Question

How do irrational numbers differ from rational numbers?

Answer 1

Hint: For the given problem where we have to establish the difference between irrational numbers and rational numbers, first we are going to see the definitions and then try to understand the differences by taking the examples of both the given quantities.

Complete step-by-step solution:
Defining rational number,
Rational numbers can be defined as any number that exists in the form of \[\dfrac{p}{q}\] where \[p\] and \[q\] both are integers and also \[q \ne 0\].
For example, \[7,\dfrac{5}{2},\dfrac{9}{{11}}\] and \[ - \dfrac{8}{3}\] all are rational numbers.
By above given examples we can also conclude that when we divide a rational number into \[\dfrac{p}{q}\] form it either terminates or gets repeated.
For example, when we divide \[\dfrac{5}{2}\] and write in its decimal form, it can be written as 2.5 which means it gets terminated.
When we divide \[\dfrac{9}{{11}}\] and write in its decimal form, it can be written as \[0.8181818181\] which means \[81\] is consistently repeating. It can also be written as \[0.\overline {81} \].
Defining irrational number,
Any other real number is called an irrational number. It can also be said as a real number which when converted in decimal form are either non-terminating or non-repeating are called irrational numbers.
For example, \[\sqrt 3 ,\pi ,e\] are all irrational numbers.
When we write value of \[\pi \] it can be written as following,
\[ \Rightarrow \pi = 3.14159265358979323846264338327950...\]
By obtaining the value of\[\pi \] it can be easily concluded that it is neither terminating nor repeating.
When we write value of \[\sqrt 3 \] in the decimal form, it can be written as following,
\[ \Rightarrow \sqrt 3 = 1.73205080757...\]
By above obtained value of\[\sqrt 3 \] it can be again concluded that it is neither terminating nor repeating.

Note: When we have to check if a given data is rational number or irrational number then first, we try to write in its decimal form, if it terminates or repeats then it is rational number otherwise it is irrational number.