Mathematical Constants
compiled by Stanislav Sýkora, Extra Byte, Via R.Sanzio 22C, Castano Primo, Italy 20022
in Stan's Library, Ed.S.Sykora, Vol.II. First release March 31, 2008.
Permalink via DOI:  10.3247/SL2Math08.001
Stan's Library | Physics Constants Stan's LINKS | Stan's HUB
Note: This list is so far rudimentary but it keeps growing and will soon include some categories not covered elsewhere.
Basic math constants 
π, Archimedes' constant  3.141 592 653 589 793 238 462 643 ...  Circumference of a disc with unit diameter. 
e, Euler number  2.718 281 828 459 045 235 360 287 ...  Base of natural logarithms. 
γ, Euler-Mascheroni constant  0.577 215 664 901 532 860 606 512 ...  limit of [(1+1/2+1/3+...1/n)-ln(n)]. 
√2, Pythagora's constant  1.414 213 562 373 095 048 801 688 ...  Diagonal of a square with unit side. 
Φ, Golden ratio  1.618 033 988 749 894 848 204 586 ...  Φ = 1/(1-Φ) or Φ = (√5 + 1)/2. 
φ, Inverse golden ratio (often confused)  0.618 033 988 749 894 848 204 586 ...  φ = 1/Φ = 1 - Φ =(1-φ)/φ or φ = (√5 - 1)/2. 
Derived math constants 
ln(2), Natural logarithm of 2  0.693 147 180 559 945 309 417 232 ...  ex = 2
log(2), Decadic logarithm of 2  0.301 029 995 663 981 195 213 738 ...  10x = 2
ln(10), Natural logarithm of 10  2.302 585 092 994 045 684 017 991 ...  ex = 10
ln2(10), Binary logarithm of 10  3.321 928 094 887 362 347 870 319 ...  2x = 10
log(e), Decadic logarithm of e  0.434 294 481 903 251 827 651 128 ...  10x = e
ln2(e), Binary logarithm of e  1.442 695 040 888 963 407 359 924 ...  2x = e
Square root of golden ratio  1.272 0196 495 140 689 642 524 224 ...  [(√5 + 1)/2]1/2. Relates sides of squares with golden-ratio areas.
Square root of inverse golden ratio  0.786 151 377 757 423 286 069 559 ...  [(√5 - 1)/2]1/2.
Cubic root of golden ratio  1.173 984 996 705 328 509 966 683 ...  [(√5 + 1)/2]1/3. Relates sides of cubes with golden-ratio volumes.
Cubic root of inverse golden ratio  0.851 799 642 079 242 917 055 213 ...  [(√5 - 1)/2]1/3.
Classical, named math constants 
ζ(3), Apery' constant  1.202 056 903 159 59 ...  Special value of the Riemann function ζ(x). 
B, Brun' constant  1.902 160 58 ...  Sum of the reciprocals of twin primes 
C, Catalan's constant  0.915 965 594 177 219 015 ...  C = Sum(n=0,1,2,...)[(-1)^n/(2n+1)^2] 
α, Feigenbaum constant  2.502 907 875 0 ...  Appears in the theory of chaos. 
δ, Feigenbaum constant  4.669 201 609 1 ...  Appears in the theory of chaos. 
M, Gauss' lemniscate constant  1.198140 234 735 59...  Length of lemniscate [r2=cos(2θ)] is 2π/M.
A, Gleisher-Kinkelin constant  1.282 427 129 ...  Appears in number theory. 
λ, Golomb-Dickman constant  0.624 329 989 ...  Longest cycle distribution in random permutations. 
λ, Laplace limit constant  0.662 743 419 3...  Let η = √(1+λ2); then λeη = 1+η.
M3, Madelung's constant  -1.747 564 594 633 ...  M3 = Sum(i,j,k)[(-1)^(i+j+k)/sqr(i^2+j^2+k^2)] 
A, Moving sofa constant  2.219 531 668 871 ...  Greatest sofa that can turn a hallway corner. 
α, Otter's constant  2.955 765 285 652 ...  Appears in tree enumeration. 
β, Otter's constant  0.534 949 606 142 ...  Appears in tree enumeration. 
p3, Polya's random-walk constant  0.340 537 339 287 ...  Probability to return back in a 3D-lattice random walk. 
m, Rényi's parking constant  0.747 597 920 253 ...  Density of randomly parked cars in a street. 
G, Wilbraham-Gibbs constant  1.851 937 051 982 466 170 4...  G = Integral of sin(θ)/θ from 0 to π 
Geometry constants 
ρ2, 2D close packing ratio  0.906 899 682 117 089 252 970 392 ...  ρ2 = π / 2√3. Best covering of 2D plane by disks. 
ρ3, 3D close packing ratio  0.740 480 489 693 061 041 169 313 ...  ρ3 = π / 3√2. Best covering of 3D space by spheres. 
Minimum area of a 2D figure of unit width  0.704 770 923 010 457 972 467 598 ...  (pi - sqrt(3))/2. See the link to Reuleaux triangle. 
Cube: body diagonal / side ratio  1.732 050 807 568 877 293 527 446 ...  √3. Diagonal of a cube with unit side. 
Cube: body diagonal / face diagonal  1.224 744 871 391 589 049 098 642 ...  √(3/2). 
Cube: angle between body diagonal and a side  54.735 610 317 245 345 684 622 999 ...  cos(φm) = √(1/3) in degrees. See magic angle (below) 
Cube: angle between body and face diagonals  35.264 389 682 754 654 315 377 000 ...  90 - magic angle; in degrees. 
Solid angle fraction of the magic-angle cone  0.211 324 865 405 187 117 745 425 ...  (1-1/√3)/2; Simple cone with polar angle = magic angle 
Polar angle of the golden-ratio cone  76.345 415 254 024 494 986 936 602 ...  Simple cone that cuts the solid angle in golden ratio 
   Same as above, but in radians  1.332 478 864 985 030 510 208 009 ...  cosθ = 2φ-1; φ = Golden ratio 
Statistics and probability constants 
Normal probability density: Maximum  0.398 942 280 401 432 677 939 946 ...  1/√(2π); for normal pdf with std.dev. σ = 1 
Normal probability density: 75% Percentile  0.674 ...  x/σ for which Integral[-∞,x] N(x,σ) = 0.75 
Normal probability density: 90% Percentile  1.281 ...  x/σ for which Integral[-∞,x] N(x,σ) = 0.90 
Normal probability density: 95% Percentile  1.645 ...  x/σ for which Integral[-∞,x] N(x,σ) = 0.95 
Normal probability density: 98% Percentile  2.054 ...  x/σ for which Integral[-∞,x] N(x,σ) = 0.98 
Normal probability density: 99% Percentile  2.326 ...  x/σ for which Integral[-∞,x] N(x,σ) = 0.99 
Normal probability density: 99.9% Percentile  3.090 ...  x/σ for which Integral[-∞,x] N(x,σ) = 0.999 
Normal probability density: 99.99% Percentile  3.720 ...  x/σ for which Integral[-∞,x] N(x,σ) = 0.9999 
Other interesting math constants 
Minimum of Γ(x) for positive x  0.885 603 194 410 889 ...  For x > 0, the Gamma function minimum is unique 
   Location of the above minimum  1.461 632 144 968 362 ...   
Minimum value of xx, x ≥ 0 0.692 200 627 555 346 353 865 421 ...  = e-1/e. The minimum is unique. 
   Location of the above minimum  0.367 879 441 171 442 321 595 523 ...  xmin = 1/e. 
Solution of x = e-x 0.567 143 290 409 783 872 999 968 ...  Also solution of x = -ln(x). 
Math constants useful in physical sciences 
Root of   e-x + x/5 - 1 = 0  4.965 1 ...  Related to: Planck's radiation law maximum. 
Integral of   x3/[ex - 1] over (0,∞)  4.493 8 ...  Related to: Planck's radiation law integral. 
φm, Magic angle in degrees  54.735 610 317 245 345 684 622 999 ...  cos(φm) = √(1/3) = angle between body diagonal & a side in a cube. 
φm, Magic angle in radians  0.955 316 618 124 509 278 163 857 ...  Solution of P2 = (1-3.cos2(φ))/2 = 0. 
Area of a Lorentzian line / hw  1.570 796 326 794 896 619 231 321 ...  = π / 2. Here h = height, w = half-height width. 
Area of a Gaussian line / hw  1.064 467 019 431 226 179 315 267 ...  = sqrt[π /(4ln2)]. Here h = height, w = half-height width. 
ξ0, First root of Bessel J0(x)  2.404 825 557 695 ...  Also first root of sinc(0,x). 
ξ2, First root of Bessel J1(x)  3.831 705 970 207 ...  Also first root of sinc(2,x). 
ξ3, First root of [sin(x)/x - cos(x)]  4.493 409 457 909 ...  Also first root of sinc(3,x). 
ξ4, First root of [2J1(x)/x - J0(x)]  5.135 622 301 840 ...  Also first root of sinc(4,x). 
Engineering math constants 
1 dB power ratio  1,258 925 411 794 167 210 423 954 ...  101/10
1 dB inverse power ratio  0.794 328 234 724 281 502 065 918 ...  10-1/10
1 dB amplitude ratio  1.122 018 454 301 963 435 591 038 ...  101/20
1 dB inverse amplitude ratio  0.891 250 938 133 745 529 953 108 ...  10-1/20
3 dB power ratio  1.995 262 314 968 879 601 352 455 ...  103/10
3 dB inverse power ratio  0.501 187 233 627 272 285 001 554 ...  10-3/10
3 dB amplitude ratio  1.412 537 544 622 754 302 155 607 ...  103/20
3 dB inverse amplitude ratio  0.707 945 784 384 137 910 802 214 ...  10-3/20
Conversion constants 
1 rad, Radian in degrees  57.295 779 513 082 320 876 798 154 ...  180/π 
1°, Degree in radians  0.017 453 292 519 943 295 769 237 ...  π/180 

References sorted by year and by the first author

  • Finch S.R.,
    Mathematical Constants,
    Cambridge University Press 2003. ISBN 0-521-81805-2. more >>

Web links

TOP | Stan's Library | Physics Constants Stan's LINKS | Stan's HUB | TOP
   
Copyright ©2008 Stanislav Sýkora    DOI: 10.3247/SL2Math08.001 Designed by Stan Sýkora