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Square Root By Hand Calculator

Work out square root by hand instantly with clear inputs, formula shown and shareable results.

About Square Root By Hand Calculator

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.

Formula

Newton iteration

xₖ₊₁ = ½(xₖ + v / xₖ)

Worked example

With the default values below, Square Root By Hand Calculator returns Square root of 45.00000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Numberminimum 0
2,025
Newton iterations to showaccepts 1–20
4

Results

Square root
45.00000000
Newton's method estimate
73.56949229
Integer part of the root
45
Remainder after the integer part
0.000000

Step by step

  1. 1Values used
  2. 2Newton iteration
  3. 3Square root= 45.00000000
  4. 4Newton's method estimate= 73.56949229
  5. 5Integer part of the root= 45
  6. 6Remainder after the integer part= 0.000000

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is √2025?

Exactly 45, since 45² = 2025 and the remainder is zero.

How fast does Newton's method converge?

Quadratically. Four iterations from a rough start already give around ten correct digits.

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