------
## 最终结论
```python
def babylonian_squareroot(number):
    guess = number / 2.0
    tolerance = 1e-10
    while True:
        new_guess = (guess + number / guess) / 2
        if abs(new_guess - guess) < tolerance:
            break
        guess = new_guess
    return guess

# Test cases
assert babylonian_squareroot(10)==3.162277660168379
assert babylonian_squareroot(2)==1.414213562373095
assert babylonian_squareroot(9)==3.0
```