Work out square root by hand instantly with clear inputs, formula shown and shareable results.
The by-hand long-division method pairs digits from the decimal point outwards. Newton's iteration x ← ½(x + v/x) reaches the same answer and doubles the number of correct digits each step.
Newton iteration
xₖ₊₁ = ½(xₖ + v / xₖ)
Exactly 45, since 45² = 2025 and the remainder is zero.
Quadratically. Four iterations from a rough start already give around ten correct digits.