow 2}
Here 2 is a natural number.
Hence some rational number are part of natural numbers.
Conclusion
Not all rational numbers are part of natural number. It depends on condition to condition.