Question

Root 20 is irrational Prove it ​

Answer 1

Answer:- Let us assume that

root20 is an irrational number.

That implies:- root20 is a rational number.

Then there exist a co-prime positive integer (a)&(b) such that root20 = a/b.

= root20 = a/b

( squaring on both sides)

= (root20)square = a/b hole square

[ root and square cancel ]

= 20 = a^2/b^2

= 20b^2 = a^2 ...............(1)

= b^2 = a^2/20

We know that from the theorem is ( if given that p divides a^2. Therefore the fundamental theorem of arithmetic it follows that p is one of the prime factors of a^2. Also we realise that the only prime factors of a^2 are p1 p2 ....... pn. So, p is one of p1,p2,........., pn.

since p divides a^2

p divides a

that rule if 20 divides a^2

20 divides a.

Let a/20 = c

a = 20c

substituting (a) value in (1) we

get:- 20b^2 = a^2

20b^2 = (20)^2

20b^2 = 2o^2 c^2

20b^2 = 400c^2

b^2 = 400c^2/ 20

b^2 = 20c^2

we know that if 20 divides b^2 then 20 divides b.

from the above 2 cases both a&b have 20

as their factor.

But a&b are co-primes integer.

which are contradiction.

our assumption is wrong.

Therefore root20 is an irrational number.

I hope it is helps you.