Question

Show that 5 - V3 is irrational ​

Answer 1

⭐SOLUTION⭐

___________________

Let us assume 5-√3 as rational, then we have

⇒5-√3 is rational

⇒5-(5-√3 ) is rational [difference of rational is always rational]

⇒√3 is rational,

Now √3 is rational so we know that rational number can be express in the form of a/b where b is not equal to zero and a and b have no common factor other than 1

⇒√3=a/b

Squaring both sides

⇒(√3)²=a²/b²

⇒3=a²/b²

⇒3b²=a² ‐-----(1)

Thus 3 divides a² and also 3 divides a.

Let a=3c for some numbers c .

Putting a²=3b² from (1)

⇒3b²=9c²

⇒b²=3c²

Thus 3 divides b² and also divides b.

Thus 3 is common factor of a and b .

But, this contradicts the fact that a and b have no common factor other than 1

This wrong contradiction arises by our wrong assumption √3 as rational , thus √3 is irrational.

Now we know that √3 is irrational but as we have assumed 5-√3 as rational then √3 becomes rational, So this all contradiction arises be our wrong assumptions.

Thus 5-√3 is irrational.

____________________