Question

Prove that root 5 is a irrational number? Why.

Answer 1

Answer

=> ✓5

Let us assume that ✓5 is an rational number

So , it can be written in the form p/q where q not equal to 0.

= ✓5 = a/b

Squaring both sides

= ✓5² =a/b²

= 5² =a²/b²

= 5b² =a²_______(1)

Here we can see that a² is divisible by 5.

Now we can write

a = 5c

Put the value in equation (1)

5b² = 5c²

5b² = 25c²

Divide by 25 we get

b²/5 = c²

here b will divide by 5 & we know a is divide by 5.

But a & b are co primes so it arise contradiction

Hence ✓5 us an irrational number

Answer 2

Answer:

let us assume that root 5 is rational

√5=a/b(a and b are co prime no)

√5b=a.

squaring on both side

(√5b)^2=a^2

5b^2=a^2

a^2/b^2

so 5 divides a square

so 5 also divides a also

we can say that

a/5=c where c is some integer

a=5c

now we know that

5b square=a square

putting a =5c

5b^2=25c^2

b^2=5c square

b=5c

b is also divided by 5

but we know that a and b are co prime so this contradiction is arisen because of our wrong assumption so √5 is irrational nuber