Calculator Form
Example Data Table
| n | n² | √(n²) | Perfect Square? | Last Digit |
|---|---|---|---|---|
| 1 | 1 | 1 | Yes | 1 |
| 2 | 4 | 2 | Yes | 4 |
| 3 | 9 | 3 | Yes | 9 |
| 4 | 16 | 4 | Yes | 6 |
| 5 | 25 | 5 | Yes | 5 |
Formula Used
Square of a number: n² = n × n
Principal square root: √x, valid for x ≥ 0
Nearest lower square: floor(√x)², valid for x ≥ 0
Nearest higher square: ceil(√x)², valid for x ≥ 0
Range sum: Σ(xᵢ²) across all generated points
Consecutive positive integer identity: 1² + 2² + ... + m² = m(m + 1)(2m + 1) / 6
This calculator evaluates custom ranges directly, so it works for decimals, negatives, and non-unit steps. Perfect-square tests apply only to nonnegative integers.
How to Use This Calculator
- Enter the main number you want to analyze.
- Set the range start, range end, and step size.
- Choose the number of decimal places.
- Press Calculate to show the result above the form.
- Review the summary cards, generated table, and Plotly chart.
- Use the export buttons to download CSV or PDF output.
Frequently Asked Questions
1. What does this calculator do?
It squares a selected number, checks perfect-square status, estimates nearby square numbers, and builds a custom range table with graph and exports.
2. Can I use decimal inputs?
Yes. Decimal inputs are supported for the main number, range bounds, and step size. Perfect-square classification remains meaningful only for nonnegative integers.
3. Does it work with negative numbers?
Yes. Negative numbers can be squared and included in range tables. Real square roots are shown only when the value is zero or positive.
4. What is a perfect square?
A perfect square is a nonnegative integer formed by multiplying an integer by itself, such as 1, 4, 9, 16, and 25.
5. Why does the range size have a limit?
A limit keeps the page fast, readable, and export friendly. Large datasets are better handled by increasing the step size or narrowing the range.
6. What does the graph show?
The chart plots x against x² for every generated point. It helps visualize how squared values grow as input values move away from zero.
7. What is included in the CSV file?
The CSV includes summary metrics first, then the full generated range table with squared values, roots, parity, and digit-based fields.
8. When should I use the sum of squares formula?
Use it for consecutive positive integers starting at one. The calculator still computes direct sums for custom ranges, decimals, and irregular steps.