How to find the square root of a number using only addition, subtraction, multiplication, and division operations?

To calculate the square root of a number without using built-in functions, you can apply Newton's method (the tangent method), using only addition, subtraction, multiplication, and division operations.

Method idea:

• Start with an initial approximation x (for example, x = number / 2).

• Iteratively improve the approximation using the formula:

  x = (x + number / x) / 2

• Repeat until the difference between the current and previous values becomes sufficiently small.

Example in Python:

number = 25.0
x = number / 2.0
epsilon = 1e-10
while True:
    prev_x = x
    x = (x + number / x) / 2.0
    if abs(x - prev_x) < epsilon:
        break
print(f"Square root of {number} ≈ {x}")

This method converges quickly and does not require special functions.