Decimal to Square Root Calculator

Turn any decimal into clean square root notation quickly. Choose exact, nearest integer, or simplified rational radicand outputs. Control precision, tolerance, and denominator bounds for accuracy tuning. Export CSV or PDF, and share calculated steps. Fast, reliable, transparent results with clear educational notes included.

Enter Decimal & Options

Supports standard or scientific notation.
Used for approximating x² as m/n.
Marks “exact” if relative error ≤ given ppm.
We search squarefree b ≤ 100 for a compact a√b approximation.

Quick Notes

  • Negative decimals are supported; results include the sign.
  • Nearest √N picks N ≈ x² to minimize absolute error.
  • Rational √(m/n) uses continued fractions bounded by Max denominator.
  • We simplify to a canonical form K·√C with C squarefree.

Results

Input decimal
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Nearest √N
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Canonical K·√C
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Relative error
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Form Expression Value Absolute Error Relative Error (%) Actions
Enter a decimal and click Calculate to see results.

Example Data

Click a row to load the decimal into the calculator.

Decimal Typical Radical Notes
1.414213562≈ √2Square root of 2 approximation
2.236067977≈ √5Square root of 5 approximation
0.5= √(1/4)Exact: equals 1/2
-3.16227766≈ -√10Negative value supported
3.0= √9Perfect square when squared

Formulas Used

  1. Given decimal x, compute y = x².
  2. Nearest √N: choose N = round(y). Then x ≈ sgn(x)·√N. Error: |x − sgn(x)·√N|.
  3. Rational radicand: approximate y ≈ m/n with small integers using continued fractions (denominator ≤ Max).
  4. Simplify √(m/n) to canonical K·√C, where C is squarefree:
    • Let m = s²·r, n = t²·u with r,u squarefree.
    • √(m/n) = (s/t)·√(r/u) = (s/(t·u))·√(r·u).
    • Extract any square factor from r·u into K; remaining under-root part is C.
  5. Relative error (%) = 100·|approx − x|/max(1,|x|).

How to Use

  1. Enter a decimal (positive or negative).
  2. Adjust precision, tolerance, and max denominator if needed.
  3. Click Calculate to compute all forms.
  4. Review the nearest √N and canonical K·√C representation.
  5. Copy expressions or export your results to CSV or PDF.

FAQs

It means a coefficient K (often a reduced fraction) multiplied by a square root of a squarefree integer C. No square factors remain inside C.

If x² is a rational number whose simplified radical becomes a rational multiple of √1, the radical part vanishes and the value is exact rational.

Representing a decimal as the square root of a nearby integer is compact and widely recognized. It also gives an intuitive feel for magnitude.

An integer is squarefree if no prime square (like 4, 9, 25) divides it. Keeping the radicand squarefree is a standard simplification rule.

Smaller tolerance flags equality more strictly; smaller denominators force shorter fractions and may increase approximation error. Increase denominators for tighter fits.

Yes. We compute on |x| then apply the sign to the final expression, e.g., −√N or −K·√C.

They may use different maximum denominators, rounding strategies, or stopping rules in continued fractions, producing different yet valid approximations.

Reference: Common Decimals → Radical Forms

Handy mappings to check your results. Values rounded to 9–10 decimals.

Decimal Suggested Radical Canonical K·√C Note
0.353553391√(1/8)¼·√2Same as 1/(2√2)
0.500000000√(1/4)1/2Exact rational
0.707106781√(1/2)½·√2Squarefree C=2
1.414213562√2√2Irrational
1.732050807√3√3Irrational
2.000000000√42Perfect square
2.236067977√5√5Squarefree C=5
2.828427125√82·√2Extract square factor 4
3.162277660√10√10Squarefree C=10
-3.605551275-√13−√13Negative supported

Squarefree Radicands (2–30) Cheat Sheet

Squarefree C (no square factors) with prime factorization and √C approximations.

C Prime factors √C (≈) Tip
221.41421356Very common constant
331.73205081√3 in trig/geometry
552.23606798Appears in golden ratio
62·32.44948974Product of 2 and 3
772.64575131Prime radicand
102·53.16227766Squarefree composite
11113.31662479Prime radicand
13133.60555128Prime radicand
142·73.74165739Squarefree composite
153·53.87298335Squarefree composite
17174.12310563Prime radicand
19194.35889894Prime radicand
213·74.58257569Squarefree composite
222·114.69041576Squarefree composite
23234.79583152Prime radicand
262·135.09901951Squarefree composite
29295.38516481Prime radicand
302·3·55.47722558Squarefree composite
© Decimal to Square Root Calculator

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