Question

Prove that root 5 - root 2 is an irrational number

Answer 1

Answer:

Step-by-step explanation:

So

First assume that ✓5 is a rational number

Therefore

✓5 =p\q (where p and q are integers and co prime numbers and q not equal to 0)

Then squaring both side

(✓5)^2 =p^2\q^2

5 = p^2\q^2

q^2= p^2\5 ( if p square is divisible by 5 then p is also divisible by 5)

Then, Take p =5m

q^2= (5m)^2\5

q^2= 25m^2\5

25 will be canceld by 5 and it becomes

= 5m^2\q^2 (if q square is divisible by 5 then q is also divisible by 5)

Therefore, we assumed √5 as a co prime no but is is not.....

So our assumption was wrong

=> √5 Is an irrational numbers

And as we know the sum of irrational and irrational numbers gives an irrational numbers

Therefore √5 -√2 is an irrational numbers