Show that 5 - 2√3 is an irrational number.

1.	Show that 5 - 2√3 is an irrational number.
Answer» 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.

Answer»

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.

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