Question

Find a rational number whose square is arbitrarily closer to 2

Answer 1

Solution :-

We know that (1)² = 1 and (2)² = 4

To find the correct answer of this question, we have to choose rational numbers between 1 and 2, e.g. 1.1, 1.2, 1.3.......1.9

Since (1.4)² = 1.96 and (1.5)² = 2.25, we consider rational numbers between 1.4 and 1.5, e.g. 1.41, 1.42, 1.43......1.49

Continuing in this manner, we can obtain closer and closer rational approximation, e.g. (1.414213562)² is less than 2

While, (1.414213563)² is more than 2.

So, the required rational number will be 1.414213562, whose square is arbitrarily closer to 2.

Answer 2

We restrict ourselves to positive rational numbers. since (1)² = 1 and (2)² = 4 , we led to choose rational numbers between 1 and 2.
e.g., 1.1 , 1.2 , 1.3, 1.4 ......... 1.9.

since (1.4)² = 1.96 and (1.5)² = 2.25
so, we consider rational number between 1.4 and 1.5. e.g., 1.41, 1.42, 1.43.......1.49.

continuing in this manner we can obtain closer and closer rational approximations.
e.g., (1.4142135624) is less than 2 and (1.4142135624)² is greater than 2.e.g., (1.4142135624)² = 2.0000000001 > 2