hey user!!
your answer is here...
first of all, lemme tell you what are rational and irrational numbers.
rational numbers are those numbers which can be represented in the form of p/w and q ≠ 0
irrational numbers are those numbers which cannot be represented in the form of p/q
there are two types of numbers, which are :-
# TERMINATING
# NON-TERMINATING
in non-terminating numbers, there are two more parts..
>> REPEATING
>> NON-REPEATING
now, which can be represented in the form of p/q
terminating are those numbers which terminates at the end. means remainder come 0 at the end..
so, terminating numbers can be represented in the form of p/q hence, it's rational.
now, come to non-terminating numbers..
these are numbers, which never terminate.. it goes to infinity.
but repeating numbers can be represented in the form of p/q so it's also a rational number but the non-terminating non-repeating numbers can't be represented in the form of p/q so it's irrational.
ok.. now we learnt which are rational and which are irrational.. we came to know that terminating and non-terminating repeating numbers can be represented in the form of p/q so it's rational.
and other numbers non-terminating non-repeating can't be represented in the form of p/q so it's irrational.
now come to ur question...
given :-
▶pi (π) = 3.1415926...
here it's going till infinity so it's non-terminating, next we can clearly see here that it's not repeating.. so it's a non-terminating non-repeating number and hence, irrational number.
▶0.7777...
it is also going till infinity.. so non-terminating but see the numbers are repeating. here "7" is repeating.. so it's non-terminating repeating number. as I told u before that if it's repeating, then it can be represented in the form of p/q so it's rational number.
let's verify it ;)
let x be = 0.777777... --------(1)
multiplying both sides by 10 since only one number is repeating.
10 × x = 10 × 7.777777...
10x = 7.777777... ---------(2)
subract equation (1) from equation (2)
10x - x = 7.77777... - 0.777777...
9x = 7
x = 7/9
hence, p/q form of 0.77777... is 7/9
and we proved it that non-terminating repeating numbers are rational.
▶0.36458121...
it is also a irrational number as it's non-terminating non-repeating.
▶ √5 = 2.23606798...
irrational number.. numbers are not repeating and going till infinity.
so, pi = irrational
0.77777... = rational
0.36458121... = irrational
√5 = irrational
cheers!!
