Jacobi/Legendre symbol with deterministic Miller–Rabin primality, Tonelli–Shanks square roots for prime moduli, and brute-force resolution for small composite moduli.

Math & Numbers

M/M/1 Queue Calculator (L, Lq, W, Wq)

Steady-state M/M/1 metrics: ρ, P0, L, Lq, W, Wq, state probabilities, and Little's-law cross-checks; unstable λ ≥ μ inputs are rejected.

Math & Numbers

M/M/c Multi-Server Queue Calculator

Steady-state M/M/c metrics with the Erlang-C formula: offered load, P0, Pw, Lq, L, W, Wq plus Little's-law cross-checks; unstable λ ≥ cμ inputs are rejected.

Math & Numbers

Radioactivity Converter (Bq / Ci / dpm)

Convert radioactivity (radionuclide activity) between becquerel (Bq, SI base = 1 decay/s), kilo/mega/gigabecquerel, curie (Ci = 3.7×10^10 Bq, exact 1950 ICRU definition, originally the activity of 1 g of radium-226), milli/microcurie, and disintegrations per minute (dpm = 1/60 Bq, exact). Converts via becquerel to the target unit and lists the equivalent value in all eight units. Reference: banana (K-40) ≈ 15 Bq, smoke detector (Am-241) ≈ 37 kBq = 1 µCi.

Math & Numbers

Radiation Absorbed Dose Converter (Gy / rad)

Convert radiation absorbed dose between Gy (gray = J/kg, SI base), kGy (1 = 1000 Gy), cGy (1 = 0.01 Gy), mGy (1 = 0.001 Gy), and rad (1 = 0.01 Gy). Converts via Gy to the target unit and lists the equivalent value in all five units. ⚠️ Absorbed dose is NOT equivalent dose: converting Gy → Sv requires multiplying by the radiation weighting factor w_R (X/γ/β = 1, neutrons 5–20, alpha 20) — use the dose-equivalent converter for that side. Reference values: chest X-ray ≈ 1e-4, CT scan ≈ 1e-2, radiotherapy fraction ≈ 2, whole-body LD50/60 ≈ 4–6 Gy.

Math & Numbers

Radiation Equivalent Dose Converter (Sv / rem)

Convert radiation equivalent dose between Sv (sievert, SI base), mSv (1 = 0.001 Sv), µSv (1 = 1e-6 Sv), rem (1 = 0.01 Sv), and mrem (1 = 1e-5 Sv). Converts via Sv to the target unit and lists the equivalent value in all five units. ⚠️ Equivalent dose is NOT absorbed dose: equivalent dose = absorbed dose (Gy) × radiation weighting factor w_R (X/γ/β = 1, neutrons 5–20, alpha 20). Converting Sv → Gy requires dividing by w_R — use the absorbed-dose converter for that side. Reference values: natural background ≈ 3 mSv/yr, chest CT ≈ 7 mSv, annual occupational limit ≈ 50 mSv/yr, ARS threshold ≈ 1 Sv.

Math & Numbers

Refrigerant Pressure-Temperature Lookup (R-22 / R-410A / R-134a)

Saturated pressure ↔ temperature lookup for common HVAC refrigerants via the Antoine fit ln(P) = A − B/(T_C + C): R-22 (HCFC-22), R-134a (HFC-134a), and R-410A (R-32/R-125 near-azeotropic blend). Coefficients fitted against the ASHRAE Handbook / NIST REFPROP saturation tables. Two modes: solve saturation pressure from temperature, or solve saturation temperature from pressure. Pressure may be absolute or gauge (atmospheric default 1.01325 bar) in bar/MPa/kPa/PSI; temperature in °C/K/°F. Engineering estimate only — does not replace the manufacturer's P-T chart.

Math & Numbers

Orifice Flow Calculator (Bernoulli, Q=Cd·A·√(2ΔP/ρ))

Compute the flow rate through a thin-plate orifice for an incompressible fluid (ISO 5167 Bernoulli form with discharge coefficient C_d). Orifice area A=π·d²/4; volumetric flow Q=C_d·A·√(2·ΔP/ρ); mass flow ṁ=C_d·A·√(2·ρ·ΔP); throat velocity v_orifice=C_d·√(2·ΔP/ρ). If the upstream pipe diameter D is supplied the diameter ratio β=d/D is also returned. ΔP≥0, ρ>0, C_d in 0..1 (default 0.61 for a sharp-edged plate). Diameter in m/cm/mm, pressure in Pa/kPa/bar/atm/psi, density in kg/m³/g/cm³; volumetric flow reported in m³/s, L/s, L/min and m³/h.

Math & Numbers

Overall Heat Transfer Coefficient (1/U=ΣR)

Compute the overall heat transfer coefficient for series thermal resistances (flat-wall model): total resistance R_total = Σ R_i (K/W), thermal conductance G = 1/R_total (W/K), area-based coefficient U = 1/(R_total·A) (W/(m²·K)), and if a temperature difference is given the heat flow rate Q = ΔT/R_total (W). Each R_i is a layer resistance already containing the area factor (e.g. convection R_conv=1/(h·A), conduction R_cond=d/(k·A), fouling R_foul=R_f''/A). Enter one resistance (K/W) per line, at least one, all positive. ΔT may be negative (reverse heat flow); a °C difference equals a K difference, a °F difference is converted by ×5/9.

Math & Numbers

Passive Earth Pressure (Rankine)

Calculate the Rankine passive earth pressure on a smooth vertical wall backfilled with cohesionless soil. K_p = tan²(45°+φ/2). Optional uniform surcharge q. Returns the base pressure p₀, the total force P, and the resultant height. Note: theoretical K_p is an upper bound; designers often apply a reduction factor for wall friction.

Math & Numbers

Rebar Area Calculator

Calculate the total cross-sectional area of reinforcing steel: A = n·π·D²/4. Choose a standard metric diameter (6–40 mm) or a custom diameter and the number of bars. Also returns the single-bar area, perimeter, theoretical unit weight (ρ=7850 kg/m³), and total weight.