Question

What is a rational or irrational number?

Answer 1

Hint: A rational number is a real number that can be represented as a ratio of two integers, or in the form of a simplest fraction, where the denominator should not be equal to zero. Similarly, the vice versa is for an irrational number, a number which cannot be expressed in the form of fractions is an irrational number.

Complete step-by-step answer:
Rational Numbers:
A rational number is a number that can be represented in the simplest fraction or \[\dfrac{p}{q}\], where p and q are integers and q should not be equal to zero (\[q \ne 0\]).
Some examples of rational numbers are: \[2,1.2, - 8\]etc.
We can see that \[2\] can be written in the form of its simplest fraction as \[\dfrac{2}{1}\], where \[1 \ne 0\], so it satisfies the condition for rational numbers.
Similarly, \[1.2\] and \[ - 8\] can be expressed in their simplest fraction in terms of \[\dfrac{p}{q}\], so they are rational numbers.

Irrational Numbers:
An irrational number is the opposite of rational numbers, a number that cannot be expressed in terms of simplest fractions that is in the form of \[\dfrac{p}{q}\].
Some examples of irrational numbers are: value of pi\[\left( \pi \right)\],\[\sqrt 2 \];
The examples are \[\left( \pi \right)\] and \[\sqrt 2 \] because their values are \[\pi = 3.14159265358.....\] and it goes on, so it cannot be expressed in terms of fractions. Similarly, for the value of \[\sqrt 2 \], the value of \[\sqrt 2 = 1.414213562 \ldots ..\] the decimal expansion goes on expanding without any repetition of digits after the decimal. So, \[\sqrt 2 \] is an irrational number.

Note: Since, we sometimes write \[\pi = \dfrac{{22}}{7}\] in solving, but pi is an irrational number, whereas \[\dfrac{{22}}{7}\] is a rational number, as because they don’t have same values, we just approximate the decimal values of \[\pi \] with \[\dfrac{{22}}{7}\] to simplify our calculations while solving.