Question

Show how to find rational
numbers whose squares can be arbitarily close to 2?

Answer 1

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