Question

prove that 3 + 2√5is irrational number

Answer 1

Prove that 3 + 2√5 is irrational.

       Let take that 3 + 2√5 is a rational number.

      So we can write this number as

                                    3 + 2√5           = a/b

      Here a and b are two co prime number and b is not equal to 0

      Subtract 3 both sides we get

                                    2√5                 = a/b – 3

                                    2√5                 = (a-3b)/b

      Now divide by 2 we get

                                    √5                    = (a-3b)/2b

      Here a and b are integer so (a-3b)/2b is a rational number so √5 should be a rational number But √5 is a irrational number so it contradict the fact   

                                    Hence result is 3 + 2√5 is a irrational number

Answer 2

Let take that 3 + 2√5 is a rational number.
So we can write this number as
3 + 2√5 = a/b
Here a and b are two co prime number and b is not equal to 0
Subtract 3 both sides we get
2√5 = a/b – 3
2√5 = (a-3b)/b
Now divide by 2, we get
√5 = (a-3b)/2b
Here a and b are integer so (a-3b)/2b is a rational number so √5 should be a rational number But √5 is a irrational number so it contradicts.
Hence, 3 + 2√5 is a irrational number