formula.expert

Mathematical constants

16 values, each with its units, its uncertainty, and where it came from.

Mathematical 16

Pi exact

π=3.141592653589793\pi = 3.141592653589793

dimensionlessThe ratio of a circle's circumference to its diameter, 3.14159265358979 — an irrational and transcendental number defined, never measured.

Tau (2π) exact

τ=6.283185307179586\tau = 6.283185307179586

dimensionlessThe ratio of a circle's circumference to its radius, τ = 2π = 6.28318530717959 — one full turn, and a proposed replacement for π.

Euler's Number exact

e=2.718281828459045e = 2.718281828459045

dimensionlessThe base of the natural logarithm, e = 2.71828182845905 — the unique number whose exponential function is its own derivative.

Golden Ratio exact

φ=1.618033988749895\varphi = 1.618033988749895

dimensionlessThe golden ratio φ = (1 + √5)/2 = 1.61803398874989 — the number that satisfies φ² = φ + 1 and the limit of Fibonacci ratios.

Euler–Mascheroni Constant exact

γ=0.5772156649015329\gamma = 0.5772156649015329

dimensionlessThe limiting gap between the harmonic series and the natural logarithm, γ = 0.577215664901533 — still not known to be irrational.

Square Root of Two exact

2=1.4142135623730951\sqrt{2} = 1.4142135623730951

dimensionlessThe diagonal of a unit square, √2 = 1.41421356237310 — the first number ever proved irrational, and the ratio behind A4 paper.

Square Root of Three exact

3=1.7320508075688772\sqrt{3} = 1.7320508075688772

dimensionlessThe constant of Theodorus, √3 = 1.73205080756888 — the height of an equilateral triangle of side 2 and the ratio in three-phase power.

Natural Logarithm of Two exact

ln⁡2=0.6931471805599453\ln 2 = 0.6931471805599453

dimensionlessThe natural logarithm of 2, ln 2 = 0.693147180559945 — the number that converts a decay constant into a half-life.

Natural Logarithm of Ten exact

ln⁡10=2.302585092994046\ln 10 = 2.302585092994046

dimensionlessThe natural logarithm of 10, ln 10 = 2.30258509299405 — the factor that converts between natural and common logarithms.

Common Logarithm of e exact

log⁡10e=0.4342944819032518\log_{10} e = 0.4342944819032518

dimensionlessThe base-10 logarithm of e, 0.434294481903252 — the modulus of common logarithms, and the reciprocal of ln 10.

Degrees per Radian exact

180π=57.29577951308232 ∘/rad\frac{180}{\pi} = 57.29577951308232\ ^{\circ}\text{/rad}

°/radThe size of one radian in degrees, 57.2957795130823° — an exact conversion factor, since the degree is defined as exactly π/180 radians.

Radians per Degree exact

π180=0.017453292519943295 rad\frac{\pi}{180} = 0.017453292519943295\ \text{rad}

radOne degree expressed in radians, π/180 = 0.0174532925199433 rad — the exact factor for converting degrees into radians.

Gradians per Degree exact

109=1.1111111111111112 gon/∘\frac{10}{9} = 1.1111111111111112\ \text{gon/}^{\circ}

gon/°The exact ratio of gradians to degrees, 10/9 = 1.11111111111111 — a right angle is 100 gon, so 90° maps onto 100 gon.

Catalan's Constant exact

G=0.915965594177219G = 0.915965594177219

dimensionlessCatalan's constant G = 0.915965594177219 — the alternating sum of reciprocal odd squares, of unknown irrationality after 160 years.

Apéry's Constant exact

ζ(3)=1.2020569031595942\zeta(3) = 1.2020569031595942

dimensionlessApéry's constant ζ(3) = 1.20205690315959 — the sum of reciprocal cubes, proved irrational in 1978 and still with no closed form.

Feigenbaum Delta exact

δ=4.66920160910299\delta = 4.66920160910299

dimensionlessThe Feigenbaum bifurcation constant δ = 4.66920160910299 — the universal ratio at which period doubling accelerates into chaos.