Start with defining the term real number and its parts and try to understand the term rational, irrational numbers
. Rational numbers
are those numbers in which an Integer is divided by another integer. Mathematically we can say that if a and b are two integers such that they can be represented in form a/b then a/b is a rational number, where b can never be equal to zero. Irrational numbers are those numbers in which two numbers a and b can never be written in the form a/b. Examples of Irrational Numbers are pi , √2 , √5, etc
Now, let us check if ¾ is a rational number.
Assume that 3/4 is a rational number. Since every rational number can be expressed as a/b form which do not have any common factor so we can say that
a/b = 3/4
Squaring both side we get:
a2/b2 = 9/16
a2 = (9/16) b2
Or we can say b2 = a2(16/9)
a2 is divided by 9.
So we can say that a is divided by 9. It implies that there exist some number c such that a/9 = c
a = 9c
By squaring both side we get a2 = 81 c2
Put the value of a2 from the equation a2 = (9/16) b2 we get:
(9/16)b2 = 81c2 this tends to b2/ 9 = 16 c2
It implies that b2 is divided by 9.
So we can say that b is divided by 9.
This implies that a and b have a common factor 9.
Since a and b have common factor therefore 3/4 is not a irrational number. Hence 3/4 is a rational number.