How to compute square roots by hand using long division or newton's method for numbers like 123?
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Executive summary
Two reliable hand methods exist to compute square roots: Newton’s method (the iterative averaging formula often called Heron’s method) quickly refines an initial guess and is what calculators use in practice [1] [2], while the classical digit‑by‑digit "long division" square‑root algorithm produces digits one at a time but is not described in the provided reporting; therefore its stepwise description is outside the dataset used here.
1. Newton’s method: the idea and the formula that drives convergence
Newton’s method finds roots of f(x)=0 by iterating x{n+1} = xn − f(x_n)/f′(x_n); for square roots set f(x)=x^2 − N which yields the familiar iteration x{n+1} = 0.5*(xn + N/x_n) — an averaging of the guess and N divided by the guess — a specialization historically attributed to Heron and widely documented as the standard algorithm used in calculators [3] [1] [2].
2. Why Newton’s method converges fast (and what can go wrong)
When f′ at the root is nonzero and f is smooth, Newton’s method converges quadratically: roughly doubling the number of correct digits each iteration once close enough to the true root; this is why only a handful of iterations usually give high accuracy [3]. Failure modes noted in standard texts include encountering an iterate where the derivative is zero or poor global behavior leading to a different root, but for positive N and the square‑root specialization these problems are rare with a reasonable initial guess [4] [3].
3. Hands‑on: computing √123 by Newton iterations (worked example)
Start by setting N = 123 and choosing an initial guess; a sensible integer guess is 11 because 11^2 = 121 is close to 123, then apply x{n+1} = 0.5*(xn + 123/x_n) repeatedly (the formula and approach are the standard Newton/Heron specialization [1] [2]). With x0 = 11, x1 = 0.5(11 + 123/11) ≈ 0.5(11 + 11.181818) ≈ 11.090909; x2 = 0.5(11.090909 + 123/11.090909) ≈ 11.0907225, and a third iteration yields x3 ≈ 11.0905365, which is already accurate to many decimals — this rapid improvement is the quadratic convergence behavior textbooks describe [3] [5].
**4. Practical stopping rules and numeric hygiene**
Texts and programming examples recommend stopping when successive iterates change by less than a chosen tolerance or when |guess^2 − N| is small relative to the magnitude of the guess (a relative tolerance) to avoid needless iterations; SICP and other instructional sources implement a relative‑tolerance test |guess^2 − N| < guess tolerance to balance small and large N [2] [5]. Implementations also guard against divides by zero and choose initial guesses (N or a nearby perfect square) that avoid pathological trajectories [6] [5].
5. The long‑division (digit‑by‑digit) square‑root algorithm: reporting limits and where to look
The classical digit‑by‑digit “long division” method yields one decimal (or integer) digit per pass and is valuable for extracting exact decimal digits by hand; however, the provided reporting and sources do not include a worked description or authoritative steps for that long‑division algorithm, so a sourced, stepwise manual procedure cannot be reproduced from this dataset — interested readers should consult a numerical methods or elementary arithmetic text for the annotated digit‑by‑digit algorithm (reporting limitation).
6. How to choose between methods in practice
For hand calculation aiming for a few decimal places, Newton’s method is simple, fast and requires only arithmetic and division; for formal digit‑by‑digit generation of every decimal digit without intermediate rounding, the long‑division style algorithm is appropriate but was not detailed in the provided sources [1] [2]. For programming or calculator work, Newton’s method is the standard because of its simplicity and quadratic convergence which is emphasized in numerical‑analysis and educational sources [1] [3].