Question

Explain how irrational numbers differ from rational numbers.

Answer 1

Hint:- Rational numbers are those which can be expressed as the ratio of two numbers while irrational numbers cannot be expressed as the ratio of two numbers.

Complete step-by-step answer:
Fractional numbers are those which are written in the form of \[\dfrac{A}{B}\] where A will be the numerator and can be any integer while B will be the denominator and is also an integer except zero.
So, now those real numbers which can be expressed as fractional numbers are known as rational numbers but the real numbers which cannot be expressed as fractional numbers are known as irrational numbers.
Now rational numbers are those numbers which are finite and recurring in nature or in other words the number whose number of digits are finite or they repeat after some interval like 3.2 or 5.89898989….. .
But the numbers which are non-terminating and non-repeating in nature or in other words the number of digits in a number are not finite and cannot occur after the same interval like 3.23557845620…… are known as irrational numbers.
Now rational numbers include the numbers which are perfect squares like 1, 4, 9, 16, 25, 36 etc.
While the irrational numbers include the numbers which are surds like 2, 3, 5, 7, 11, 13 etc.
Both the numerator and the denominator rational numbers are whole numbers except the denominator cannot be zero.
But the irrational numbers cannot be expressed in the form of numerator and denominator.

Note:- Rational numbers are not always in the form of \[\dfrac{A}{B}\], but we can change them in the form of \[\dfrac{A}{B}\]. Like 3.5 is not in the form of \[\dfrac{A}{B}\] but it can also be written as \[\dfrac{7}{2}\]. So, 3.5 is a rational number but irrational numbers cannot be written as \[\dfrac{A}{B}\] because the digits of the irrational numbers are non-terminating and non-repeating. And all irrational numbers are decimal numbers but all decimal numbers are not irrational numbers.