Question

When approximating the square root of 2, which is the best choice for a next guess?

a) (a² + b²)/2
b) (a+b)/2
c) (a-b)/2
d) (a² - b²/2)

Answer 1

To approximate the square root of 2, the best guess from the given options is the average of two approximate values (a and b), which is represented by the formula (a+b)/2. This formula is aligned with the iterative approximation techniques used in methods like the Babylonian method for square rooting.

Step-by-step explanation:

When approximating the square root of 2, one effective method is to use the process of averaging paired numbers that are close to the value being square rooted. Of the options provided, the best choice for a next guess in the approximation would be (a+b)/2, which is the average of two numbers, a and b. This method stems from the process of iterative approximation, particularly related to the Babylonian method or Newton-Raphson method for finding square roots.

Let's say you want to approximate √2, and you have two numbers where one is a bit too low (a) and the other is a bit too high (b), taking the average of these two numbers will provide a better approximation of the square root.

For instance, if a = 1 and b = 2, since 1^2 < 2 and 2^2 > 2, the average (1+2)/2 = 1.5 will be closer to the actual square root of 2.