Mathematical & Scientific Constants

Ratio of a circle's circumference to its diameter. Appears in geometry, trigonometry, probability, and throughout physics. An irrational and transcendental number.

Base of the natural logarithm. The unique number for which the derivative of eˣ equals eˣ. Fundamental in calculus, compound interest (continuous compounding), and probability.

The ratio a/b such that (a+b)/a = a/b. Found in geometry, Fibonacci sequences, art, architecture, and nature (e.g., spiral patterns in shells and plants).

The first known irrational number. Equals the diagonal of a unit square. Appears in the Pythagorean theorem and signal processing.

Diagonal of a unit cube face. Used in equilateral triangle geometry, hexagonal packing, and three-phase electrical power calculations.

Appears naturally in the formula for the golden ratio (φ = (1 + √5) / 2) and in the diagonal of a 1×2 rectangle.

Limiting difference between the harmonic series and the natural logarithm. Appears in number theory, probability, and the analysis of algorithms. Whether it is rational or irrational remains unknown.

Used in the rule of 70 for doubling time, binary entropy, and half-life calculations. Equals the alternating harmonic series 1 − 1/2 + 1/3 − 1/4 + …

Conversion factor between natural and common (base-10) logarithms: log₁₀(x) = ln(x) / ln(10). Essential in decibel and pH calculations.

Value of the Riemann zeta function at 3. Proved irrational by Roger Apéry in 1979. Appears in quantum electrodynamics and the calculation of electron magnetic moment.

Sum of the series 1 − 1/3² + 1/5² − 1/7² + … Appears in combinatorics, lattice models, and certain integrals in physics. Rationality remains an open question.

The real root of x³ = x + 1. The unique number satisfying ρ = ∛(ρ + ∛(ρ + ∛(…))). Used in generalisations of Fibonacci-type sequences.

Maximum speed at which information or matter can travel. Exact by definition since 1983. Foundational to special relativity (E = mc²) and the definition of the meter.

Proportionality constant in Newton's law of universal gravitation (F = Gm₁m₂/r²). One of the least precisely measured fundamental constants.

Relates the energy of a photon to its frequency (E = hf). Cornerstone of quantum mechanics and the uncertainty principle. Exact by definition since 2019.

h divided by 2π. Used in quantum mechanics wherever angular frequency replaces ordinary frequency (E = ℏω).

Charge of a single proton (or magnitude of an electron's charge). Exact by definition since 2019. Defines the ampere via fixed numerical value.

Rest mass of an electron. Used in atomic physics, spectroscopy, and calculations involving electromagnetic interactions.

Rest mass of a proton. About 1836 times the electron mass. Used in nuclear and particle physics.

Rest mass of a neutron. Slightly heavier than the proton; a free neutron is unstable and decays with a half-life of about 10 minutes.

Relates the average kinetic energy of particles to temperature. Bridges thermodynamics and statistical mechanics. Exact by definition since 2019.

Electric constant; relates electric field to charge density in a vacuum. Appears in Coulomb's law and Maxwell's equations.

Magnetic constant; relates magnetic field to current in a vacuum. Related to ε₀ and c by μ₀ = 1/(ε₀c²).

Dimensionless coupling constant of the electromagnetic force. Characterises the strength of interaction between charged particles and photons. Its value near 1/137 is unexplained by current theory.

Proportionality constant in the Stefan–Boltzmann law relating the total radiated power of a black body to the fourth power of its temperature (P = σAT⁴).

Relates the wavelengths of spectral lines of hydrogen. The most precisely measured fundamental constant (relative uncertainty ~1.9 × 10⁻¹²).

The quantum of magnetic flux through a superconducting loop. Equals h/(2e). Used in superconductivity and the Josephson effect.

Characteristic quantum length scale of the electron: λ_C = h/(mₑc). Below this scale, relativistic quantum effects dominate pair production.

Number of constituent particles (atoms, molecules, ions) in one mole of a substance. Exact by definition since 2019. Central to stoichiometry and molar mass calculations.

Appears in the ideal gas law (PV = nRT) and relates energy to temperature per mole of substance. Exact since 2019 (R = Nₐ · k_B).

Electric charge per mole of elementary charges (F = Nₐ · e). Used in electrochemistry to relate electrical charge to the amount of substance deposited or dissolved.

Defined as 1/12 of the mass of a carbon-12 atom. Used to express atomic and molecular masses (e.g., H ≈ 1 u, O ≈ 16 u).

Reference pressure at sea level. Used in gas law calculations, diving, and atmospheric science. Exactly 101,325 pascals by definition.

Volume occupied by one mole of an ideal gas at standard temperature and pressure (0 °C, 100 kPa). A useful approximation for real gases under mild conditions.

The anchor of the atomic mass scale. By definition, exactly 12 g/mol. All other atomic masses are measured relative to carbon-12.

Appears in Planck's law of black-body radiation: c₁ = 2πhc². Used to calculate spectral radiance as a function of wavelength.

Appears in Planck's law: c₂ = hc/k_B. Used to convert between wavelength and temperature for black-body radiation peaks.

Relates the peak wavelength of a black-body spectrum to its temperature (λ_max = b / T). Used in astrophysics to infer stellar surface temperatures from colour.

Volumetric mean radius of the Earth. Used as a baseline in geodesy, navigation, and any formula that converts angular distances to kilometres.

Total mass of the Earth. Used to compute surface gravity (g = GM⊕/R⊕²), orbital mechanics, and tidal forces.

Standard gravitational acceleration at sea level defined by ISO 80000-3. The exact value is used to define the kilogram-force and SI unit of pressure.

Minimum speed needed to escape Earth's gravitational pull from its surface without further propulsion. Derived from v_esc = √(2GM⊕/R⊕).

Photospheric radius of the Sun. About 109 times Earth's radius. Used in stellar structure models and solar luminosity calculations.

Total mass of the Sun. About 333,000 times Earth's mass. Used as the standard unit of mass in astrophysics for stars and galaxies.

Nominal total power radiated by the Sun as electromagnetic radiation. Adopted as an exact IAU nominal value. Used to classify stellar brightness.

Exact distance used to define the AU since 2012 (149,597,870,700 m). Approximately the mean Earth–Sun distance. The base unit for distances in the Solar System.

Distance light travels in one Julian year (365.25 days) in a vacuum. Commonly used for stellar and galactic distances.

Distance at which one astronomical unit subtends an angle of one arcsecond. 1 pc ≈ 3.2616 light years. The preferred unit for stellar distances in professional astronomy.

Average speed of Earth in its orbit around the Sun. Used in heliocentric velocity corrections and aberration calculations.

Time for Earth to complete one orbit relative to distant stars. Slightly longer than the tropical year due to axial precession. Used in positional astronomy.

Total mass of the Moon. About 1/81 of Earth's mass. Its gravitational pull drives ocean tides and stabilises Earth's axial tilt.

Average centre-to-centre distance between Earth and the Moon. Ranges from ~356,500 km (perigee) to ~406,700 km (apogee).

Rate of expansion of the universe. Value from Planck 2018 CMB data. There is an ongoing tension with distance-ladder measurements (~73 km/s/Mpc), known as the Hubble tension.

Exact standard value of gravitational acceleration used in engineering calculations. Defines the kilogram-force: 1 kgf = 9.80665 N.

Reference atmospheric pressure at sea level. Used in HVAC, pneumatics, fluid mechanics, and compressed-gas calculations.

The lowest theoretically possible temperature. At absolute zero, all classical thermal motion ceases. Defined as exactly −273.15 °C in the 2019 SI.

Maximum density of liquid water, occurring at 4 °C. The reference density used in specific gravity calculations and the original kilogram definition.

Density of dry air at 0 °C and 101,325 Pa. Used in aerodynamics, HVAC load calculations, and buoyancy problems.

Speed of longitudinal pressure waves in dry air at 20 °C. Increases by about 0.6 m/s per degree Celsius. Used in acoustics, sonar, and Mach number calculations.

Used in the ideal gas law (PV = nRT) and thermodynamic efficiency calculations. Connects molar heat capacities (Cₚ − C_v = R for ideal gases).

Thermal conductivity of pure copper at room temperature. One of the highest of any metal. Used in heat-sink and heat-exchanger design.

Stiffness (elastic modulus) of structural steel. Used in beam deflection, column buckling, and stress–strain calculations. Exact value depends on alloy composition.

Resistivity of annealed copper at 20 °C. Used to size conductors and estimate cable resistance. Basis for IACS (International Annealed Copper Standard).

Magnetic constant; defines the relationship between magnetic flux density and field intensity in a vacuum. Central to inductance and transformer design.

Electric constant; governs capacitance in a vacuum. Used in capacitor design and the calculation of electric field strengths.