Exports
Submit a calculation to enable CSV and PDF downloads.
Calculator Inputs
Plotly Graph
The graph shows the curve y = x², the horizontal line y = input, and the intersection point at the square root.
Example Data Table
| Decimal Number | Square Root | Squared Check | Notes |
|---|---|---|---|
| 0.04 | 0.20000000 | 0.04000000 | Small decimal value |
| 0.81 | 0.90000000 | 0.81000000 | Common decimal square |
| 2.25 | 1.50000000 | 2.25000000 | Exact rational root |
| 12.25 | 3.50000000 | 12.25000000 | Exact decimal root |
| 50.5 | 7.10633520 | 50.50000000 | Irrational square root |
| 144.50 | 12.02081528 | 144.50000000 | Larger decimal input |
Formula Used
The square root of a number N is a value x such that:
This page also uses the Newton-Raphson update formula for iterative estimation:
Each new estimate usually gets closer to the true square root very quickly. The calculator compares the iterative estimate with the standard square root result and reports the absolute and relative difference.
How to Use This Calculator
- Enter any non-negative decimal or whole number.
- Select how many decimal places you want in the result.
- Choose a method or compare both methods together.
- Optionally add an initial guess for Newton-Raphson.
- Set the number of Newton iterations.
- Click Calculate Square Root.
- Review the result summary above the form.
- Use the graph and optional step table for deeper analysis.
- Download the final output as CSV or PDF.
FAQs
1) What is a decimal square root?
A decimal square root is the number that produces the original decimal when multiplied by itself. For example, the square root of 12.25 is 3.5 because 3.5 × 3.5 equals 12.25.
2) Can this calculator handle whole numbers too?
Yes. Whole numbers are valid inputs because they are also decimal-compatible values. You can enter numbers like 4, 25, or 144 and get their square roots with your chosen precision.
3) Why are there two methods shown?
The page includes a direct square root result and a Newton-Raphson estimate. Showing both helps you compare numerical methods, check convergence, and learn how iterative approximation reaches the final answer.
4) What happens if I enter a negative number?
This calculator is limited to real-number square roots, so negative inputs are rejected. A negative number would require complex-number output, which is outside the scope of this page.
5) How much precision should I choose?
Use fewer decimal places for quick estimates and more decimal places for technical work. For classroom use, 4 to 8 decimals are often enough. Analytical tasks may need higher precision.
6) Why does Newton-Raphson need an initial guess?
Newton-Raphson starts from a guess and improves it step by step. A reasonable starting value helps it converge quickly. This calculator can generate one automatically when you leave the field empty.
7) What does the squared check mean?
The squared check multiplies the displayed square root by itself. It helps confirm that the output returns close to the original input value, especially when rounding and finite precision are involved.
8) What do the CSV and PDF downloads include?
The exports include the input, selected precision, method details, square root values, error information, and Newton-Raphson steps when available. They are useful for reports, class notes, and audit trails.