Prove that 6 is an irrational number.

Let's assume 6 is a rational number. As per the definition of rational numbers, we can express 6 = p q , {where p , q are co-primes and q ≠ 0 }. On squaring both sides, we get, 6 2 = p 2 q 2 ⇒ 6 = p 2 q 2 ⇒ p 2 = 6 q 2 . . . . . . i Which means that, p 2 is a multiple of 6 , ⇒ 6 divides p Now, let p = 6 m ⇒ p 2 = 36 m 2 Now putting this value of p 2 in equation i , 36 m 2 = 6 q 2 ⇒ 6 m 2 = q 2 ⇒ 6 divides q 2 ⇒ 6 divides q This means that, 6 is a common divisor of p and q , which contradicts the assumption done in the first step. { p , q are co-primes}. Therefore, 6 is an irrational number.

Prove that 11 is an irrational number.

Let 11 is a rational number. As per the definition of rational numbers, we can express 11 = p q , {where p , q are co-primes and q ≠ 0 }. On squaring both sides, we get, 11 2 = p 2 q 2 ⇒ 11 = p 2 q 2 ⇒ p 2 = 11 q 2 . . . . . . i Which means that, p 2 is a multiple of 11 , ⇒ 11 divides p Now, let p = 11 m ⇒ p 2 = 121 m 2 Now putting this value of p 2 in equation i , 121 m 2 = 11 q 2 ⇒ 11 m 2 = q 2 ⇒ 11 divides q 2 ⇒ 11 divides q This means that, 11 is a common divisor of p and q , which contradicts the assumption done in the first step. { p , q are co-primes}. Therefore, 11 is an irrational number.

Prove that 13 is an irrational number.

Let 13 is a rational number. As per the definition of rational numbers, we can express 13 = p q , {where p , q are co-primes and q ≠ 0 }. On squaring both sides, we get, 13 2 = p 2 q 2 ⇒ 13 = p 2 q 2 ⇒ p 2 = 13 q 2 . . . . . . i Which means that, p 2 is a multiple of 13 , ⇒ 13 divides p Now, let p = 13 m ⇒ p 2 = 169 m 2 Now putting this value of p 2 in equation i , 169 m 2 = 13 q 2 ⇒ 13 m 2 = q 2 ⇒ 13 divides q 2 ⇒ 13 divides q This means that, 13 is a common divisor of p and q , which contradicts the assumption done in the first step. { p , q are co-primes}. Therefore, 13 is an irrational number.

Prove that 3 7 is an irrational number.

Let us assume that 3 7 is a rational number Now, let 3 7 = a b , where a , b are co-primes and b ≠ 0 ⇒ 7 = a 3 b Since a , b are rational numbers, then a 3 b is also a rational number ⇒ 7 is a rational number But, we know that, 7 is an irrational number This shows our initial assumption was incorrect Therefore, 3 7 is an irrational number.

E M B I B E

Modern's abc+ of Mathematics for Class 10>Chapter 1 - Real Numbers>EXERCISE>Q 11

MEDIUM

10th CBSE

IMPORTANT

Earn 100

Find the greatest number of $6$ digits exactly divisible by $24, 15$ and $36 .$

50% studentsanswered this correctly

Important Questions on Real Numbers

MEDIUM

10th CBSE

IMPORTANT

Modern's abc+ of Mathematics for Class 10>Chapter 1 - Real Numbers>EXERCISE>Q 12

Can two positive numbers have

28

and

812

as their

HCF

and

LCM

respectively.

MEDIUM

10th CBSE

IMPORTANT

Modern's abc+ of Mathematics for Class 10>Chapter 1 - Real Numbers>EXERCISE>Q 13

Three lighthouses flash their lights every $15$ seconds, $12$ seconds and $18$ seconds respectively. If they last flashed together at $9 : 15 a . m .$ , then at what time will they next flash together?

Write the answer in $hour : minutes am$ format.

MEDIUM

10th CBSE

IMPORTANT

Modern's abc+ of Mathematics for Class 10>Chapter 1 - Real Numbers>EXERCISE>Q 14

Kiran goes to Big Bazaar every

64

days and Harpreet goes to the same every

72

days. They met each other one day. How many days later will they meet each other again?

MEDIUM

10th CBSE

IMPORTANT

Modern's abc+ of Mathematics for Class 10>Chapter 1 - Real Numbers>EXERCISE>Q 15

There is a circular path around a sports field. Sonia takes

18

minutes to drive one round of the field, while Ravi takes

12

minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. If they meet again at the starting point at