Q7 of 114 Page 35

Prove that is an irrational numbers.

Assume that 2√3 - 1 is rational
2√3 = a/b , where a and b are integers .
 ⇒ 2√3 = a/b + 1
 ⇒ 2√3 = a/b + b/b
⇒ √3 = (a + b) / 2b
we know that a, b, and 2 are integers and they are also rational {i.e RHS is rational}
therefore √3 will be rational.
but we know that √3 is irrational.
there is a contradiction
so, 2√3 - 1 is an irrational number.


More from this chapter

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5

i. Give an example of two irrationals whose sum is rational.

ii. Give an example of two irrational whose product is rational.

 6

State whether the given statement is true or false.

i. The sum of two rationals is always rational.


ii. The product of two rationals is always rational.


iii. The sum of two irrationals is always an irrational.


iv. The product of two irrationals is always an irrational.


v. The sum of rational and an irrational is irrational.


vi. The product of a rational and an irrational is irrational.

 8

Prove that is an irrational number.

 9

Prove that is an irrational number.