Prove that 1 3 is irrational.

Suppose that 1 3 is rational. Hence, 1 3 can be written in form of p q , where p , q ( q ≠ 0 ) are co-prime. 1 3 = p q ⇒ 3 = q p ⇒ q p is a rational number. ⇒ 3 is rational. But this contradicts that 3 is irrational. Hence, our supposition is wrong. Therefore, 1 3 is irrational.

Prove that 13 is irrational.

Let 13 be a rational number, then 13 = p q , where p, q are integers, q ≠ 0 and p , q have no common factors. ⇒ 13 = p 2 q 2 ⇒ p 2 = 13 q 2 . . . . i As 13 divides 13 q 2 , so 13 divides p 2 ⇒ 13 divides p Let p = 13 m, where m is an integer. Substituting this value of p in i , we get 13 m 2 = 13 q 2 ⇒ 169 m 2 = 13 q 2 ⇒ 13 m 2 = q 2 As 13 divides 13 m 2 , so 13 divides q 2 13 divides q Thus, p and q have a common factor 13 .This contradicts that p a n d q have no common factors (except 1 ) Hence, 13 is not a rational number. So, we conclude that 13 is an irrational number.

Prove that 5 - 3 2 is irrational.

Let us assume that 5 - 3 2 is a rational number, then 5 - 3 2 = r (say), where r is a rational number. ⇒ 3 2 = 5 - r ⇒ 2 = 5 - r 3 As r is a rational number, so 5 - r 3 is also rational number. ⇒ 2 is a rational number. But this contradicts that 2 is an irrational number. Hence, our assumption is wrong. Therefore, 5 - 3 2 is an irrational number.

Prove that 3 + 2 7 is irrational.

Let us assume that 3 + 2 7 is a rational number, then 3 + 2 7 = r (say), where r is a rational number. ⇒ 2 7 = r - 3 ⇒ 7 = r - 3 2 As r is a rational number, so r - 3 2 is also rational number. ⇒ 7 is a rational number. But this contradicts that 7 is an irrational number. Hence, our assumption is wrong. Therefore, 3 + 2 7 is an irrational number.

Prove that 7 - 2 3 5 is irrational.

Let us assume that 7 - 2 3 5 is a rational number, then 7 - 2 3 5 = r (say), where r is a rational number. ⇒ 7 - r = 2 3 5 ⇒ 5 = 3 7 - r 2 As r is a rational number, so 3 7 - r 2 is also rational number. ⇒ 5 is a rational number. But this contradicts that 5 is an irrational number. Hence, our assumption is wrong. Therefore, 7 - 2 3 5 is an irrational number.

Prove that 2 + 5 is irrational.

Let us assume that 2 + 5 is a rational number, then 2 + 5 = r (say), where r is a rational number. ⇒ 5 = r - 2 ⇒ 5 2 = r - 2 2 Use the identity a - b 2 = a 2 + b 2 - 2 a b ⇒ 5 = r 2 + 2 2 - 2 × r × 2 ⇒ 5 = r 2 + 2 - 2 r 2 ⇒ 2 r 2 = r 2 - 3 2 = r 2 - 3 2 r As r is a rational number and r ≠ 0 , so r 2 - 3 2 r is also rational number. ⇒ 2 is a rational number. But this contradicts that 2 is an irrational number. Hence, our assumption is wrong. Therefore, 2 + 5 is an irrational number.

E M B I B E

CBSE Syllabus Standard Mathematics for Class X>Chapter 1 - Real Numbers>Chapter Test>Q 6

EASY

10th CBSE

IMPORTANT

Earn 100

Show that $15^{n}$ cannot end with the digits $0, 2, 4, 6 o r 8$ for any natural number $n .$

Important Questions on Real Numbers

EASY

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 1 - Real Numbers>Chapter Test>Q 7

Prove that

\frac{1}{\sqrt{3}}

is irrational.

EASY

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 1 - Real Numbers>Chapter Test>Q 7

Prove that

\sqrt{13}

is irrational.

EASY

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 1 - Real Numbers>Chapter Test>Q 7

Prove that

5 - 3 \sqrt{2}

is irrational.

EASY

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 1 - Real Numbers>Chapter Test>Q 7

Prove that

3 + 2 \sqrt{7}

is irrational.

EASY

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 1 - Real Numbers>Chapter Test>Q 7

Prove that

7 - \frac{2}{3} \sqrt{5}

is irrational.

EASY

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 1 - Real Numbers>Chapter Test>Q 7

Prove that

\sqrt{2} + \sqrt{5}

is irrational.

EASY

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 1 - Real Numbers>Chapter Test>Q 8

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion. Also write the decimal expansion which have terminating decimal expansions.

$\frac{21}{16}$

EASY

10th CBSE

IMPORTANT

CBSE Syllabus Standard Mathematics for Class X>Chapter 1 - Real Numbers>Chapter Test>Q 8

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion. Also write the decimal expansion which have terminating decimal expansions.

$\frac{9}{240}$