prove that √5 is a rational number
Answers
Answer:
let root5 is irrational. so root 5 = p/q .p and q are rational . Gcd of p and q is 1
Step-by-step explanation:
q ×root 5 =p
q × root 5-p=0
s.o.b.s.
5q^2+p^2-2pqroot5=0
2pqroot5=5q^2+p^2
root5=5q^2+p^2 /2pq
lhs is irrational and rhs is rational this is not possible
hence our assumption is wrong.
so root5 is irrational number
Answer:
if p is a prime no and divides a² then p divides a,where a greater than 0
Step-by-step explanation:
if possible,suppose that√5 be a rational no.
~ we can find 2 co- prime no's a and b,b is not equal to 0
such that a/ b = √5
a = b√5→ 1
squaring on both sides
a²= (b√5)²
a² = b²5
5 divides a² → x
therefore,a= 5c → 2
put value of a in equation 1 we get
5c = b√5
squaring on both sides
5c² = (b√5)²
25c² = b²5
dividing 5 on both sides
5c² = b²
5 divides b²,
therefore,5 divides b²→ xx
from x and xx we get
5 is a factor of a as well as b
so,a and b are not co- primes
therefore our supposition is wrong
hence√5 is a irrational no.