To prove:
5 - 2√3 is an irrational number.
Solution:
Let assume that 5 - 2√3 is rational.
Therefore it can be expressed in the form of \(\frac{p}{q}\), where p and q are integers and q ≠ 0
Therefore we can write 5 - 2√3 = \(\frac{p}{q}\)
2√3 = 5 - \(\frac{p}{q}\) ⇒ √3 = \(\frac{5q-p}{2q}\)
\(\frac{5q-p}{2q}\)is a rational number as p and q are integers.
This contradicts the fact that √3 is irrational, so our assumption is incorrect.
Therefore 5 - 2√3 is irrational.
Note:
Sometimes when something needs to be proved, prove it by contradiction.
Where you are asked to prove that a number is irrational prove it by assuming that it is rational number and then contradict it.
