Prove that 5 is irrational.

Let us assume, on the contrary, that 5 is a rational number. Therefore, we can find two co-prime integers a , b ( b ≠ 0 ) such that 5 = a b . 5 = a b ⇒ a = 5 b ⇒ a 2 = 5 b 2 Therefore, a 2 is divisible by 5 , then a is also divisible by 5 . So, a 2 = 5 k 2 = 5 5 k 2 = 5 b 2 ⇒ b 2 = 5 k 2 This means that b 2 is divisible by 5 and hence, b is divisible by 5 . This implies that a and b have 5 as a common factor. And this is a contradiction to the fact that a and b are co-prime. So, our assumption that 5 is rational is wrong. Hence, 5 cannot be a rational number. Therefore, 5 is irrational.

Prove that 5 is irrational.

Let us assume, on the contrary, that 5 is a rational number. Therefore, we can find two co-prime integers a , b ( b ≠ 0 ) such that 5 = a b . 5 = a b ⇒ a = 5 b ⇒ a 2 = 5 b 2 Therefore, a 2 is divisible by 5 , then a is also divisible by 5 . So, a 2 = 5 k 2 = 5 5 k 2 = 5 b 2 ⇒ b 2 = 5 k 2 This means that b 2 is divisible by 5 and hence, b is divisible by 5 . This implies that a and b have 5 as a common factor. And this is a contradiction to the fact that a and b are co-prime. So, our assumption that 5 is rational is wrong. Hence, 5 cannot be a rational number. Therefore, 5 is irrational.

Prove that 3 + 2 5 is irrational.

Let us assume, on the contrary, that 3 + 2 5 is rational. Therefore, we can find two integers a , b ( b ≠ 0 ) such that 3 + 2 5 = a b 2 5 = a b - 3 5 = 1 2 a b - 3 Since a and b are integers, 1 2 a b - 3 will also be rational and therefore, 5 is rational. This contradicts the fact that 5 is irrational. Hence, our assumption that 3 + 2 5 is rational is false. Therefore, 3 + 2 5 is irrational.

Prove that the following is irrational. 1 2

Given : 1 2 Let us assume that 1 2 is rational. Therefore, we can consider co-primes as a , b ( b ≠ 0 ) such that 1 2 = a b 2 = b a b a is rational as a and b are integers. Therefore, 2 is rational. This contradicts the fact that 2 is irrational. Hence, our assumption is false and 1 2 is irrational.

Prove that the following are irrational. 7 5

Given, 7 5 Let us assume that 7 5 is rational. Therefore, we can find two integers a , b ( b ≠ 0 ) such that 7 5 = a b ∴ 5 = a 7 b a 7 b is rational as a and b are integers. Therefore, 5 should be rational. This contradicts the fact that 5 is irrational. Therefore, our assumption that 7 5 is rational is false. Hence, 7 5 is irrational.

Prove that the following is irrational. 6 + 2

Let 6 + 2 be rational. Therefore, we can find two integers a , b b ≠ 0 such that, 6 + 2 = a b 2 = a b - 6 Since a and b are integers, a b - 6 is a rational number and hence, 2 should be rational. This contradicts the fact that 2 is irrational. Therefore, our assumption is false and hence, 6 + 2 is irrational.

E M B I B E

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.3>Q 1

HARD

10th Assam Board

IMPORTANT

Earn 100

Prove that $\sqrt{5}$ is irrational.

Important Questions on Real Numbers

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.3>Q 2

Prove that

3 + 2 \sqrt{5}

is irrational.

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.3>Q 3

Prove that the following is irrational.

\frac{1}{\sqrt{2}}

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.3>Q 3

Prove that the following are irrational.

7 \sqrt{5}

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.3>Q 3

Prove that the following is irrational.

6 + \sqrt{2}

EASY

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.4>Q 1

Without actually performing the long division state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

\frac{13}{3125}

EASY

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.4>Q 1

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

\frac{17}{8}

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.4>Q 1

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

\frac{64}{455}

EASY

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.4>Q 1

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

\frac{15}{1600}

Learn and Prepare for any exam you want !

|

|

|

|

|

|

|

|