Exercise: Loops and Functions

Let’s put loops and functions to work: implement a square root function without using math.Sqrt. Given a number x, find z such that z² is as close to x as possible.

The trick is Newton’s method. Start with a guess for z, then refine it with:

z -= (z*z - x) / (2*z)

Each iteration, the guess gets closer. A starting value of z = 1 works fine for any input.

Give it a shot:

1. Implement Sqrt in the provided function. Run the loop 10 times and print z on each step.
2. Watch how fast it converges for x = 2, x = 3, etc.
3. Then change the loop to stop when z barely changes between iterations — you’ll likely need fewer than 10 steps.

To initialize a float, use either:

z := 1.0
z := float64(1)

Compare your final result to math.Sqrt — they should match closely.

For the curious: z² − x measures how far off your guess is, and dividing by 2z (the derivative of z²) scales the correction. This is Newton’s method — it converges remarkably fast for square roots.