| Basic math constants |
| Zero and One (and i, and ...) |
0 and 1 (and √(-1), and ...) |
Can anything be more basic than these two ? (or three, or ...) |
| π, Archimedes' constant |
3.141 592 653 589 793 238 462 643 ••• |
Circumference of a disc with unit diameter. |
| e, Euler number, Napier's constant |
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[n→∞]{(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 = (√5 + 1)/2 = 2.cos(π/5). Diagonal of a unit side pentagon. |
| φ, Inverse golden ratio (often confused with Φ) |
0.618 033 988 749 894 848 204 586 ••• |
φ = 1/Φ = Φ -1 =(1-φ)/φ or φ = (√5 - 1)/2. |
| δs, Silver ratio / mean |
2.414 213 562 373 095 048 801 688 ••• |
δs = 1+√2. |
| Constants derived from the basic ones |
| Conversions between logarithms for bases 2, e, 10: |
| 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 |
| Spin-offs of the imaginary unit i. Formally, i is a solution of z2 = -1 and of z = e zπ/2. For any integer k and any z, i 4k+z = i z. i 4f = e i2πf |
| De Moivre numbers ei2πk/n |
cos(2πk/n) + i.sin(2πk/n) |
for any integer k and n≠0. |
| ii = e-π/2 , the imaginary unit elevated to itself |
0.207 879 576 350 761 908 546 955 ••• |
A transcendental real number |
| i-i = (-1)-i/2 = eπ/2 |
4.810 477 380 965 351 655 473 035 ••• |
Inverse of the above. Square root of Gelfond's constant. |
| ln(i) / i = π/2 |
1.570 796 326 794 896 619 231 321 ••• |
This value could also be classified as a π spin-off |
| i! = Γ(1+i) = i*Γ(i) (see Gamma function) |
0.498 015 668 118 356 042 713 691 ••• |
-i 0.154 949 828 301 810 685 124 955 ••• |
| | i! | absolute value of the above |
0.521 564 046 864 939 841 158 180 ••• |
arg( i! ) = - 0.301 640 320 467 533 197 887 531 ••• rad |
| i^i^i^... infinite power tower of i; solution of z = i z |
0.438 282 936 727 032 111 626 975 ••• |
+i 0.360 592 471 871 385 485 952 940 ••• |
| | i^i^i | absolute value of the above |
0.567 555 163 306 957 825 384 613 ••• |
arg( i^i^i^... ) = 0.688 453 227 107 702 130 498 767 ••• rad |
| Basic roots of i, up to a term of 4k in the exponent (like i4k+1/4 = i1/4, for any integer k): |
| i1/2 = √i = (1 + i)/√2 = cos(π/4) + i.sin(π/4) |
0.707 106 781 186 547 524 400 844 ••• |
+i 0.707 106 781 186 547 524 400 844 ••• |
| i1/3 = (√3 + i)/2 = cos(π/6) + i.sin(π/6) |
0.866 025 403 784 438 646 763 723 ••• |
+i 0.5 |
| i1/4 = cos(π/8) + i.sin(π/8) |
0.923 879 532 511 286 756 128 183 ••• |
+i 0.382 683 432 365 089 771 728 459 ••• |
| i1/5 = cos(π/10) + i.sin(π/10) |
0.951 056 516 295 153 572 116 439 ••• |
+i 0.309 016 994 374 947 424 102 293 ••• |
| i1/6 = cos(π/12) + i.sin(π/12) |
0.965 925 826 289 068 2867 497 431 ••• |
+i 0.258 819 045 102 520 762 348 898 ••• |
| i1/7 = cos(π/14) + i.sin(π/14) |
0.974 927 912 181 823 607 018 131 ... |
+i 0.222 520 933 956 314 404 288 902 ... |
| i1/8 = cos(π/16) + i.sin(π/16) |
0.980 785 280 403 230 449 126 182 ... |
+i 0.195 090 322 016 128 267 848 284 ... |
| i1/9 = cos(π/18) + i.sin(π/18) |
0.984 807 753 012 208 059 366 743 ... |
+i 0.173 648 177 666 930 348 851 716 ••• |
| i1/10 = cos(π/20) + i.sin(π/20) |
0.987 688 340 595 137 726 190 040 ••• |
+i 0.156 434 465 040 230 869 010 105 ••• |
| e spin-offs; note also: e = (e1/e)^(e1/e)^(e1/e)^... = PowerTower(e1/e) |
| 2e |
5.436 563 656 918 090 470 720 574 ••• |
1/e = 0.367 879 441 171 442 321 595 523 ••• |
| cosh(1) = (e + 1/e)/2 |
1.543 080 634 815 243 778 477 905 ••• |
sinh(1) = (e - 1/e)/2 = 1.175 201 193 643 801 456 882 381 ••• |
| e2 |
7.389 056 098 930 650 227 230 427 ••• |
1/e2 = 0.135 335 283 236 612 691 893 999 ••• |
| √e |
1.648 721 270 700 128 146 848 650 ••• |
1/√e = 0.606 530 659 712 633 423 603 799 ••• |
| e±i = cos(1) ± i sin(1) = cosh(i) ± sinh(i) |
0.540 302 305 868 139 717 400 936 ••• |
±i 0.841 470 984 807 896 506 652 502 ••• |
| ee |
15.154 262 241 479 264 189 760 430 ••• |
e-e = 0.065 988 035 845 312 537 0767 901 ••• |
| e±ie = cos(e) ± i.sin(e) |
- 0.911 733 914 786 965 097 893 717 ••• |
±i 0.410 781 290 502 908 695 476 009 ••• |
| ie = cos(eπ/2) ± i.sin(eπ/2) |
-0.428 219 773 413 827 753 760 262 ••• |
±i -0.903 674 623 776 395 536 600 853 ••• |
| e1/e |
1.444 667 861 009 766 133 658 339 ••• |
e-1/e = 0.692 200 627 555 346 353 865 421 ••• |
| e±i/e = cos(1/e) ± i.sin(1/e) |
0.933 092 075 598 208 563 540 410 ••• |
±i 0.359 637 565 412 495 577 0382 503 ••• |
| Infinite power tower of 1/e |
0.567 143 290 409 783 872 999 968 ... |
(1/e)^(1/e)^(1/e)^...; also solution of x = e-x |
| Ramanujan's number: 262537412640768743 + |
0.999 999 999 999 250 072 597 198 ••• |
exp(π√163). Closest approach of exp(π√n) to integer. |
| π spin-offs |
| 2π |
6.283 185 307 179 586 476 925 286 ••• |
1/π = 0.318 309 886 183 790 671 537 767 ••• |
| π2 |
9.869 604 401 089 358 618 834 490 ••• |
1/π2 = 0.101 321 183 642 337 771 443 879 ••• |
| √π |
1.772 453 850 905 516 027 298 167 ••• |
1/√π = 0.564 189 583 547 756 286 948 079 ••• |
| ln(π) |
1.144 729 885 849 400 174 143 427 ••• |
log10(π) = 0.497 149 872 694 133 854 351 268 ••• |
| ln(π).π |
3.596 274 999 729 158 198 086 001 ... |
ln(π)/π = 0.364 378 839 675 906 257 049 587 ... |
| π±i = cos(ln(π)) ± i.sin(ln(π)) |
0.413 292 116 101 594 336 626 628 ... |
±i 0.910 598 499 212 614 707 060 044 ... |
| iπ = cos(π2/2) + i.sin(π2/2) |
0.220 584 040 749 698 088 668 945 ... |
- i 0.975 367 972 083 631 385 157 482 ... |
| ππ |
36.462 159 607 207 911 770 990 826 ••• |
π-π = 0.027 425 693 123 298 106 119 556 ••• |
| π±iπ = cos(π.ln(π)) ± i.sin(π.ln(π)) |
-0.898 400 579 757 743 645 668 580 ... |
±i -0.439 176 955 555 445 894 369 454 ... |
| π1/π |
1.439 619 495 847 590 688 336 490 ••• |
π-1/π = 0.694 627 992 246 826 153 124 383 ••• |
| π±i/π = cos(ln(π)/π) ± i.sin(ln(π)/π) |
0.934 345 303 678 637 694 262 240 ... |
±i 0.356 368 985 033 313 899 907 691 ... |
| e and π combinations, except trivial ones like, for any integer k, eiπk = (-1)k, cosh(iπk) = (-1)k, sinh(iπk) = 0 |
| eπ |
8.539 734 222 673 567 065 463 550 ••• |
√(eπ) = 2.922 282 365 322 277 864 541 623 ••• |
| e/π |
0.865 255 979 432 265 087 217 774 ••• |
π/e = 1.155 727 349 790 921 717 910 093 ••• |
| eπ = (-1)-i, Gelfond's constant |
23.140 692 632 779 269 005 729 086 ••• |
e-π = 0.043 213 918 263 772 249 774 417 ••• |
| e1/π |
1.374 802 227 439 358 631 782 821 ••• |
e-1/π = 0.727 377 349 295 216 469 724 148 ... |
| e±i/π = cos(1/π) ± i.sin(1/π) |
0.949 765 715 381 638 659 994 406 ... |
±i 0.312 961 796 207 786 590 745 276 ... |
| πe |
22.459 157 718 361 045 473 427 152 ••• |
π-e = 0.044 525 267 266 922 906 151 352 ••• |
| π1/e |
1.523 671 054 858 931 718 386 285 ••• |
π-1/e = 0.656 309 639 020 204 707 493 834 ... |
| γ spin-offs and some combinations (for basic definition of γ, see the Basic Constants section) |
| 2γ |
1.154 431 329 803 065 721 213 024 ... |
1/γ = 1.732 454 714 600 633 473 583 025 ••• |
| ln(γ) |
-0.549 539 312 981 644 822 337 661 ••• |
log10(γ) = -2.386 618 912 168 323 894 602 884 ... |
| eγ |
1.569 034 853 003 742 285 079 907 ... |
πγ = 1.813 376 492 391 603 499 613 134 ••• |
| eγ |
1.781 072 417 990 197 985 236 504 ••• |
e-γ = 0.561 459 483 566 885 169 824 143 ••• |
| e±iγ = cos(γ) ± i sin(γ) |
0.837 985 287 880 196 539 954 992 ••• |
±i 0.545 692 823 203 992 788 157 356 ••• |
| Infinite power tower of γ |
0.685 947 035 167 428 481 875 735 ... |
γ^γ^γ^...; also solution of x = γx |
| Golden ratio spin-offs and combinations (for basic definition of Φ and its inverse φ, see the Basic Constants section) |
| Complex golden ratio Φc = 2.eiπ/5 |
1.618 033 988 749 894 848 204 586 ••• |
+i 1.175 570 504 584 946 258 337 411 ••• |
| Associate of Φ = imaginary part of Φc |
1.175 570 504 584 946 258 337 411 ••• |
2.sin(π/5), while Φ = 2.cos(π/5) = real part of Φc |
| Square root of Φ |
1.272 019 649 514 068 964 252 422 ••• |
√Φ; relates the sides of squares with golden-ratio areas. |
| Square root of the inverse φ |
0.786 151 377 757 423 286 069 559 ••• |
1/√Φ |
| Cubic root of Φ |
1.173 984 996 705 328 509 966 683 ••• |
√Φ1/3. Relates edges of cubes with golden-ratio volumes. |
| Cubic root of the inverse φ |
0.851 799 642 079 242 917 055 213 ... |
1/√Φ1/3 |
| π/Φ = π.φ |
1.941 611 038 725 466 577 346 865 ••• |
Area of golden ellipse with semi_axes {1,φ} |
| ln(Φ) = - ln(φ) |
0.481 211 825 059 603 447 497 758 ••• |
Natural logarithm of Φ |
| Φ 2/π, such as in golden spiral |
1.358 456 274 182 988 435 206 180 ••• |
(2/π) ln(Φ) = 0.306 348 962 530 033 122 115 675 ••• |
| Infinite power tower of the inverse φ |
0.710 439 287 156 503 188 669 345 ... |
φ^φ^φ^...; also solution of x = φx = Φ-x |
| Miscellaneous derived constants: |
| 2^√2, the Gelfond - Schneider constant |
2.665 144 142 690 225 188 650 297 ••• |
a transcendental number ... |
| √2^√2 = 2^(1/√2) |
1.632 526 919 438 152 844 773 495 ••• |
... and its root, also transcendental |
| Classical, named math constants |
| Apéry's constant ζ(3) |
1.202 056 903 159 594 285 399 738 ••• |
A special value of the Riemann function ζ(x) |
| Artin's constant |
0.373 955 813 619 202 288 054 728 ••• |
Product of factors [1-1/p(p-1)], p prime |
| Bernstein's constant β |
0.280 169 499 023 869 133 036 436 ••• |
From the theory of function approximations by polynomials |
| Blazy's constant |
2.566 543 832 171 388 844 467 529 ... |
its Don Blazy's expansion generates all prime numbers |
| Brun's constant for twin primes B4 |
1.902 160 583 104 ••• (?) |
Sum of reciprocals of prime pairs (p,p+2) |
| Brun's constant for prime cousins B4 |
1.197 0449 ••• (?) |
Sum of reciprocals of prime pairs (p,p+4) |
| Brun's constant for prime quadruplets B'4 |
0.870 588 380 ... (?) |
Sum of reciprocals of prime quadruplets (p,p+2,p+4,p+6) |
| Champernowne constant C10 |
0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... |
String concatenation of dec expansions of natural numbers |
| Catalan's constant C |
0.915 965 594 177 219 015 054 603 ••• |
C = Sum[n=0,∞]{(-1)^n/(2n+1)^2} |
| Continued fractions constant |
1.030 640 834 100 712 935 881 776 ••• |
(1/6)π2/(ln(2)ln(10)). Mean c.f.terms per decimal digit |
| Conway's constant λ(3) |
1.303 577 269 034 296 391 257 099 ••• |
Growth rate of derived look-and-say strings |
| Delian's constant |
1.259 921 049 894 873 164 767 210 ••• |
21/3 |
| Embree - Trefethen constant β |
0.70258 ••• (?) |
Theory of 2nd order recurrences with random add/subtract |
| Erdös - Borwein constant |
1.606 695 152 415 291 763 783 301 ••• |
Sum[n=1,∞]{1/(2^n -1)} |
| Feigenbaum reduction parameter α |
-2.502 907 875 095 892 822 283 902 ••• |
Appears in the theory of chaos |
| Feigenbaum
bifurcation velocity δ |
4.669 201 609 102 990 671 853 203 ••• |
Appears in the theory of chaos |
| Gauss' constant G |
0.834 626 841 674 073 186 814 297 ••• |
1/agm(1,√2); agm ... arithmetic-geometric mean |
| Gauss-Kuzmin-Wirsing constant λ1 |
0.303 663 002 898 732 658 597 448 ••• |
2nd eigenvalue of GKW functional operator (first is 1) |
| Gelfond's constant |
23.140 692 632 779 269 005 729 086 ••• |
e^π = (-1)^(-i) |
| Gelfond - Schneider constant |
2.665 144 142 690 225 188 650 297 ••• |
2^√2, a transcendental number |
| Gibbs constant G |
1.178 979 744 472 167 270 232 028 ••• |
(2/π)(Wilbraham-Gibbs constant G'); see below. |
| Glaisher-Kinkelin constant A |
1.282 427 129 100 622 636 875 342 ••• |
Appears in number theory |
| Golomb-Dickman constant λ |
0.624 329 988 543 550 870 992 936 ••• |
Longest cycle distribution in random permutations |
| Gompertz constant G |
0.596 347 362 323 194 074 341 078 ••• |
-e.Ei(-1), Ei(x) being the exponential integral |
| Grossmann's constant |
0.737 338 303 369 29 ••• (?) |
The only x for which {a0=1; a1=x; an+2=an/(1+an+1)} converges |
| Khinchin-Lévy constant γ |
3.275 822 918 721 811 159 787 681 ••• |
exp(π2/(12.ln2)); unstable nomenclature |
| Lévy constant γ |
1.186 569 110 415 625 452 821 722 ••• |
π2/(12.ln2). Logarithm of Khinchin-Lévy's |
| Knuth's random-generators constant |
0.211 324 865 405 187 117 745 425 ... |
(1-sqrt(1/3))/2; also solid angle of a cone with magic polar angle |
| Kolakoski constant γ |
0.794 507 192 779 479 276 240 362 ••• |
Related to Kolakoski sequence |
| Lemniscate constant L |
2.622 057 554 292 119 810 464 839 ••• |
L = πG, where G is the Gauss' constant |
| First lemniscate constant L1 |
1.311 028 777 146 059 905 232 419 ••• |
L1 = L/2 = πG/2 |
| Second lemniscate constant L2 |
0.599 070 117 367 796 103 337 484 ••• |
L2 = 1/(2G) |
| Landau-Ramanujan constant |
0.764 223 653 589 220 662 990 698 ••• |
Related to the density of sums of two integer squares |
| Laplace limit constant λ |
0.662 743 419 349 181 580 974 742 ••• |
Let η = √(1+λ2); then λeη = 1+η
Click here for more |
| Liouville's constant |
0.110 001 000 000 000 000 000 001 ••• |
Sum[n=0,∞]{10^(-n!)} |
| Madelung's constant M3 |
-1.747 564 594 633 182 190 636 212 ••• |
M3 = Sum[i,j,k]{(-1)^(i+j+k)/sqr(i^2+j^2+k^2)} |
| Meissel-Merten's constant B1 |
0.261 497 212 847 642 783 755 426 ••• |
Limit[n→∞]{Sum[prime p≤n]{1/p}-ln(ln(n))} |
| MRB constant |
0.187 859 642 462 067 120 248 517 ••• |
Sum[n=1,2,...]{(-1)^n (n^(1/n) - 1)} |
| MKB constant |
0.687 652 368 927 694 369 809 3 ••• |
Limit[n→∞]{abs(Intg[1,2n]{(-1)^x x^(1/x) dx]}} |
| Omega constant W(1) |
0.567 143 290 409 783 872 999 968 ••• |
Root of [x - e-x] or [x + ln(x)]. See also. |
| Otter's constant α |
2.955 765 285 651 994 974 714 817 ••• |
Appears in enumeration of rooted and unrooted trees: |
| Otter's asymptotic constant βu for unrooted trees |
0.534 949 606 1(?) ••• |
UT(n) ~ βu αn n-5/2 |
| Otter's asymptotic constant βr for rooted trees |
0.439 924 012 571 (?) ••• |
RT(n) ~ βr αn n-3/2 (V. Kotesovec) |
| Plastic number ρ (or silver constant) |
1.324 717 957 244 746 025 960 908 ••• |
Real root of x3 = x + 1 |
| Pogson's ratio |
2.511 886 431 509 580 111 085 032 ••• |
1001/5; in astronomy 1 stellar magnitude brightness ratio |
| Polya's random-walk constant p3 |
0.340 537 329 550 999 142 826 273 ••• |
Probability a 3D-lattice random walk returns back. See also |
| Prévost's constant |
3.359 885 666 243 177 553 172 011 ••• |
Sum of reciprocals of Fibonacci numbers |
| Reciprocal even Fibonacci constant |
1.535 370 508 836 252 985 029 852 ••• |
Sum of reciprocals of even-indexed Fibonacci numbers |
| Reciprocal odd Fibonacci constant |
1.824 515 157 406 924 568 142 158 ••• |
Sum of reciprocals of odd-indexed Fibonacci numbers |
| Rényi's parking constant m |
0.747 597 920 253 411 435 178 730 ••• |
Density of randomly parked cars in a street |
| Sierpinski's constant K |
2.584 981 759 579 253 217 065 893 ••• |
For explanation, click also here |
| Soldner's constant μ |
1.451 369 234 883 381 050 283 968 ••• |
Root of logarithmic integral li(x) |
| Somos' quadratic recurrence constant |
1.661 687 949 633 594 121 295 818 ••• |
√(1√(2√(3√(4 ...)))) |
| Shall-Wilson or twin primes constant C2 |
0.660 161 815 846 869 573 927 812 ••• |
Product[twin_primes p,p+2]{p(p-2)/(p-1)2} |
| Theodorus' constant |
1.732 050 807 568 877 293 527 446 ••• |
√3. |
| Viswanath's constant |
1.131 988 248 794 3 ••• (?) |
Growth of Fibonacci-like sequences with random add/subtract |
| Wilbraham-Gibbs constant G' |
1.851 937 051 982 466 170 361 053 ••• |
Intg[0,π]{sin(θ)/θ dθ}. |
| Some other, notable math constants |
| (1-1/2)*(1-1/4)*(1-1/8)*(1-1/16)* ... |
0.288 788 095 086 602 421 278 899 ••• |
Product [k=1,∞](1-xk), for x=1/2 |
| 1+1/22+1/33+1/44+ ... |
1.291 285 997 062 663 540 407 282 ••• |
Sum [k=1,∞](1/kk) |
| Infinite power towers for inverses of some sequences a(n): iPT{a(n)} = Lim[n→∞]{(1/a(k)^(1/a(k+1)^(1/a(k+2)^...^(1/a(n))}, k being the first index such that a(k)>1 |
| iPT{2}, the infinite power tower of 1/2 |
0.641 185 744 504 985 984 486 200 ... |
(1/2)^(1/2)^(1^2)^...; also solution of x = 2-x |
| iPT{n} |
0.690 347 126 114 964 319 467 328 ... |
(1/2)^(1/3)^(1/4)^(1/5)^... |
| iPT{2^n} |
0.570 203 397 398 373 262 211 917 ... |
(1/2)^(1/4)^(1/8)^(1/16)^... |
| iPT{prime(n)} |
0.719 405 031 245 092 118 809 081 ... |
(1/2)^(1/3)^(1/5)^(1/7)^(1/11)^... |
| iPT{n!} |
0.885 946 913 356 773 110 697 240 ... |
(1/2!)^(1/3!)^(1/4!)^(1/5!)^(1/6!)^... |
| Selected natural and integer numbers and their sequences |
| Large integers |
| Googol |
10100 = 10^100 |
A large integer ... |
| Googolplex |
10googol = 10^10^100 |
... a larger integer ... |
| Googolplexplex |
10googolplex = 10^10^10^100 |
... and a still larger one. |
| Graham's number (last 20 digits) |
... 04575627262464195387 |
3^^^...^^^3, with 64 power operators; too big to write |
| Skewes' numbers |
10^14 < n < e^e^e^79 |
Bounds on the first integer for which π(n) < li(n) |
| Polygonal (2D) figurate numbers. See also on OEIS: numbers which are (A090466) or aren't (A090467) "some" polygonal number. See the link for more! |
| Triangualar numbers |
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ••• |
n(n+1)/2 |
| Square numbers |
1, 4, 9, 16, 25, 36, 49, 64, 81,100,121, ••• |
n*n |
| Square triangular numbers |
1,36,1225,41616,1413721,48024900, ••• |
[[(3+2√2)k-(3-2√2)k]/(4√2)]2; both triangular and square |
| Pentagonal numbers |
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ••• |
n(3n-1)/2 |
| Hexagonal numbers |
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, ••• |
n(2n-1); also cornered hexagonal numbers |
| Heptagonal numbers |
1, 7, 18, 34, 55, 81, 112, 148, 189, ••• |
n(5n-3)/2 |
| Octagonal numbers |
1, 8, 21, 40, 65, 96, 133, 176, 225, ••• |
n(3n-2) |
| Centered polygonal (2D) figurate numbers |
| Centered triangular numbers |
1, 4, 10, 19, 31, 46, 64, 85, 109, 136, ••• |
(3n2+3n+2)/2. See also: centered triangular primes ••• |
| Centered square numbers |
1, 5, 13, 25, 41, 61, 85, 113, 145, 181, ••• |
2n2-2n+1. See also: centered square primes ••• |
| Centered pentagonal numbers |
1, 6, 16, 31, 51, 76, 106, 141, 181, 226, ••• |
(5(n-1)2+5(n-1)+2)/2. See also: centered pentagonal primes ••• |
| Centered hexagonal numbers |
1, 7, 19, 37, 61, 91, 127, 169, 217, ••• |
n3- (n-1)3 = 3n(n-1)+1; also hex numbers |
| Centered heptagonal numbers |
1, 8, 22, 43, 71, 106, 148, 197, 253, ••• |
(7n2-7n+2)/2 |
| Centered octagonal numbers |
1, 9, 25, 49, 81, 121, 169, 225, 289, ••• |
(2n-1)2; squares of odd numbers |
| Polyhedral (3D) figurate numbers |
| Tetrahedral numbers Tn |
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ••• |
n(n+1)(n+2)/6. Only three are squares: 1, 4, 19600 |
| Cubic numbers, cubes |
1, 8, 27, 64, 125, 216, 343, 512, 729, ••• |
n3 |
| Pentatopic (or pentachoron) numbers |
0, 1, 5, 15, 35, 70, 126, 210, 330, 495, ••• |
n(n+1)(n+2)(n+3)/24; n=0,1,2,... |
| Octahedral numbers On |
1, 6, 19, 44, 85, 146, 231, 344, 489, ••• |
n(2n2+1)/3. |
| Combinatorial numbers and basic counting |
| Binomial coefficients C(n,m) = n!/(m!(n-m)!) (ways to pick m among n labelled elements); C(n,m)=0 if m<0 or m>n; C(n,0)=1; C(n,1)=n; C(n,m)=C(n,n-m): |
| m = 2, n = 4,5,6,... |
6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ••• |
n(n-1)/2; shifted triangular numbers |
| m = 3, n = 6,7,8,... |
20, 35, 56, 84, 120, 165, 220, 286, 364, ••• |
n(n-1)(n-2)/3!; shifted tetrahedral numbers |
| m = 4, n = 8,9,10,... |
70, 126, 210, 330, 495, 715, 1001, 1365, ••• |
n(n-1)(n-2)(n-3)/4!; for n < 2m, use C(n,n-m) |
| m = 5, n = 10,11,12... |
252, 462, 792, 1287, 2002, 3003, 4368, ••• |
n(n-1)(n-2)(n-3)(n-4)/5! |
| m = 6, n = 12,13,14,... |
924, 1716, 3003, 5005, 8008, 12376, ••• |
n(n-1)(n-2)(n-3)(n-4)(n-5)/6! = n(6)/6! |
| m = 7, n = 14,15,16... |
3432, 6435, 11440, 19448, 31824, ••• |
n(7)/7!; all cases up to n=14 are covered |
| Central binomial coefficients C(2n,n) = (2n)!/n!2 |
1, 2, 6, 20, 70, 252, 924, 3432, 12870, ••• |
C(2n,n) = Sum[k=0,n]{C2(n,k)}: Franel number of order 2 |
| Factorials n! = 1*2*3...*n |
1, 1, 2, 6, 24, 120, 720, 5040, 40320, ••• |
Permutations of an ordered set of n labelled elements |
| Quadruple factorials (2n)!/n! |
1, 2, 12, 120, 1680, 30240, 665280, ••• |
in terms of Catalan numbers, equal to (n+1)!C(n) |
| Factorions in base 10 |
1, 2, 145, 40585 (that's all) |
Equal to the sum of factorials of their dec digits |
| Factorions in base 16 |
1, 2, 2615428934649 (that's all) |
Equal to the sum of factorials of their hex digits |
| Franel numbers of order 3 |
1, 2, 10, 56, 346, 2252, 15184, 104960, ••• |
Sum[k=0,n]{C3(n,k)} |
| Lah numbers L(n,m) (unsigned); signed L(n,m) = (-1)nL(n,m); They expand rising factorials in terms of falling factorials and vice versa. L(n,1) = n!
|
| m = 2, n = 2,3,4,... |
1, 6, 36, 240, 1800, 15120, 141120, ••• |
|
| m = 3, n = 3,4,5,... |
1, 12, 120, 1200, 12600, 141120, ••• |
General formula: L(n,m)=C(n,m)(n-1)!/(m-1)! |
| m = 4, n = 4,5,6,... |
1, 20, 300, 4200, 58800, 846720, ••• |
|
| Stirling numbers of the first kind c(n,m) (unsigned); signed s(n,m) = (-1)n-mc(n,m); number of permutations of n distinct elements with exactly m cycles.
|
| m = 1, n = 1,2,3,... |
1, 1, 2, 6, 24, 120, 720, 5040, 40320, ••• |
(n-1)! Note: by convention, c(n,0) = 1 |
| m = 2, n = 2,3,4,... |
1, 3, 11, 50, 274, 1764, 13068, 109584, ••• |
a(n+1)=n*a(n)+(n-1)! |
| m = 3, n = 3,4,5,... |
1, 6, 35, 225, 1624, 13132, 118124, ••• |
|
| m = 4, n = 4,5,6,... |
1, 10, 85, 735, 6769, 67284, 723680, ••• |
A definition of s(n,m): |
| m = 5, n = 5,6,7,... |
1, 15, 175, 1960, 22449, 269325, ••• |
x(n) = x(x-1)(x-2)...(x-(n-1)) = Sum{m=0:n}[s(n,m).xm] |
| m = 6, n = 6,7,8,... |
1, 21, 322, 4536, 63273, 902055, ••• |
See also OEIS A008275 |
| m = 7, n = 7,8,9,... |
1, 28, 546, 9450, 157773, 2637558, ••• |
|
| m = 8, n = 8,9,10,... |
1, 36, 870, 18150, 357423, 6926634, ... |
|
| m = 9, n = 9,10,11,... |
1, 45, 1320, 32670, 749463, 16669653, ... |
|
| Stirling numbers of the second kind S(n,m); number of partitions of n distinct elements into m non-empty subsets. S(n,1) = 1. By convention, S(0,0) = 1.
|
| m = 2, n = 2,3,4,... |
1, 3, 7, 15, 31, 63, 127,255, 511, 1023, ••• |
2(n-1)-1 |
| m = 3, n = 3,4,5,... |
1, 6, 25, 90, 301, 966, 3025, 9330, ••• |
|
| m = 4, n = 4,5,6,... |
1, 10, 65, 350, 1701, 7770, 34105, ••• |
A definition of S(n,m): |
| m = 5, n = 5,6,7,... |
1, 15, 140, 1050, 6951, 42525, 246730, ••• |
xn = Sum{m=0:n}[S(n,m).x(m)] |
| m = 6, n = 6,7,8,... |
1, 21, 266, 2646, 22827, 179487, ••• |
See also OEIS A008277 |
| m = 7, n = 7,8,9,... |
1, 28, 462, 5880, 63987, 627396, ••• |
|
| m = 8, n = 8,9,10,... |
1, 36, 750, 11880, 159027, 1899612, ••• |
|
| m = 9, n = 9,10,11,... |
1, 45, 1155, 22275, 359502, 5135130, ••• |
|
| More counting (enumeration) of various objects |
| Subsets of a set of n labelled elements |
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ••• |
2n, for n=0,1,2,... |
| Composition numbers |
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... |
1 for n=0, else 2^(n-1); Distinct compositions of number n=0,1,2,... |
| Catalan numbers C(n) |
1, 1, 2, 5, 14, 42, 132, 429,1430,4862, ••• |
C(n)=C(2n,n)/(n+1); ubiquitous in number theory |
| Bell numbers B(n) |
1, 1, 2, 5, 15, 52, 203, 877, 4140, ••• |
Partitions of a set of n=0,1,2,... labelled elements |
| Ordered Bell numbers or Fubini numbers |
1, 1, 3, 13, 75, 541, 4683, 47293, ••• |
Weakly ordered partitions of n=0,1,2,... labelled elements |
| Partition numbers p(n) |
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, ••• |
Partitions of a set of n=0,1,2,... unlabelled elements |
| Free labelled trees |
1, 1, 3, 16, 125, 1296,16807,262144, ••• |
nn-2 (Cayley formula), n=1,2,3,... labelled vertices |
| Rooted labelled trees |
1, 2, 9, 64, 625, 7776, 117649, ••• |
nn-1, n=1,2,3,... labelled vertices |
| Free unlabelled trees |
1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235 ••• |
for n=1,2,3,... unlabeled vertices |
| Rooted unlabelled trees |
1, 2, 4, 9, 20, 48, 115, 286, 719, ••• |
for n=1,2,3,... unlabelled vertices |
| Simple connected graphs |
1, 1, 2, 6, 21, 112, 853, 11117, 261080, ••• |
for n = 1,2,3,... unlabelled nodes |
| Compositions of powers of integer numbers |
| Pythagorean triples (a,b,c), a2 + b2 = c2 |
(3,4,5) (5,12,13) (7,24,25) (8,15,17) |
(9,40,41) (11,60,61) (12,35,37) (13,84,85) (16,63,65) ... |
| Pythagorean quadruples, a2 + b2 + c2 = d2 |
(1,2,2,3) (2,3,6,7) (4,4,7,9) (1,4,8,9) |
(6,6,7,11) (2,6,9,11) (10,10,23,27) (7,14,22,23) ... |
| Pythagorean quintuples |
(1,2,4,10,11) (1,2,8,10,13) ... |
etc; there is an infinity of them in each category |
Taxicab numbers
Ta(1), Ta(2), Ta(3), Ta(4),
Ta(5),
Ta(6) |
2, 1729, 87539319, 6963472309248,
48988659276962496,
24153319581254312065344, ••• |
Ta(n) is the smallest T(n) number, one that can be written as a
sum of two positive cubes in n different ways. Only six are known
and only Ta(1), Ta(2) are cubefree (not divisible by a cube). |
| Hardy-Ramanujan number |
1729 = 13+123 = 93+103 |
Smallest cubefree taxicab number T(2) |
| Vojta's number |
15170835645 |
Smallest cubefree T(3) number (see the link) |
| Gascoigne's number |
1801049058342701083 |
Smallest cubefree T(4) number (see the link) |
| Special numbers related to divisors; σ(n) is the divisor function (sum of all divisors of n), and σ(n)-n is the sum of all proper divisors of n |
| Perfect numbers |
6, 28, 496, 8128, 33550336, ••• |
n = equals the sum of its own proper divisors = σ(n) - n |
| Abundant numbers |
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, ••• |
n exceeds the sum of its own proper divisors; n > σ(n) - n |
| Deficient numbers |
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, ••• |
n is less than the sum of its own proper divisors; n < σ(n) - n |
| Odd abundant numbers |
945, 1575, 2205, 2835, 3465, 4095, ••• |
Funny the smallest one is so large |
| Odd abundant numbers not divisible by 3 |
5391411025, 26957055125, ••• |
see also |
| Last integer not composed of two abundants |
20161 |
Exactly 1456 integers are the sum of two abundants |
| Amicable number pairs |
(220,284); (1184,1210); (2620,2924); ••• |
n = σ(m) - m, m = σ(n) - n |
| Superperfect numbers |
2, 4, 16, 64, 4096, 65536, 262144, ••• |
n = σ(σ(n)) - n |
| Special numbers related to absolute and/or relative primes, natural number factorizations, etc |
| Prime numbers |
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ••• |
A prime number is divisible only by 1 and itself; excluding 1 |
| Twin prime numbers |
3, 5, 11, 17, 29, 41, 59, 71, 101, 107, ••• |
For each prime p in this list, p+2 is also a prime |
| Euler's totient function Φ(n) for n=1,2,3,... |
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, ••• |
Number of k's smaller than n and relatively prime to it |
| Möbius function μ(n) for n=1,2,3,... |
1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, ••• |
μ(n) = (-1)^k if n has k different prime factors; else μ(n) = 0 |
| Mersenne primes (first 7 for p = 2,3,5,7,13,17,19) |
3, 7, 31, 127, 8191, 131071, 524287, ••• |
Some M(p) = 2p-1; p prime; Largest known: M(43112609) |
| Pseudoprimes to base 2 (Sarrus numbers) |
341, 561, 645, 1105, 1387, 1729, ••• |
Composite odd n such that 2n-1 = 1 (mod n) |
| Pseudoprimes to base 3 |
91, 121, 286, 671, 703, 949, 1105, ••• |
Composite odd n such that 3n-1 = 1 (mod n) |
| Carmichael's pseudoprimes |
561, 1105, 1729, 2465, 2821, 6601, ••• |
Composite odd n such that bn-1 = 1 (mod n) for any coprime b |
| Euler's pseudoprimes in base 2 |
341, 561, 1105, 1729, 1905, 2047, ••• |
Composite odd n such that 2(n-1)/2 = ±1 (mod n) |
| Ishango bone prime quadruplet |
11, 13, 17, 19 |
Crafted in the paleolithic Ishango bone |
| Named sequences of binary numbers {0,1} or {-1,+1}; n-th term often refers to the digits di of the binary expansion B(n) of n. |
| Baum - Sweet sequence |
1,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1,1,0,0 ••• |
1 if B(n) contains no block of 0's of odd length |
| Fibonacci words |
0, 01, 010, 01001, 01001010, ... |
Like Fibonacci recurrence, but using string concatenation |
| Infinite Fibonacci word |
010010100100101001010010010 ••• |
Infinite continuation of the above |
| Golay - Rudin - Shapiro sequence |
+1,+1,+1,-1,+1,+1,-1,+1,+1,+1,+1,-1 ••• |
(-1)^Sum[i]{didi+1} |
| Thue - Morse sequence |
0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0 ••• |
1 if B(n) contains an odd number of ones (parity = 1) |
| Other named sequences of integers |
| Euler numbers E(n) for n=0,2,4,... |
1, -1, 5, -61, 1385, -50521, 2702765, ••• |
1/cosh(t) = Sum[n=0,∞]{tn(E(n)/n!)}; the odd ones are 0 |
| Fibonacci numbers F(n) |
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ••• |
Fn = Fn-1 + Fn-2; F0 = 0, F1 = 1 |
| Golomb's sequence (or Silverman's sequence) |
1, 2,2, 3,3, 4,4,4, 5,5,5, 6,6,6,6, 7,7,7 ••• |
a(1)=1, a(n) is the (least possible) number of times n occurs |
| Heegner numbers h |
1, 2, 3, 7, 11, 19, 43, 67, 163 (full set) |
The quadratic ring Q(√(-h)) has class number 1 |
| Lucas numbers L(n) |
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ••• |
Fn = Fn-1 + Fn-2; F0 = 2, F1 = 1 |
| Sylvester's (or Ahmes') sequence |
2, 3, 7, 43, 1807, 3263443, ••• |
sn is the product of previous members, plus 1 |
| Selected rational numbers and their sequences |
| Bernoulli numbers B0 = 1, B1 = -1/2, B2k+1= 0 for k>1, Bm = δm,0 - Sum{k=0...(m-1)}[C(m,k)Bk/(m-k+1)]; x/(ex-1) = Sum{k=0,...}[Bnxn/n!]; Example: B10 = 5/66 |
| Bn = N/D; n = 2,4,6,... |
N: 1,-1, 1, -1, 5, -691, 7, -3617, 43867, ••• |
D: 6, 30, 42, 30, 66, 2730, 6, 510, 798, ••• |
| Other |
| Rationals ≤ 1, sorted by denominators:nominators |
1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, ••• |
Take the inverse values for rationals ≥ 1 |
| Farey sequences of fractions Fn (example n=5) |
0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1, ... |
F1 = { 0/1,1/1 }; for higher orders, interpolate ( a/c,b/d ) → a+b/c+d |
| Stern - Brocot sequence (example n=4) |
1/1, 1/2, 2/1, 1/3, 2/3, 3/2, 3/1, 1/4, 2/5, 3/5, 3/4, ... |
Wraps up the binary Stern - Brocot tree |
| Useful function-related values and constants |
| Function gamma Γ(x) = Intg[0,x]{tx-1e-t dt}; Γ(x+1) = xΓ(x); Γ(1) = Γ(2) = 1 |
| Γ(x) minimum location; x ≥ 0 |
1.461 632 144 968 362 341 262 659 ••• |
Also the positive root of digamma function ψ(x) |
| Γ(x) minimum value; x ≥ 0 |
0.885 603 194 410 888 700 278 815 ••• |
For x > 0, the Gamma function minimum is unique |
| Γ(1/2) |
1.772 453 850 905 516 027 298 167 ••• |
√π, crops up very often |
| Γ(1/3) |
2.678 938 534 707 747 633 655 692 ••• |
Γ(2/3) = 1.354 117 939 426 400 416 945 288 ••• |
| Γ(1/4) |
3.625 609 908 221 908 311 930 685 ••• |
Γ(3/4) = 1.225 416 702 465 177 645 129 098 ••• |
| Γ(i) (real and imaginary parts) |
-0.154 949 828 301 810 685 124 955 ... |
- i 0.498 015 668 118 356 042 713 691 ... |
| Function digamma ψ(x) = (dΓ(x)/dx)/Γ(x); ψ(x+1) = ψ(x) + 1/x; For positive root, see above. |
| ψ(1) = -γ |
- 0.577 215 664 901 532 860 606 512 ••• |
ψ(2) = 1-γ = +0.422 784 335 098 467 139 393 488 ••• |
| ψ(1/2) = -γ-2ln(2) |
- 1.963 510 026 021 423 479 440 976 ••• |
ψ(-1/2) = 0.036 489 973 978 576 520 559 024 ... |
| Riemann zeta function ζ(x) = Sum[k=0,1,2,...]{k^(-x)} and analytic continuations which exist everywhere except at x = 1 |
| ζ(-1/2) = -ζ(3/2)/(4π) |
-0.207 886 224 977 354 566 017 306 ••• |
ζ(0) = -0.5 exact; ζ(-1) = -1/12 exact |
| ζ(1/2) |
-1.460 354 508 809 586 812 889 499 ••• |
ζ(3/2) = 2.612 375 348 685 488 343 348 567 ••• |
| ζ(2) = π2 /6 |
1.644 934 066 848 226 436 472 415 ••• |
ζ(3) = 1.202 056 903 159 594 285 399 738 •••, see Apéry's |
| ζ(4) = π4 /90 |
1.082 323 233 711 138 191 516 003 ••• |
ζ(5) = 1.036 927 755 143 369 926 331 365 ••• |
| ζ(6) = π6 /945 |
1.017 343 061 984 449 139 714 517 ••• |
ζ(7) = 1.008 349 277 381 922 826 839 797 ••• |
| ζ(8) = π8 /9450 |
1.004 077 356 197 944 339 378 685 ••• |
ζ(9) = 1.002 008 392 826 082 214 417 852 ••• |
| ζ(10) = π10 /93555 |
1.000 994 575 127 818 085 337 145 ••• |
ζ(11) = 1.000 494 188 604 119 464 558 702 ••• |
| ζ(12) = π12 691/638512875 |
1.000 246 086 553 308 048 298 637 ••• |
ζ(13) = 1.000 122 713 347 578 489 146 751 ••• |
| ζ(i) (real and imaginary parts) |
0.003 300 223 685 324 102 874 217 ••• |
-i 0.418 155 449 141 321 676 689 274 ••• |
| 1st root, imaginary part |
14.134 725 141 734 693 790 457 251 ••• |
Note: all listed roots have real parts +0.5 |
| 2nd root |
21.022 039 638 771 554 992 628 479 ••• |
3rd root: 25.010 857 580 145 688 763 213 790 ••• |
| 4th root; for more, see OEIS Wiki |
30.424 876 125 859 513 210 311 897 ••• |
5th root: 32.935 061 587 739 189 690 662 368 ••• |
| Dawson integral F(x) = e-x^2 Intg[0,x]{e-t^2 dt} |
| F(x) maximum: Location xmax |
0.924 138 873 004 591 767 012 823 ••• |
F(x) being an odd function; there is a minimum at -xmax |
| F(x) maximum: Value |
0.541 044 224 635 181 698 472 759 ••• |
F(xmax) = 1/(2xmax). |
| F(x) inflection: Location xi |
1.501 975 268 268 611 498 860 348 ••• |
Dawson integral: see above. |
| F(x) inflection: Value at xi |
0.427 686 616 017 928 797 406 755 ... |
F(xi) = xi/(2xi2-1). |
| Exponential integral Ei(x) = Intg[1,∞]{exp(-xt)/t dt} and Logarithmic integral Li(x) = Intg[0,x]{dt/ln(t)}, x ≥ 0 (use principal value when crossing t = 1) |
| Real root of Li(x) |
1.451 369 234 883 381 050 283 968 ••• |
μ; this is the Soldner's constant |
| Li(2) |
1.045 163 780 117 492 784 844 588 ••• |
Li(0) = Li(μ) = 0; Li(1) = -∞; Li(∞) = ∞ |
| Real root of Ei(x) |
0.372 507 410 781 366 634 461 991 ••• |
ln(μ) |
| Ei(1) |
1.895 117 816 355 936 755 466 520 ••• |
Ei(0) = -∞; Ei(ln(μ)) = 0; Ei(-∞) = 0; Ei(∞) = ∞ |
| Ei(-1) |
- 0.219 383 934 395 520 273 677 163 ••• |
Equals -(Gompertz constant)/e ... |
| Function y(x) = xx = exp(x.ln(x)), x ≥ 0: |
| Location of minimum |
0.367 879 441 171 442 321 595 523 ••• |
xmin = 1/e. The minimum is unique. |
| Value at minimum |
0.692 200 627 555 346 353 865 421 ••• |
equals e-1/e. |
| Function sinc(x) = sin(x)/x and its Hilbert transform Hsinc(x) = [1-cos(x)]/x, appearing in spectral theory (transient data truncation artifacts) |
| sinc(x): Half-height argument |
1.895 494 267 033 980 947 144 035 ••• |
Solution of sinc(x) = 1/2 |
| sinc(x): First minimum location |
4.493 409 457 909 064 175 307 880 ••• |
A solution of tan(x) = x |
| sinc(x): First minimum value |
-0.217 233 628 211 221 657 408 279 ••• |
|
| Hsinc(x): First maximum location |
2.331 122 370 414 422 613 667 835 ... |
A solution of x.sin(x) = 1-cos(x) |
| Hsinc(x): First maximum value |
0.724 611 353 776 708 475 738 990 ... |
|
| Geometry constants |
| Magic angle φm and Tetrahedral angle θm. Notes: φm = acos(1/√3) = atan(√2) = π/2 - asin(1/√3); π-θm = acos(1/3) = atan(2√2) |
| Magic angle φm |
0.955 316 618 124 509 278 163 857 ••• |
Degrees: 54.735 610 317 245 345 684 622 999 ... |
| Complementary magic angle φ'm = π/2 - φm |
0.615 479 708 670 387 341 067 464 ••• |
Degrees: 35.264 389 682 754 654 315 377 000 ... |
| Tetrahedral angle θm = 2φm |
1.910 633 236 249 018 556 327 714 ••• |
Degrees: 109.471 220 634 490 691 369 245 999 ... |
| Complemetary tetrahedral angle θ'm = π - θm |
1.230 959 417 340 774 682 134 929 ••• |
Degrees: 70.528 779 365 509 308 630 754 000 ... |
| Sphere, the Queen of all bodies |
| Volume / radius3 |
4.188 790 204 786 390 984 616 857 ••• |
4π/3; volume of a sphere with unit radius |
| Radius / Volume1/3 |
0.620 350 490 899 400 016 668 006 ••• |
(3/(4π))1/3; radius of a sphere with unit volume |
| Surface / radius2 |
12.566 370 614 359 172 953 850 573 ••• |
4π; surface of a sphere with unit radius |
| Radius / Surface1/2 |
0.282 094 791 773 878 143 474 039 ••• |
1/(4π)1/2; radius of a sphere with unit surface |
| Platonic solids: Tetrahedron, regular, 4 vertices, 6 edges, 4 faces, 3 edges/vertex, 3 edges/face, 3 faces/vertex. |
| Volume / edge3 |
0.117 851 130 197 757 920 733 474 ••• |
(√2)/12 |
| Surface / edge2 |
1.732 050 807 568 877 293 527 446 ••• |
√3 |
| Height / edge |
0.816 496 580 927 726 032 732 428 ••• |
(√6)/3 |
| Angle between an edge and a face |
0.955 316 618 124 509 278 163 857 ••• |
magic angle φm (see above) |
| Dihedral angle (between adjacent faces) |
1.230 959 417 340 774 682 134 929 ... |
complementary tetrahedral angle θ'm (see above) |
| Tetrahedral angle (vertex-center-vertex) |
1.910 633 236 249 018 556 327 714 ... |
θm (see above) |
| Circumscribed sphere radius / edge |
0.612 372 435 695 794 524 549 321 ... |
Circumradius = (√6)/4, congruent with vertices |
| Midsphere radius / edge |
0.353 553 390 593 273 762 200 422 ... |
Midradius = 1/√8, tangent to edges |
| Inscribed sphere radius / edge |
0.204 124 145 231 931 508 183 107 ... |
Inradius = (√6)/12, tangent to faces; Circumradius/Inradius = 3 |
| Vertex solid angle |
0.551 285 598 432 530 807 942 144 ... |
acos(23/27) steradians |
| Polar angle of circumscribed cone |
0.615 479 708 670 387 341 067 464 ... |
complementary magic angle φ'm (see above) |
| Solid angle of circumscribed cone |
1.152 985 986 532 130 094 749 141 ... |
2π(1-sqrt(2/3)) steradians |
| Platonic solids: Octahedron, regular, 6 vertices, 12 edges, 8 faces, 4 edges/vertex, 3 edges/face, 4 faces/vertex. |
| Volume / edge3 |
0.471 404 520 791 031 682 933 896 ... |
(√2)/3 |
| Surface / edge2 |
3.464 101 615 137 754 587 054 892 ... |
2√3 |
| Dihedral angle (between adjacent faces) |
1.910 633 236 249 018 556 327 714 ... |
tetrahedral angle (see above) |
| Circumscribed sphere radius / edge |
0.707 106 781 186 547 524 400 844 ••• |
Circumradius = 1/√2, congruent with vertices |
| Midsphere radius / edge |
0.5 exact |
Midradius, tangent to edges |
| Inscribed sphere radius / edge |
0.408 248 290 463 863 016 366 214 ... |
1/√6; Circumradius/Inradius = √3 |
| Vertex solid angle |
1.359 347 637 816 487 748 385 570 ... |
4asin(1/3) steradians |
| Polar angle of circumscribed cone |
0.785 398 163 397 448 309 615 660 ... |
π/4; Degrees: 45 exact |
| Solid angle of circumscribed cone |
1.840 302 369 021 220 229 909 405 ... |
2π(1-sqrt(1/2)) steradians |
| Platonic solids: Cube, or Hexahedron, 8 vertices, 12 edges, 6 faces, 3 edges/vertex, 4 edges/face, 3 faces/vertex: |
| Body diagonal / edge |
1.732 050 807 568 877 293 527 446 ... |
√3. Diagonal of a cube with unit side |
| Body diagonal / Face diagonal |
1.224 744 871 391 589 049 098 642 ... |
sqrt(3/2) |
| Angle between body diagonal and an edge |
0.955 316 618 124 509 278 163 857 ... |
magic angle φm (see above) |
| Angle between body and face diagonals |
0.615 479 708 670 387 341 067 464 ... |
complementary magic angle φ'm (see above) |
| Circumscribed sphere radius / edge |
0.866 025 403 784 438 646 763 723 ... |
Circumradius = (√3)/2, congruent with vertices |
| Midsphere radius / edge |
0.707 106 781 186 547 524 400 844 ••• |
Midradius = 1/√2, tangent to edges |
| Inscribed sphere radius / edge |
0.5 exact |
Circumradius/Inradius = √3 |
| Vertex solid angle |
1.570 796 326 794 896 619 231 321 ... |
π/2 steradians |
| Polar angle of circumscribed cone |
0.955 316 618 124 509 278 163 857 ... |
magic angle φm (see above) |
| Solid angle of circumscribed cone |
2.655 586 578 711 150 775 737 130 ... |
2π(1-sqrt(1/3)) steradians |
| Platonic solids: Icosahedron, regular, 12 vertices, 30 edges, 20 faces, 5 edges/vertex, 3 edges/face, 5 faces/vertex. |
| Volume / edge3 |
2.181 694 990 624 912 373 503 822 ... |
5Φ2/6 = 5(3 + √5)/12, where Φ is the golden ratio |
| Surface / edge2 |
8.660 254 037 844 386 467 637 231 ... |
5√3 |
| Dihedral angle (between adjacent faces) |
2.411 864 997 362 826 875 007 846 ... |
2atan(Φ2); Degrees: 138.189 685 104 221 401 934 142 083 ... |
| Circumscribed sphere radius / edge |
0.951 056 516 295 153 572 116 439 ... |
ξΦ/2 = (sqrt(10)+2sqrt(5))/4, ξ being the associate of Φ |
| Midsphere radius / edge |
0.809 016 994 374 947 424 102 293 ... |
Midradius = Φ/2, tangent to edges |
| Inscribed sphere radius / edge |
0.755 761 314 076 170 730 480 133 ... |
Inradius = Φ2/(2√3) = sqrt(42+18√5)/12 |
| Vertex solid angle |
2.634 547 026 044 754 659 651 303 ... |
2π - 5asin(2/3) steradians |
| Polar angle of circumscribed cone |
1.017 221 967 897 851 367 722 788 ... |
acos(sqrt((5-√5)/10)); Degrees: 58.282 525 588 538 994 675 ... |
| Solid angle of circumscribed cone |
2.979 919 307 985 462 371 739 387 ... |
2π(1-sqrt((5-√5)/10)) steradians |
| Platonic solids: Dodecahedron, regular, 20 vertices, 30 edges, 12 faces, 3 edges/vertex, 5 edges/face, 3 faces/vertex. |
| Volume / edge3 |
7.663 118 960 624 631 968 716 053 ... |
(5Φ3)/(2ξ2) = (15+7√5)/4 |
| Surface / edge2 |
20.645 728 807 067 603 073 108 143 ... |
15Φ/ξ = 3.sqrt(25+10√5) |
| Dihedral angle (between adjacent faces) |
2.034 443 935 795 702 735 445 577 ... |
2atan(Φ); Degrees: 116.565 051 177 077 989 351 572 193 ... |
| Circumscribed sphere radius / edge |
1.401 258 538 444 073 544 676 677 ... |
Circumradius = Φ(√3)/2 = (sqrt(15)+sqrt(3))/4 |
| Midsphere radius / edge |
1.309 016 994 374 947 424 102 293 ... |
Midradius = Φ2/2, tangent to edges |
| Inscribed sphere radius / edge |
1.113 516 364 411 606 735 194 375 ... |
Inradius = Φ2/(2ξ) = sqrt(250+110√5)/20 |
| Vertex solid angle |
2.961 739 153 797 314 967 874 090 ... |
π - atan(2/11) steradians |
| Polar angle of circumscribed cone |
1.311 325 654 302 976 339 315 817 ... |
acos(sqrt(1-Φ/√3)); Degrees: 75.133 425 558 791 741 514 ... |
| Solid angle of circumscribed cone |
4.671 114 867 409 770 916 675 651 ... |
2π(1-sqrt(1-Φ/√3)) steradians |
| Surface-to-Volume indices for CLOSED 3D bodies: σ3 = Surface/Volume2/3, sorted by value: |
| Sphere |
4.835 975 862 049 408 922 150 900 ... |
(36π)1/3; the absolute minimum for closed bodies |
| Icosahedron, regular |
5.148 348 556 199 515 646 330 812 ... |
(5√3)/[5(3+√5)/12]2/3; Platonic icosahedron |
| Dodecahedron, regular |
5.311 613 997 069 083 669 796 666 ... |
(3√(25+10√5))/[(15+7√5)/4]2/3; Platonic dodecahedron |
| Cylinder, closed, with minimum σ3 |
5.535 810 445 932 085 257 290 411 ... |
(54π)1/3; attained for Height = Diameter |
| Octahedron, regular |
5.719 105 757 981 619 442 544 453 ... |
(2√3)/[(√2)/3]2/3; Platonic octahedron |
| Cube |
6 |
exact |
| Cone (closed) with minimum σ3 |
6.092 947 785 379 555 603 436 316 ... |
(72π)1/3; attained for Height = (Base diameter)√2 |
| Tetrahedron, regular |
7.205 621 731 056 016 360 052 792 ... |
(√3)/[(√2)/12]2/3; Platonic tetrahedron |
| Surface-to-Volume indices for OPEN 3D bodies: σ3 = Outer_Surface/Volume2/3: |
| Cup (half-closed cylinder) with minimum σ3 |
4.393 775 662 684 569 789 060 427 ... |
3π1/3; attained for Height = Radius |
| Tube (open cylinder) for length = diameter = 1 |
3.690 540 297 288 056 838 193 607 ... |
(16π)1/3; scales as (Length/Diameter)1/3 |
| Cone (open) with minimum σ3 |
4.188 077 948 623 138 128 725 597 ... |
((√3)27π/2)1/3; attained for Height = (Base radius)√2 |
| Solid angle fractions f cut-out by cones with a given polar angle θ, f = (1 - cosθ)/2; the subtended solid angle is 4πf |
| θ = θ'm, the complementary tetrahedral angle |
0.333 333 333 333 333 333 333 333 ... |
1/3 exact |
| θ = 60 degrees |
0.25 |
1/4 exact |
| θ = 1 radian |
0.229 848 847 065 930 141 299 531 ... |
(1-cos(1))/2 |
| θ = φm, the magic angle |
0.211 324 865 405 187 117 745 425 ... |
(1-sqrt(1/3))/2; also the Knuth's constant |
| θ = 45 degrees |
0.146 446 609 406 726 237 799 577 ... |
(1-1/√2)/2 |
| θ = φ'm, the complementary magic angle |
0.091 751 709 536 136 983 633 785 ... |
(1-sqrt(2/3))/2 |
| θ = 30 degrees |
0.066 987 298 107 780 676 618 138 ... |
(1-sqrt(3/4))/2 |
| θ = 15 degrees |
0.017 037 086 855 465 856 625 128 ... |
(1-sqrt((1+sqrt(3/4))/2))/2 |
| θ = 0.5 degrees (base of 1 degree diameter) |
1.903 846 791 435 563 132 241 ...e-5 |
Steradians: 2.392 444 437 413 785 769 530 ...e-4 |
| Polar angles θ of cones cutting a given fraction f of the full solid angle, θ = acos(1-2f) |
| f = (Φ-1)/Φ, where Φ is the golden-ratio |
1.332 478 864 985 030 510 208 009 ... |
Degrees: 76.345 415 254 024 494 986 936 602 ... |
| f = 1/3 |
1.230 959 417 340 774 682 134 929 ... |
Degrees: 70.528 779 365 509 308 630 754 000 ... |
| f = 1/4 |
1.047 197 551 196 597 746 154 214 ... |
Degrees: 60 |
| f = 0.01 ( 1%) |
0.200 334 842 323 119 592 691 046 ... |
Degrees: 11.478 340 954 533 572 625 029 817 ... |
| f = 1e-6 ( 1 ppm) |
0.002 000 000 333 333 483 333 422 ... |
Degrees: 0.114 591 578 124 766 407 153 079 ... |
| Various solid angles in (ste)radians |
| Square on a sphere with 1 degree sides |
3.046 096 875 119 366 637 825 ...e-4 |
4 asin(sin(α/2)sin(β/2)); α = β = 1 degree = π/180 |
| Triangle on a sphere with 1 degree sides |
1.319 082 346 912 923 487 761 ...e-4 |
See Huilier's formula |
| Packing ratios (monodispersed) |
| ρ2, 2D disks, thickest |
0.906 899 682 117 089 252 970 392 ... |
ρ2 = π / 2√3. Best covering of infinite 2D plane. |
| 2D disks, closest random |
0.772 ± 0.002 |
Empirical & theoretical |
| ρ3, 3D spheres, thickest |
0.740 480 489 693 061 041 169 313 ... |
ρ3 = π / 3√2. Best covering of infinite 3D space |
| 3D spheres, closest random |
0.634 ± 0.007 |
Empirical & theoretical; in practice: a vibrated bed |
| Perimeters of ellipses with major semi-axis 1, and minor semi-axis b (area = πab). Special cases: b=0 ... flat ellipse, b = 1 ... circle. |
| b = 1/Φ, where Φ is the golden-ratio |
5.154 273 178 025 879 962 492 835 ... |
Golden ellipse |
| b = 0.613 372 647 073 913 744 075 540 ... |
π+2 = mean of flat ellipse and circle |
Mid-girth ellipse differs from golden ellipse by < 1% |
| b = 1/√2 |
5.402 575 524 190 702 010 080 698 ... |
Balanced ellipse (interfocal_distance = minor_axis) |
| b = 1/2 (Midway ellipse) |
4.844 224 110 273 838 099 214 251 ... |
b = 1/3: 4.454 964 406 851 752 743 376 500 ... |
| b = 3/4 |
5.525 873 040 177 376 261 321 396 ... |
b = 2/3: 5.288 479 863 096 863 263 777 221 ... |
| b = 1/4 |
4.289 210 887 578 417 111 478 604 ... |
b = 1/5: 4.202 008 907 937 800 188 939 832 ... |
| b = 1/6 |
4.150 013 265 005 047 157 825 880 ... |
b = 1/7: 4.116 311 284 366 438 220 003 847 ... |
| b = 1/8 |
4.093 119 575 024 437 585 615 711 ... |
b = 1/9: 4.076 424 191 956 689 482 335 178 ... |
| b = 1/10 |
4.063 974 180 100 895 742 557 793 ... |
b = 0.01: 4.001 098 329 722 651 860 747 464 ... |
| b = 0.001 |
4.000 015 588 104 688 244 610 756 ... |
b = 0.0001: 4.000 000 201 932 695 375 419 076 ... |
| Moving sofa constants for largest sofa that can turn a unit-width hallway corner (it can't exceed 2√2) |
| Gerver's |
2.219 531 668 871 97 (?) |
So far the largest |
| Hammersley's |
2.207 416 099 162 477 962 306 856 ... |
π/2 + 2/π. This was a nice attempt |
| Other, more or less named, geometry constants: |
| Gravitoid constant |
1.240 806 478 802 799 465 254 958 ••• |
2√(2/(3√3)). Width/Depth ratio of gravitoid curve and gravidome |
| Minimum area of a constant-width figure |
0.704 770 923 010 457 972 467 598 ••• |
(pi - sqrt(3))/2 for width = 1. See Reuleaux triangle |
| Moser's worm constant |
0.232 239 210 ••• ? |
Area of smallest region accomodating any curve of length 1 |
| Square drill constant |
0.987 700 390 736 053 460 131 999 ••• |
Portion of square area covered by a Reuleaux drill |
| Universal parabolic constant |
2.295 587 149 392 638 074 034 298 ••• |
ln(1+√2)+√2. Arc-to-latus_rectum ratio in any parabola. |
| Area doubling (Pythagora's) constant |
1.414 213 562 373 095 048 801 688 ... |
√2. Area-doubling scale factor |
| Area tripling (Theodorus') constant |
1.732 050 807 568 877 293 527 446 ... |
√3. Area-tripling scale factor |
| Volume doubling (Delian's) constant |
1.259 921 049 894 873 164 767 210 ... |
21/3. Volume-doubling scale factor |
| Volume tripling constant |
1.442 249 570 307 408 382 321 638 ... |
31/3. Volume-tripling scale factor |
| Statistics and probability constants |
| Normal probability distribution with density N(x,σ): |
| Density maximum * σ |
0.398 942 280 401 432 677 939 946 ... |
1/√(2π), attained at x = 0 |
| Percentiles: x/σ for which Intg[-∞,x]{N(x,σ)dx} = P, Intg[-x,x]{N(x,σ)dx} = 2P-1 |
| 75% |
0.674 489 750 196 081 743 202 227 ••• |
Probable error: x/σ for which Intg[-x,x]{N(x,σ)dx} = 0.5 |
| 80% |
0.841 621 233 572 914 205 178 706 ... |
85% ... 1.036 433 389 493 789 579 713 244... |
| 90% |
1.281 551 565 544 600 466 965 103 ... |
95% ... 1.644 853 626 951 472 714 863 848 ... |
| 98% |
2.053 748 910 631 823 052 937 351 ... |
99% ... 2.326 347 874 040 841 100 885 606 ... |
| 99.9% |
3.090 232 306 167 813 541 540 399 ... |
99.99% ... 3.719 016 485 455 680 564 393 660 ... |
| Math constants useful in Sciences |
| Planck's radiation law prl(x) = x3/(ex - 1), or prl(λ) = λ-5(e1/λ - 1)-1 |
| Root of (e-x + x/5 - 1) = 0 |
4.965 114 231 744 276 303 698 759 ... |
Related to prl maximum. |
| Integral of x3/(ex - 1) over [0,∞] |
6.493 939 402 266 829 149 096 022 ... |
π4/15; related to prl integral. |
| Magic angle. Root of P2 = (1-3.cos2(φ))/2, P2(x) being the 2nd-order Legendre polynomial. |
| φm |
0.955 316 618 124 509 278 163 857 ... |
acos(1/√3); Degrees: 54.735 610 317 245 345 684 622 999 ... |
| φm complement (π/2 - φm) |
0.615 479 708 670 387 341 067 464 ... |
asin(1/√3); Degrees: 35.264 389 682 754 654 315 377 000 ... |
| Ideal gas statistics with n particles per unit volume |
| 1st Chandrasekhar constant c = Γ(4/3)/(4π/3)1/3 |
0.553 960 278 365 090 204 701 121 ... |
Mean distance to nearest neighbor = c/n1/3 |
| 2nd Chandrasekhar constant C = (2π)-1/3 |
0.541 926 070 139 289 008 744 561 ... |
Most probable distance to nearest neighbor = C/n1/3 |
| Spectral lines (peaks) of height H and half-height width W: |
| Area of a Lorentzian peak / HW |
1.570 796 326 794 896 619 231 321 ... |
π / 2 |
| Area of a Gaussian peak / HW |
1.064 467 019 431 226 179 315 267 ... |
sqrt(π /(4ln2)) |
| Area of a Sinc peak / HW |
0.828 700 120 129 003 061 896 869 ... |
π/(2η), η being defined by sinc(η) = 1/2 (see sinc function) |
| The Bloembergen-Purcell-Pound function, bpp(x) = x/(1+x2) + 4x/(1+4x2), ubiquitous in the theory of 2nd rank relaxation processes |
| bpp(x) maximum: Location xmax |
0.615 795 146 961 756 244 755 982 ... |
bpp(x) being an odd function; there is a minimum at -xmax |
| bpp(x) maximum: Value at xmax |
1.425 175 719 086 501 535 329 674 ... |
For first term only: bpp1,max(y) = 0.5, for y = 1 |
| First roots ξn of sinc(n,x) for n = 0, 1, 2, 3, 4 (nD Fourier transform of an n-dimensional unit sphere): |
| ξ0 |
2.404 825 557 695 772 768 621 631 ... |
sinc(0,x) = J0(x), the Bessel function |
| ξ1 |
3.141 592 653 589 793 238 462 643 ... |
sinc(1,x) = sin(x)/x = sinc(x) = j0(x), 1st kind spherical Bessel |
| ξ2 |
3.831 705 970 207 512 315 614 435 ... |
sinc(2,x) = 2J1(x)/x |
| ξ3 , also location of 1st negative lobe of sinc(1,x) |
4.493 409 457 909 064 175 307 880 ... |
sinc(3,x) = 3[sin(x)/x - cos(x)]/x2 = 3j1(x)/x |
| ξ4 |
5.135 622 301 840 682 556 301 401 ... |
sinc(4,x) = 8J2(x)/x3 |
| Engineering constants; click here for conventional physical constants instead |
| Amplitude / Effective_Amplitude |
1.414 213 562 373 095 048 801 688 ... |
√2, holds only for harmonic functions |
| Power factor of 2 (or 0.5) in dB |
±3.010 299 956 639 811 952 137 388 ... |
±10.log10(2); corresponding amplitudes ratio is √2 : 1 |
| Amplitude factor of 2 (or 0.5) in dB |
±6.020 599 913 279 623 904 274 777 ... |
±20.log10(2) |
| ±1 dB ratios: |
| Power |
1.258 925 411 794 167 210 423 954 ... |
10+1/10 |
| Inverse power |
0.794 328 234 724 281 502 065 918 ... |
10-1/10 |
| Amplitude |
1.122 018 454 301 963 435 591 038 ... |
10+1/20 |
| Inverse amplitude |
0.891 250 938 133 745 529 953 108 ... |
10-1/20 |
| ±3 dB ratios: |
| Power |
1.995 262 314 968 879 601 352 455 ... |
10+3/10 +3 dB in power or +6 dB in amplitude |
| Inverse power |
0.501 187 233 627 272 285 001 554 ... |
10 -3/10 -3 dB in power or -6 dB in amplitude |
| Amplitude |
1.412 537 544 622 754 302 155 607 ... |
10+3/20 |
| Inverse amplitude |
0.707 945 784 384 137 910 802 214 ... |
10-3/20 |
| Music and acoustics: |
| Half-note frequency ratio |
1.059 463 094 359 295 264 561 825 ••• |
21/12 |
| "Perfect fifth" ratio |
3/2, exact |
also 2/3 |
| Pythagorean comma |
1.013 643 264 770 507 8125 |
(3/2)12/27, frequency ratio of 12 perfect fifth to 7 octaves |
| Computer and Software Engineering constants |
| Decadic-to-binary precision/capacity factor |
3.321 928 094 887 362 347 870 319 ••• |
ln2(10); Example: 7 decadic digits require 23+ binary bits |
| Binary-to-decadic precision/capacity factor |
0.301 029 995 663 981 195 213 738 ••• |
log10(2); Example: 31 binary bits require 9+ decimal digits |
| Unsigned integer data types maximum values (for signed integers see the 3rd column) |
| byte (8 bits) 2^8-1 |
255 |
signed max = 2^7-1 = +127 |
| word (16 bits) 2^16-1 |
65'535 |
signed max = 2^15-1 = +32'767 |
| dword (32 bits) 2^32-1 |
4'294'967'295 |
signed max = 2^31-1 = +2'147'483'647 |
| qword (64 bits) 2^64-1 |
18'446'744'073'709'551'615 |
signed max = 2^63-1 = +9'223'372'036'854'775'807 |
| Bit configurations which can't be used as signed integers since, though formally negative, aritmetic negation returns the same value (weird numbers) |
| 8 bits |
hex 0x80 |
signed -2^7 = -128 |
| 16 bits |
hex 0x8000 |
signed -2^15 = -32'768 |
| 32 bits |
hex 0x80000000 |
signed -2^31 = -2'147'483'648 |
| 64 bits |
hex 0x8000000000000000 |
signed -2^63 = -9'223'372'036'854'775'808 |
| Floating point (real) data types: |
| float (1+8+23 bits): Maximum value |
3.402823669209384634633746...e+38 |
2^(2^(8-1)); IEEE 754; bits are for: sign, exponent, mantissa |
| float (1+8+23 bits): minimum value |
1.401298464324817070923729...e-45 |
2*2^(-2^(8-1))*2^(-(23-1)) |
| float (1+8+23 bits): epsilon value |
1.1920928955078125 e-7 |
2^(-23), for x < ε, 1+x → 1 |
| double (1+11+52 bits): Maximum value |
1.79769313486231590772930...e+308 |
2^(2^(11-1)); IEEE 754; bits are for: sign, exponent, mantissa |
| double (1+11+52 bits): minimum value |
4.94065645841246544176568...e-324 |
2*2^(-2^(11-1))*2^(-(52-1)) |
| double (1+11+52 bits): epsilon value |
2.220446049250313080847263...e-16 |
2^(-52), for x < ε, 1+x → 1 |
| Conversion constants |
| 1 rad (radian) in degrees |
57.295 779 513 082 320 876 798 15 ... |
180/π; planar angle; 57° 17' 44.806247...'' |
| 1° (degree) in radians |
0.017 453 292 519 943 295 769 237 ... |
π/180 |
| 1 degree2 in sr (steradians) |
0.000 304 617 419 786 708 599 346 ... |
(π/180)2; used in astronomy; infinitesimal area limit |
| 1 sr (steradian) in degree2 |
3282.806 350 011 743 794 781 694 ... |
(180/π)2; inverse of the above; valid for infinitesimal areas |
| Full solid angle of 4π steradians in degrees2 |
41252.961 249 419 271 031 294 671 ... |
4π/(π/180)2 = 3602/π |
