Methods & Limitations

Understand how calculations are performed, what is exact, and what is only a model.

Published September 15, 2026

Arithmetic and numerical precision

The Integer, Squaring, Reciprocal, and Four-Function calculators use exact rational arithmetic based on arbitrary-size integers. Adding Decimals uses exact scaled-integer addition. Most other tools use standard double-precision numerical calculations and round results for display. Display precision is not a promise of measurement accuracy.

Polynomial and algebra scope

The zeros tool handles polynomials through degree 6 and only reports complex roots for quadratics. The rational-function tool accepts numerator and denominator degrees up to 2. Composition accepts two polynomials through degree 4 each. Area-under-graph calculations use polynomial antiderivatives through degree 6. These are intentionally scoped tools, not unrestricted symbolic-algebra systems.

Graphs are supporting views

Curves are sampled at finite points. Declared tangent asymptotes and rational-function exclusions break the path. Extreme values may be clipped to keep a graph readable. Coordinate transformation plots use equal scales. Use numeric features and the stated method alongside the visualization.

Financial calculations

Growth projections convert an effective annual return into a monthly rate using (1+r)^(1/12)-1. Regular contributions arrive at the end of each month. Money-lasting calculations explicitly model the selected timing and inflation convention. All return inputs are assumptions, not forecasts. Taxes, fees, market variation, and inflation are omitted unless a particular input explicitly models them.

Retirement projections are not plan-compliance engines

The 403(b), 457(b), and TSP tools project supplied contribution amounts. They do not enforce legal limits, catch-up rules, eligibility, tax treatment, withdrawal requirements, or vesting. Employer and agency amounts are entered manually. Governmental and nongovernmental 457(b) rules are not interchangeable even though simple growth math is the same.

Manual paycheck worksheets

The ten state-labeled worksheets subtract withholding and deduction dollar amounts supplied by the user. No current-year state or federal tax tables are embedded. These pages are marked noindex and are absent from the XML sitemap pending a validated state-specific implementation. The state label does not change the arithmetic.

Checking results

Examples and automated checks exercise the core calculations, ordinary inputs, and selected boundary cases. Testing cannot establish correctness for every possible input or replace a qualified review. Poorly conditioned algebra, unrealistic financial assumptions, and out-of-scope tasks can produce misleading outcomes. Verify important results independently.

References

Tool pages link to relevant background sources, including NIST, OpenStax, the IRS, TSP, and Investor.gov. These organizations have not certified or endorsed this site. Source links are provided for reading and verification, not as a claim of official affiliation.