prove that (-2+√5) is irrational
Answers
(-2+✓5) let us assume that it is rational so
(-2+✓5)=a/b (where b is not equal to 0 )
send -2 to RHS then it becomes +2
✓5=a/b+2
✓5=a+2b/2
here still a,b are integers so a+2b/2 is rational so ✓5 is also rational but it contradict s the fact that ✓5 is irrational
this contradiction had arisen because of our incorrect assumption so our assumption is wrong (-2+✓5) is rational
Let (-2+√5) be rational.
Let (-2+√5) = r, where 'r' is an integer and r≠0
√5 = r + 2
Next, we have to prove that, √5 is irrational.
Let √5 be rational.
√5 = p/q, where 'p' and 'q' are integers, co primes and q≠0
Squaring, 5 = p²/q²
p² = 5q² -------------(1)
5 divides p², 5 divides p also.
If p=5m, where 'm' is an integer.
Substituting p=5m in (1)
(5m)² = 5q²
25m² = 5q²
5m² = q²
5 divides q², 5 divides q also.
⇒5 is a common factor of p & q which is a contradiction.
∴ Our assumption is wrong.
∴ √5 is irrational.
√5 = r + 2
Since we proved that √5 is irrational, An irrational number equal to a rational number is not possible.
∴ Our assumption is wrong.
∴ (-2+√5) is irrational.