Decimal Square Root Calculator

Calculator Inputs

Decimal Number

Enter zero or a positive decimal value.

Precision Digits

Controls displayed decimal places.

Maximum Iterations

Higher values allow tighter convergence.

Tolerance

Smaller tolerance increases accuracy checks.

Initial Guess

Leave blank for an automatic guess.

Rounding Mode

Choose how the displayed root is rounded.

Iteration Table

Useful for learning and verification.

Result Label

Appears in exported files.

Reset

Formula Used

The calculator applies the Newton-Raphson method to estimate the square root of a decimal number N.

x_n+1 = 0.5 × (x_n + N / x_n)

It starts from an initial guess and keeps refining the estimate until the absolute error or the change between estimates drops below the chosen tolerance.

Absolute Error = |x² − N|

Relative Error = |x² − N| / |N|

After convergence, the raw estimate is rounded using the selected mode and precision so you can compare the displayed root with its squared check.

How to Use This Calculator

Enter the positive decimal number whose square root you need.
Choose the number of decimal places to display.
Set tolerance and maximum iterations for the accuracy target.
Optionally provide an initial guess to guide convergence.
Select a rounding mode for the final displayed value.
Choose whether to show the iteration table.
Press the calculate button to show the result above the form.
Use the export buttons to save the summary as CSV or PDF.

Example Data Table

Decimal Number	Expected Square Root	Check	Use Case
2.25	1.5	1.5² = 2.25	Exact terminating decimal root
12.5	3.53553391	3.53553391² ≈ 12.5	Irrational decimal estimate
0.04	0.2	0.2² = 0.04	Small decimal verification
98.76	9.93780660	9.93780660² ≈ 98.76	Engineering and measurement checks

Try the table values in the calculator to compare exact roots, irrational roots, and small-number behavior under different precision settings.

Frequently Asked Questions

1. What does this calculator solve?

It finds the real square root of a nonnegative decimal number, then reports the rounded value, raw estimate, convergence data, and verification metrics.

2. Can it calculate roots for negative decimals?

No. This version returns only real-number square roots. Negative inputs would need complex-number handling, which is outside this calculator’s scope.

3. Why does the calculator use iterations?

Many decimal roots are irrational, so an exact finite decimal does not exist. Iteration produces a highly accurate approximation and shows how quickly the estimate converges.

4. What does tolerance control?

Tolerance sets the stopping threshold. Smaller tolerance values usually require more iterations but produce a tighter estimate and a lower final error.

5. Why is the squared check useful?

It squares the displayed result so you can compare it with the original decimal input. That makes rounding effects easy to inspect.

6. When should I enter an initial guess?

Use an initial guess when you want to study convergence or speed up solving around a known range. Leaving it blank still works well.

7. Which rounding mode should I pick?

Standard rounding fits most classroom and reporting needs. Floor, ceiling, and truncate help when you must control how the final decimal display behaves.

8. Can I export my result for records?

Yes. After calculating, use the CSV and PDF buttons to save the summary, input settings, and iteration steps for review.