Question 10 of 10intermediate🔍 AnalyzeLong Answer6 marks

Prove that the following are irrationals:

(i)

1/2\sqrt{2}

Answer

Assume 12\frac{1}{\sqrt{2}} is rational.

12=ab\frac{1}{\sqrt{2}} = \frac{a}{b} 2=ba\sqrt{2} = \frac{b}{a}

Since ba\frac{b}{a} is rational, 2\sqrt{2} would be rational, contradicting Theorem 1.3.

∴ 12\frac{1}{\sqrt{2}} is irrational.

(ii)

75\sqrt{5}

Answer

Assume 757\sqrt{5} is rational.

75=ab7\sqrt{5} = \frac{a}{b} 5=a7b\sqrt{5} = \frac{a}{7b}

Since a7b\frac{a}{7b} is rational, 5\sqrt{5} would be rational, contradicting the fact that 5\sqrt{5} is irrational.

∴ 757\sqrt{5} is irrational.

(iii)

6 + 2\sqrt{2}

Answer

Assume 6+26 + \sqrt{2} is rational.

6+2=ab6 + \sqrt{2} = \frac{a}{b} 2=a−6bb\sqrt{2} = \frac{a - 6b}{b}

Since a−6bb\frac{a - 6b}{b} is rational, 2\sqrt{2} would be rational, contradicting Theorem 1.3.

∴ 6+26 + \sqrt{2} is irrational.

Explanation

(i) The reciprocal of an irrational number is irrational. If 12\frac{1}{\sqrt{2}} were rational, then 2\sqrt{2} would also be rational, which contradicts the proven fact that 2\sqrt{2} is irrational.

(ii) The product of a non-zero rational number and an irrational number is irrational. If 757\sqrt{5} were rational, then 5\sqrt{5} would be rational, which is false.

(iii) The sum of a rational number and an irrational number is irrational. If 6+26 + \sqrt{2} were rational, then 2\sqrt{2} would be rational, contradicting the proven fact that 2\sqrt{2} is irrational.

Previous