7275:
6184:
7270:{\displaystyle {\begin{aligned}\sin {\frac {\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}&\quad \sin {\frac {3\pi }{16}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}&\quad \sin {\frac {11\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}}}\\\sin {\frac {\pi }{16}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}&\quad \sin {\frac {7\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}}&\quad \sin {\frac {3\pi }{8}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2}}}}\\\sin {\frac {3\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}}}&\quad \sin {\frac {\pi }{4}}&={\tfrac {1}{2}}{\sqrt {2}}&\quad \sin {\frac {13\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}}}\\\sin {\frac {\pi }{8}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2}}}}&\quad \sin {\frac {9\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}}&\quad \sin {\frac {7\pi }{16}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}\\\sin {\frac {5\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}}}&\quad \sin {\frac {5\pi }{16}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}&\quad \sin {\frac {15\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}\end{aligned}}}
12088:
34:
10582:
11830:
9131:
5223:
1438:
3485:
10146:
10130:
8616:
1194:
7847:
9841:
3364:
8091:
6016:
8290:
9799:
432:
4257:
7579:
1433:{\displaystyle {\begin{alignedat}{3}a_{1}&={\tfrac {3}{2}}&&=\mathbf {1} .5,\\a_{2}&={\tfrac {17}{12}}&&=\mathbf {1.41} 6\ldots ,\\a_{3}&={\tfrac {577}{408}}&&=\mathbf {1.41421} 5\ldots ,\\a_{4}&={\tfrac {665857}{470832}}&&=\mathbf {1.41421356237} 46\ldots ,\\&\qquad \vdots \end{alignedat}}}
9120:
8832:
7590:
10125:{\displaystyle {\begin{aligned}{\sqrt {2}}&={\tfrac {3}{2}}-2\left({\tfrac {1}{4}}-\left({\tfrac {1}{4}}-{\bigl (}{\tfrac {1}{4}}-\cdots {\bigr )}^{2}\right)^{2}\right)^{2}\\&={\tfrac {3}{2}}-4\left({\tfrac {1}{8}}+\left({\tfrac {1}{8}}+{\bigl (}{\tfrac {1}{8}}+\cdots {\bigr )}^{2}\right)^{2}\right)^{2}.\end{aligned}}}
7858:
3481:). However, these squares on the diagonal have positive integer sides that are smaller than the original squares. Repeating this process, there are arbitrarily small squares one twice the area of the other, yet both having positive integer sides, which is impossible since positive integers cannot be less than 1.
8611:{\displaystyle {\sqrt {2}}=\sum _{k=0}^{\infty }(-1)^{k+1}{\frac {(2k-3)!!}{(2k)!!}}=1+{\frac {1}{2}}-{\frac {1}{2\cdot 4}}+{\frac {1\cdot 3}{2\cdot 4\cdot 6}}-{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 8}}+\cdots =1+{\frac {1}{2}}-{\frac {1}{8}}+{\frac {1}{16}}-{\frac {5}{128}}+{\frac {7}{256}}+\cdots .}
5862:
10181:
times longer than its short edge could be folded in half and aligned with its shorter side to produce a sheet with exactly the same proportions as the original. This ratio of lengths of the longer over the shorter side guarantees that cutting a sheet in half along a line results in the smaller sheets
9584:
6145:
10610:
In the brain there are lattice cells, discovered in 2005 by a group led by May-Britt and Edvard Moser. "The grid cells were found in the cortical area located right next to the hippocampus At one end of this cortical area the mesh size is small and at the other it is very large. However, the
877:
of
Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it, though this has little to any substantial evidence in traditional historian
4046:
798:
7395:
8948:
660:
8635:
7842:{\displaystyle {\sqrt {2}}=\prod _{k=0}^{\infty }{\frac {(4k+2)^{2}}{(4k+1)(4k+3)}}=\left({\frac {2\cdot 2}{1\cdot 3}}\right)\!\left({\frac {6\cdot 6}{5\cdot 7}}\right)\!\left({\frac {10\cdot 10}{9\cdot 11}}\right)\!\left({\frac {14\cdot 14}{13\cdot 15}}\right)\cdots }
5212:
8250:
3904:
While the proofs by infinite descent are constructively valid when "irrational" is defined to mean "not rational", we can obtain a constructively stronger statement by using a positive definition of "irrational" as "quantifiably apart from every rational". Let
1150:
10500:
5584:
8086:{\displaystyle {\sqrt {2}}=\prod _{k=0}^{\infty }\left(1+{\frac {1}{4k+1}}\right)\left(1-{\frac {1}{4k+3}}\right)=\left(1+{\frac {1}{1}}\right)\!\left(1-{\frac {1}{3}}\right)\!\left(1+{\frac {1}{5}}\right)\!\left(1-{\frac {1}{7}}\right)\cdots .}
4907:
6011:{\displaystyle {\frac {2}{\pi }}={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}}}\cdots ,}
229:
9794:{\displaystyle \left|{\sqrt {2}}-{\frac {p}{q}}\right|={\frac {|2q^{2}-p^{2}|}{q^{2}\!\left({\sqrt {2}}+{\frac {p}{q}}\right)}}={\frac {1}{q^{2}\!\left({\sqrt {2}}+{\frac {p}{q}}\right)}}\thickapprox {\frac {1}{2{\sqrt {2}}q^{2}}}}
5820:
9278:
6027:
906:
technique. It consists basically in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to
6189:
5424:
5097:
4252:{\displaystyle \left|{\sqrt {2}}-{\frac {a}{b}}\right|={\frac {|2b^{2}-a^{2}|}{b^{2}\!\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac {1}{b^{2}\!\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac {1}{3b^{2}}},}
690:
3537:
construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the same algebraic proof as in the previous paragraph, viewed geometrically in another way.
3039:, and in particular, would have to have the factor 2 occur the same number of times. However, the factor 2 appears an odd number of times on the right, but an even number of times on the left—a contradiction.
9846:
11374:
Bishop, Errett (1985), Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI.
5336:
7574:{\displaystyle {\frac {1}{\sqrt {2}}}=\prod _{k=0}^{\infty }\left(1-{\frac {1}{(4k+2)^{2}}}\right)=\left(1-{\frac {1}{4}}\right)\!\left(1-{\frac {1}{36}}\right)\!\left(1-{\frac {1}{100}}\right)\cdots }
9115:{\displaystyle {\sqrt {2}}={\frac {3}{2}}-{\frac {1}{2}}\sum _{n=0}^{\infty }{\frac {1}{a(2^{n})}}={\frac {3}{2}}-{\frac {1}{2}}\left({\frac {1}{6}}+{\frac {1}{204}}+{\frac {1}{235416}}+\dots \right)}
8827:{\displaystyle {\sqrt {2}}=\sum _{k=0}^{\infty }{\frac {(2k+1)!}{2^{3k+1}(k!)^{2}}}={\frac {1}{2}}+{\frac {3}{8}}+{\frac {15}{64}}+{\frac {35}{256}}+{\frac {315}{4096}}+{\frac {693}{16384}}+\cdots .}
2835:
560:
5108:
10370:
8129:
10318:
2729:
2401:
3599:
1024:
2126:
4529:
4657:
4565:
10379:
5498:
11041:
4755:
1014:; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following
2892:
2551:
2044:
2952:
2922:
2865:
4701:
3137:
3029:
2497:
2444:
2169:
11586:
11533:
11452:
10917:
10575:
10220:
10179:
9549:
8859:
7302:
6172:
5846:
5714:
5610:
5457:
5274:
5020:
4996:
4606:
4489:
4404:
3894:
3531:
3338:
3277:
3207:
3069:
2686:
2627:
2603:
2579:
2014:
1976:
1952:
1924:
1714:
1659:
1631:
964:
857:
825:
552:
516:
270:
3479:
3437:
1012:
9832:
4827:
1186:
305:
2755:
9308:
formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the
3253:
3230:
3183:
3160:
3306:
2304:
2978:
10271:
10248:
84:
5743:
9151:
6140:{\displaystyle \pi =\lim _{m\to \infty }2^{m}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}}}} _{m{\text{ square roots}}}\,.}
10895:
Although the term "Babylonian method" is common in modern usage, there is no direct evidence showing how the
Babylonians computed the approximation of
10935:
10987:, is that "the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate".
5347:
5044:
3352:
3348:
793:{\displaystyle 1+{\frac {1}{3}}+{\frac {1}{3\times 4}}-{\frac {1}{3\times 4\times 34}}={\frac {577}{408}}=1.41421{\overline {56862745098039}}.}
1199:
10746:
11179:
11866:
11650:
3279:
is irrational. This application also invokes the integer root theorem, a stronger version of the rational root theorem for the case when
3347:
that is not a perfect square is irrational. For other proofs that the square root of any non-square natural number is irrational, see
11249:
11215:
11037:
4967:
373:
9135:
3397:
of the other, place two copies of the smaller square in the larger as shown in Figure 1. The square overlap region in the middle (
5282:
10963:
1554:
The rational approximation of the square root of two derived from four iterations of the
Babylonian method after starting with
655:{\displaystyle 1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}={\frac {305470}{216000}}=1.41421{\overline {296}}.}
11020:
2760:
12130:
11494:
11308:
11146:
7386:
5207:{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}={\sqrt {\tfrac {1}{2}}}={\frac {1}{\sqrt {2}}}=\cos 45^{\circ }=\sin 45^{\circ }.}
2954:
differ by an integer, it follows that the denominators of their irreducible fraction representations must be the same, i.e.
919:
by giving them a length equal to a diagonal taken from a square, whose sides are equivalent to the intended atrium's width.
1899:
8245:{\displaystyle {\frac {1}{\sqrt {2}}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}\left({\frac {\pi }{4}}\right)^{2k}}{(2k)!}}.}
3958:
cannot be equal, since the first has an odd number of factors 2 whereas the second has an even number of factors 2. Thus
4464:
This lemma can be used to show that two identical perfect squares can never be added to produce another perfect square.
970:
or as a decimal. The most common algorithm for this, which is used as a basis in many computers and calculators, is the
11283:
10602:
827:
by approximately +0.07%, is the seventh in a sequence of increasingly accurate approximations based on the sequence of
11387:
10880:
3036:
933:
2652:, as proposition 117 of Book X. However, since the early 19th century, historians have agreed that this proof is an
11005:
1145:{\displaystyle a_{n+1}={\frac {1}{2}}\left(a_{n}+{\dfrac {2}{a_{n}}}\right)={\frac {a_{n}}{2}}+{\frac {1}{a_{n}}}.}
10280:
2691:
2325:
12135:
11859:
11724:
11643:
9284:
3567:
1468:
857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than
10654:
10621:
10611:
increase in mesh size is not left to chance, but increases by the squareroot of two from one area to the next."
2084:
682:
Increase the length by its third and this third by its own fourth less the thirty-fourth part of that fourth.
11332:
4497:
859:. Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.
11993:
12140:
11397:
10495:{\displaystyle R'={\frac {S}{L/2}}={\frac {2S}{L}}={\frac {2}{(L/S)}}={\frac {2}{\sqrt {2}}}={\sqrt {2}}=R}
5579:{\displaystyle {\frac {{\sqrt {i}}+i{\sqrt {i}}}{i}}{\text{ and }}{\frac {{\sqrt {-i}}-i{\sqrt {-i}}}{-i}}}
4615:
12087:
10931:
11738:
10327:
10154:
4536:
2653:
1155:
Each iteration improves the approximation, roughly doubling the number of correct digits. Starting with
11852:
11636:
6175:
4707:
895:
12062:
1895:
1676:
358:
10520:
3343:
The rational root theorem (or integer root theorem) may be used to show that any square root of any
2870:
2529:
2022:
873:. Little is known with certainty about the time or circumstances of this discovery, but the name of
10868:
4374:, yielding a direct proof of irrationality in its constructively stronger form, not relying on the
4263:
3534:
2927:
2897:
2840:
4663:
3094:
2991:
2459:
2406:
2131:
915:
to the corners of the original square at 45 degrees of it. The proportion was also used to design
11567:
11514:
11433:
10898:
10739:
10556:
10201:
10160:
9530:
8840:
7283:
6153:
5827:
5695:
5591:
5438:
5255:
5001:
4977:
4570:
4470:
4385:
4320:(otherwise the quantitative apartness can be trivially established). This gives a lower bound of
3875:
3512:
3319:
3258:
3188:
3050:
2667:
2608:
2584:
2560:
1995:
1957:
1933:
1905:
1695:
1640:
1612:
945:
838:
806:
533:
497:
251:
11536:
4902:{\displaystyle {\frac {1}{\sqrt {2}}}={\frac {\sqrt {2}}{2}}=\sin 45^{\circ }=\cos 45^{\circ }=}
3442:
3400:
984:
447:), the tablet also gives an example where one side of the square is 30 and the diagonal then is
11923:
11483:
Henderson, David W. (2000), "Square roots in the Śulba Sūtras", in Gorini, Catherine A. (ed.),
11176:
10984:
8101:
4798:
4375:
1662:
11948:
33:
12104:
11886:
11685:
11136:
9813:
5473:
4818:
3708:
3072:
2173:
1666:
1158:
916:
312:
275:
5849:
3385:
when he was a student in the early 1950s. Given two squares with integer sides respectively
2734:
12109:
12099:
11680:
11484:
11115:
11055:
Tom M. Apostol (Nov 2000), "Irrationality of The Square Root of Two -- A Geometric Proof",
10873:
Architecture and
Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s
10701:
3533:
is irrational. It is also an example of proof by infinite descent. It makes use of classic
3505:
3235:
3212:
3165:
3142:
2648:
2236:
2075:
2017:
11208:
3282:
2280:
1954:
is a rational number, meaning that there exists a pair of integers whose ratio is exactly
8:
11943:
11933:
11916:
11404:(1998), "Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context",
10803:
Von Fritz, Kurt (1945). "The
Discovery of Incommensurability by Hippasus of Metapontum".
8908:
5660:
3561:
3088:
3032:
2957:
1986:
330:
224:{\displaystyle 1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}}}}
11119:
10919:
seen on tablet YBC 7289. Fowler and Robson offer informed and detailed conjectures.
10692:
Fowler, David H. (2001), "The story of the discovery of incommensurability, revisited",
10581:
10253:
10230:
5815:{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{~\cdot ^{~\cdot ^{~\cdot }}}}}=2.}
869:
is incommensurable with its side, or in modern language, that the square root of two is
11834:
11700:
11549:
11471:
11349:
11105:
11072:
10828:
9142:
8904:
5242:. This diagram illustrates the circular and hyperbolic functions based on sector areas
4415:
4382:(1985, p. 18). This proof constructively exhibits an explicit discrepancy between
3382:
3076:
975:
832:
43:
9273:{\displaystyle {\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}.}
12018:
11983:
11961:
11875:
11829:
11815:
11806:
11797:
11695:
11670:
11490:
11383:
11304:
11279:
11142:
11132:
11004:
in 1826–1829 already relegates this proof to an
Appendix. The same thing occurs with
10959:
10876:
10820:
10754:
10586:
10550:
10528:
10512:
10182:
having the same (approximate) ratio as the original sheet. When
Germany standardised
8900:
8884:
8622:
3368:
971:
870:
377:
334:
20:
12055:
12050:
12028:
12013:
11896:
11792:
11787:
11782:
11777:
11705:
11690:
11659:
11555:
11463:
11413:
11341:
11064:
11017:
10812:
10640:
10626:
9130:
8862:
8275:
7309:
7308:, which is a stronger property than irrationality, but statistical analyses of its
3309:
2059:
666:
308:
10186:
at the beginning of the 20th century, they used
Lichtenberg's ratio to create the
5616:(i.e., infinite exponential tower) is equal to its square. In other words: if for
1669:
whose decimal expansions have been calculated to similarly high precision include
11743:
11675:
11366:
11183:
11024:
10750:
10697:
8626:
3817:
Hence, there is an even smaller right isosceles triangle, with hypotenuse length
3712:
2638:
1634:
11430:; Gover, T. N. (1967), "The generalized serial test and the binary expansion of
5222:
1926:
is not rational" by assuming that it is rational and then deriving a falsehood.
11911:
11901:
11563:
11401:
11327:
10850:
10846:
10598:
10549:
The celestial latitude (declination) of the Sun during a planet's astronomical
5477:
5460:
3500:
3344:
862:
46:
11276:
The Book: A Cover-to-Cover
Exploration of the Most Powerful Object of Our Time
11094:(2011), "Meaning in Classical Mathematics: Is it at Odds with Intuitionism?",
530:
digits, and is the closest possible three-place sexagesimal representation of
12124:
11966:
11928:
11748:
11729:
11719:
10824:
8097:
7305:
5419:{\displaystyle \left({\sqrt {2}}+1\right)\!\left({\sqrt {2}}-1\right)=2-1=1.}
5092:{\displaystyle \left({\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}\right)\!.}
4379:
1686:
10511:
There are some interesting properties involving the square root of 2 in the
1685:. Such computations provide empirical evidence of whether these numbers are
12033:
12003:
11758:
11753:
11710:
11418:
11096:
11091:
10663:
10191:
5430:
5027:
3628:
2731:
as an irreducible fraction in lowest terms, with coprime positive integers
1682:
1661:
was eclipsed with the use of a home computer. Shigeru Kondo calculated one
670:
10187:
11595:
11239:
10594:
9309:
5469:
5031:
3484:
3313:
828:
519:
440:
362:
326:
325:
Geometrically, the square root of 2 is the length of a diagonal across a
322:
of 2, to distinguish it from the negative number with the same property.
319:
241:
11244:"Sequence A082405 (a(n) = 34*a(n-1) - a(n-2); a(0)=0, a(1)=6)"
2062:(having no common factor) which additionally means that at least one of
11545:
11475:
11427:
11353:
11330:(2000), "Irrationality of the square root of two – A geometric proof",
11076:
10832:
10183:
10145:
3080:
39:
11844:
11361:
10590:
5613:
2633:
1015:
939:
899:
480:
11628:
11467:
11345:
11068:
10816:
10543:
10535:
7313:
5023:
3693:
3316:; for such a polynomial, all roots are necessarily integers (which
3084:
874:
484:
436:
245:
16:
Unique positive real number which when multiplied by itself gives 2
11110:
10601:, including the square root of 2 (√7 is not possible due to
911:. The system was employed to build pavements by creating a square
10524:
10195:
4801:(reciprocal) of the square root of two (i.e., the square root of
1982:
967:
912:
527:
11163:
What is mathematics? An
Elementary Approach to Ideas and Methods
3488:
Figure 2. Tom Apostol's geometric proof of the irrationality of
2253:
is necessarily even because it is 2 times another whole number.)
978:
for computing roots of arbitrary functions. It goes as follows:
19:"Pythagoras's constant" redirects here. Not to be confused with
11001:
10779:
10755:
High resolution photographs, descriptions, and analysis of the
2643:
1692:
This is a table of recent records in calculating the digits of
866:
11209:"A Compendium of BBP-Type Formulas for Mathematical Constants"
10538:
in photographic lenses, which in turn means that the ratio of
380:
of the square root of 2, here truncated to 65 decimal places:
5226:
5035:
3624:
1637:'s team. In February 2006, the record for the calculation of
908:
431:
11602:
5476:, if the square root symbol is interpreted suitably for the
11243:
10714:
5331:{\displaystyle \!\ {1 \over {{\sqrt {2}}-1}}={\sqrt {2}}+1}
5230:
4962:
3394:
3042:
2557:
Since we have derived a falsehood, the assumption (1) that
369:
7385:, along with the infinite product representations for the
3363:
439:
with annotations. Besides showing the square root of 2 in
6150:
Similar in appearance but with a finite number of terms,
5612:
is also the only real number other than 1 whose infinite
3850:
and in the same ratio, contradicting the hypothesis that
3788:
is also a right isosceles triangle. It also follows that
2262:
must be even (as squares of odd integers are never even).
1894:
One proof of the number's irrationality is the following
902:
describes the use of the square root of 2 progression or
7312:
are consistent with the hypothesis that it is normal to
2830:{\displaystyle ({\sqrt {2}}-1)({\sqrt {2}}+1)=2-1^{2}=1}
878:
practice. The square root of two is occasionally called
12072:
10553:
points equals the tilt of the planet's axis divided by
9238:
9226:
9212:
9200:
9186:
9174:
8899:
The number can be represented by an infinite series of
1670:
803:
This approximation, diverging from the actual value of
244:
that, when multiplied by itself or squared, equals the
187:
175:
161:
149:
135:
123:
109:
97:
11511:
Gourdon, X.; Sebah, P. (2001), "Pythagoras' Constant:
11238:
11038:"Paradoxes, Contradictions, and the Limits of Science"
10062:
10040:
10019:
9995:
9931:
9909:
9888:
9864:
9816:
9241:
9229:
9215:
9203:
9189:
9177:
7215:
7145:
7065:
6994:
6914:
6854:
6778:
6728:
6653:
6592:
6512:
6442:
6366:
6296:
6216:
5136:
5113:
4772:
is never even. However, this contradicts the equation
4766:
is a primitive Pythagorean triple, and from the lemma
3367:
Figure 1. Stanley Tennenbaum's geometric proof of the
1382:
1327:
1272:
1220:
190:
178:
164:
152:
138:
126:
112:
100:
11570:
11517:
11436:
10901:
10759:
tablet (YBC 7289) from the Yale Babylonian Collection
10559:
10382:
10330:
10283:
10256:
10233:
10204:
10163:
9844:
9587:
9533:
9154:
8951:
8903:, with denominators defined by 2 th terms of a
8843:
8638:
8293:
8132:
7861:
7593:
7398:
7286:
6187:
6156:
6030:
5865:
5830:
5746:
5698:
5594:
5501:
5441:
5350:
5285:
5258:
5111:
5047:
5004:
4980:
4830:
4710:
4666:
4618:
4573:
4539:
4500:
4473:
4388:
4049:
3878:
3570:
3548:
be a right isosceles triangle with hypotenuse length
3515:
3445:
3403:
3340:
is not, as 2 is not a perfect square) or irrational.
3322:
3285:
3261:
3238:
3215:
3191:
3168:
3145:
3097:
3087:
of a factor of the constant term and a factor of the
3053:
2994:
2960:
2930:
2900:
2873:
2843:
2763:
2737:
2694:
2670:
2611:
2587:
2563:
2532:
2462:
2409:
2328:
2283:
2134:
2087:
2025:
1998:
1960:
1936:
1908:
1698:
1643:
1615:
1197:
1161:
1076:
1027:
987:
948:
841:
809:
693:
563:
554:, representing a margin of error of only –0.000042%:
536:
500:
278:
254:
87:
10273:
longer length of the sides of a sheet of paper, with
9809:
The following nested square expressions converge to
4414:
This proof uses the following property of primitive
2581:
is a rational number must be false. This means that
2456:
is divisible by two and therefore even, and because
1633:
was calculated to 137,438,953,444 decimal places by
11292:
3439:) must equal the sum of the two uncovered squares (
38:The square root of 2 is equal to the length of the
11580:
11527:
11456:Journal of the Royal Statistical Society, Series A
11446:
11104:(2): 223–302 (see esp. Section 2.3, footnote 15),
10911:
10569:
10494:
10364:
10312:
10265:
10242:
10214:
10173:
10124:
9826:
9793:
9543:
9272:
9114:
8853:
8826:
8610:
8244:
8085:
7841:
7573:
7296:
7269:
6166:
6139:
6010:
5840:
5814:
5708:
5604:
5578:
5451:
5418:
5330:
5268:
5206:
5091:
5014:
4990:
4901:
4749:
4695:
4651:
4600:
4559:
4523:
4483:
4398:
4251:
3888:
3593:
3525:
3473:
3431:
3332:
3300:
3271:
3247:
3224:
3201:
3177:
3154:
3131:
3063:
3031:. Being the same quantity, each side has the same
3023:
2972:
2946:
2916:
2886:
2859:
2829:
2749:
2723:
2680:
2621:
2597:
2573:
2545:
2491:
2438:
2395:
2298:
2163:
2120:
2038:
2008:
1970:
1946:
1918:
1708:
1653:
1625:
1515:1414...) with a marginally smaller error (approx.
1432:
1180:
1144:
1006:
958:
851:
819:
792:
654:
546:
510:
299:
264:
223:
10740:Photograph, illustration, and description of the
10534:The square root of two forms the relationship of
10157:found that any sheet of paper whose long edge is
9725:
9672:
8050:
8023:
7996:
7801:
7766:
7731:
7541:
7514:
5374:
5286:
5085:
4187:
4134:
2988:As with the proof by infinite descent, we obtain
12122:
10921:Fowler and Robson, p. 376. Flannery, p. 32, 158.
10372:be the analogous ratio of the halved sheet, then
6038:
5459:can also be expressed in terms of copies of the
4574:
2983:
1495:The next two better rational approximations are
10867:
8096:The number can also be expressed by taking the
4409:
333:. It was probably the first number known to be
11561:
11489:, Cambridge University Press, pp. 39–45,
11165:, London: Oxford University Press, p. 124
11160:
11054:
3838:. These values are integers even smaller than
2642:, §I.23. It appeared first as a full proof in
2526:are both even, which contradicts step 3 (that
451:. The sexagesimal digit 30 can also stand for
11860:
11644:
10983:All that Aristotle says, while writing about
10954:
10952:
10082:
10056:
9951:
9925:
5038:with the axes in a plane has the coordinates
2867:can be expressed as the irreducible fraction
889:
11510:
11486:Geometry At Work: Papers in Applied Geometry
11396:
11035:
10313:{\displaystyle R={\frac {L}{S}}={\sqrt {2}}}
2724:{\displaystyle {\sqrt {2}}+1={\frac {q}{p}}}
2396:{\displaystyle 2b^{2}=a^{2}=(2k)^{2}=4k^{2}}
1889:
11278:. W. W. Norton & Company. p. 324.
11161:Courant, Richard; Robbins, Herbert (1941),
10934:. Numbers.computation.free.fr. 2010-08-12.
9134:The square root of 2 and approximations by
11867:
11853:
11651:
11637:
11546:The Square Root of Two to 5 million digits
11131:
11125:
10949:
10744:tablet from the Yale Babylonian Collection
8625:of this series can be accelerated with an
3739:is also a right isosceles triangle. Hence
3594:{\displaystyle {\frac {m}{n}}={\sqrt {2}}}
2659:
2605:is not a rational number; that is to say,
1884:
1443:
974:for computing square roots, an example of
32:
11482:
11426:
11417:
11360:
11298:
11250:On-Line Encyclopedia of Integer Sequences
11194:
11109:
10845:
10802:
10190:of paper sizes. Today, the (approximate)
9141:The square root of two has the following
6133:
4792:
4553:
4040:in the numerator and denominator, we get
1721:
1604:
927:
374:On-Line Encyclopedia of Integer Sequences
11377:
11089:
10580:
10144:
9129:
5221:
4440:are coprime positive integers such that
3483:
3362:
3043:Application of the rational root theorem
2980:. This gives the desired contradiction.
2121:{\displaystyle {\frac {a^{2}}{b^{2}}}=2}
665:Another early approximation is given in
430:
11326:
11273:
4524:{\displaystyle {\sqrt {2}}={a \over b}}
3139:, the only possible rational roots are
327:square with sides of one unit of length
315:. Technically, it should be called the
240:(approximately 1.4142) is the positive
12123:
11874:
11206:
10932:"Constants and Records of Computation"
10691:
5681:will be called (if this limit exists)
5233:are the same when the conic radius is
4266:being true because it is assumed that
3973:. Multiplying the absolute difference
248:. It may be written in mathematics as
11848:
11658:
11632:
11269:
11267:
11207:Bailey, David H. (13 February 2011).
10996:The edition of the Greek text of the
9125:
7324:
4652:{\displaystyle 2={a^{2} \over b^{2}}}
3899:
1727:
11303:. 2016 Kagge Forlag AS. p. 81.
10506:
2664:Assume by way of contradiction that
494:–1600 BC) gives an approximation of
10365:{\displaystyle R'={\frac {L'}{S'}}}
5429:This is related to the property of
4560:{\displaystyle a,b\in \mathbb {Z} }
1188:, the subsequent iterations yield:
13:
11538:Numbers, Constants and Computation
11264:
10153:In 1786, German physics professor
9136:convergents of continued fractions
9001:
8865:. BBP-type formulas are known for
8665:
8320:
8164:
7888:
7620:
7430:
7319:
6048:
3358:
2322:in the second equation of step 4:
1981:If the two integers have a common
865:discovered that the diagonal of a
14:
12152:
11504:
11057:The American Mathematical Monthly
9759:
4750:{\displaystyle b^{2}+b^{2}=a^{2}}
3037:fundamental theorem of arithmetic
2688:were rational. Then we may write
2271:is even, there exists an integer
1985:, it can be eliminated using the
1900:proof of a negation by refutation
1860:
1835:
1810:
1785:
1760:
1735:
934:Methods of computing square roots
526:, which is accurate to about six
357:857) is sometimes used as a good
12086:
11828:
11044:from the original on 2016-06-30.
9804:
6021:which is related to the formula
3872:cannot be both integers; hence,
3381:A simple proof is attributed to
2656:and not attributable to Euclid.
1448:A simple rational approximation
1402:
1347:
1292:
1240:
922:
831:, which can be derived from the
11611:2 billion searchable digits of
11232:
11221:from the original on 2011-06-10
11200:
11188:
11169:
11154:
11082:
11048:
11029:
11011:
10990:
10977:
10966:from the original on 2022-04-07
10938:from the original on 2012-03-01
10924:
10603:Legendre's three-square theorem
10135:
7185:
7115:
6964:
6884:
6748:
6703:
6562:
6482:
6336:
6266:
3917:be positive integers such that
3860:is in lowest terms. Therefore,
3075:, which states that a rational
1422:
11556:Square root of 2 is irrational
10889:
10861:
10839:
10796:
10772:
10763:
10731:
10707:
10685:
10622:List of mathematical constants
10519:The square root of two is the
10455:
10441:
9656:
9625:
9323:). The first convergents are:
9028:
9015:
8725:
8715:
8688:
8673:
8388:
8379:
8368:
8353:
8335:
8325:
8230:
8221:
8182:
8172:
8104:. For example, the series for
7688:
7673:
7670:
7655:
7644:
7628:
7468:
7452:
6045:
4589:
4577:
4118:
4087:
3462:
3449:
3420:
3404:
3295:
3289:
3107:
3101:
2887:{\displaystyle {\frac {p}{q}}}
2799:
2783:
2780:
2764:
2546:{\displaystyle {\frac {a}{b}}}
2505:is also even which means that
2368:
2358:
2039:{\displaystyle {\frac {a}{b}}}
1665:decimal places in 2010. Other
376:consists of the digits in the
1:
11562:Grime, James; Bowley, Roger.
11333:American Mathematical Monthly
11320:
11000:published by E. F. August in
10140:
9453:and the convergent following
5217:
3945:satisfies these bounds). Now
3560:as shown in Figure 2. By the
3393:, one of which has twice the
2984:Proof by unique factorization
2947:{\displaystyle {\sqrt {2}}+1}
2917:{\displaystyle {\sqrt {2}}-1}
2860:{\displaystyle {\sqrt {2}}-1}
1543:012) with an error of approx
674:
488:
12131:Quadratic irrational numbers
11301:The Book: Hjernen er sternen
7389:, leads to products such as
5252:One interesting property of
4696:{\displaystyle 2b^{2}=a^{2}}
4410:Proof by Pythagorean triples
3132:{\displaystyle p(x)=x^{2}-2}
3083:, if it exists, must be the
3024:{\displaystyle a^{2}=2b^{2}}
2632:This proof was hinted at by
2492:{\displaystyle 2k^{2}=b^{2}}
2439:{\displaystyle b^{2}=2k^{2}}
2164:{\displaystyle a^{2}=2b^{2}}
782:
644:
7:
11739:Quadratic irrational number
11725:Pisot–Vijayaraghavan number
11581:{\displaystyle {\sqrt {2}}}
11528:{\displaystyle {\sqrt {2}}}
11447:{\displaystyle {\sqrt {2}}}
10960:"Records set by y-cruncher"
10912:{\displaystyle {\sqrt {2}}}
10875:. Birkhäuser. p. 204.
10871:; Ostwald, Michael (2015).
10615:
10570:{\displaystyle {\sqrt {2}}}
10215:{\displaystyle {\sqrt {2}}}
10174:{\displaystyle {\sqrt {2}}}
10155:Georg Christoph Lichtenberg
10149:The A series of paper sizes
9544:{\displaystyle {\sqrt {2}}}
8854:{\displaystyle {\sqrt {2}}}
7297:{\displaystyle {\sqrt {2}}}
6167:{\displaystyle {\sqrt {2}}}
5841:{\displaystyle {\sqrt {2}}}
5709:{\displaystyle {\sqrt {2}}}
5605:{\displaystyle {\sqrt {2}}}
5452:{\displaystyle {\sqrt {2}}}
5269:{\displaystyle {\sqrt {2}}}
5015:{\displaystyle {\sqrt {2}}}
4991:{\displaystyle {\sqrt {2}}}
4601:{\displaystyle \gcd(a,b)=1}
4484:{\displaystyle {\sqrt {2}}}
4399:{\displaystyle {\sqrt {2}}}
3889:{\displaystyle {\sqrt {2}}}
3526:{\displaystyle {\sqrt {2}}}
3349:Quadratic irrational number
3333:{\displaystyle {\sqrt {2}}}
3272:{\displaystyle {\sqrt {2}}}
3202:{\displaystyle {\sqrt {2}}}
3064:{\displaystyle {\sqrt {2}}}
2681:{\displaystyle {\sqrt {2}}}
2622:{\displaystyle {\sqrt {2}}}
2598:{\displaystyle {\sqrt {2}}}
2574:{\displaystyle {\sqrt {2}}}
2009:{\displaystyle {\sqrt {2}}}
1971:{\displaystyle {\sqrt {2}}}
1947:{\displaystyle {\sqrt {2}}}
1919:{\displaystyle {\sqrt {2}}}
1902:: it proves the statement "
1709:{\displaystyle {\sqrt {2}}}
1654:{\displaystyle {\sqrt {2}}}
1626:{\displaystyle {\sqrt {2}}}
959:{\displaystyle {\sqrt {2}}}
852:{\displaystyle {\sqrt {2}}}
820:{\displaystyle {\sqrt {2}}}
547:{\displaystyle {\sqrt {2}}}
511:{\displaystyle {\sqrt {2}}}
475:is approximately 0.7071065.
265:{\displaystyle {\sqrt {2}}}
10:
12157:
11240:Sloane, N. J. A.
10737:Fowler and Robson, p. 368.
10655:Gelfond–Schneider constant
8861:can be represented with a
5022:, is a common quantity in
4467:Suppose the contrary that
3474:{\displaystyle 2(a-b)^{2}}
3432:{\displaystyle (2b-a)^{2}}
1007:{\displaystyle a_{0}>0}
931:
896:ancient Roman architecture
890:Ancient Roman architecture
426:
18:
12095:
12084:
11882:
11824:
11666:
4998:, also the reciprocal of
3692:coincide. Therefore, the
2403:, which is equivalent to
1896:proof by infinite descent
1890:Proof by infinite descent
78:
59:
54:
31:
11558:, a collection of proofs
11378:Flannery, David (2005),
11299:Nordengen, Kaja (2016).
10857:, Copernicus, p. 25
10679:
10527:interval in twelve-tone
9827:{\textstyle {\sqrt {2}}}
8837:It is not known whether
7280:It is not known whether
4491:is rational. Therefore,
3535:compass and straightedge
1588:; its square is ≈
1580:) is too large by about
669:mathematical texts, the
361:with a reasonably small
329:; this follows from the
11274:Houston, Keith (2016).
11195:Good & Gover (1967)
11141:, Dover, pp. 4–6,
10985:proofs by contradiction
10542:between two successive
10320:as required by ISO 216.
6176:trigonometric constants
3732:is half a right angle,
3503:made another geometric
2660:Proof using reciprocals
2239:because it is equal to
1885:Proofs of irrationality
1444:Rational approximations
1181:{\displaystyle a_{0}=1}
435:Babylonian clay tablet
300:{\displaystyle 2^{1/2}}
12136:Mathematical constants
11835:Mathematics portal
11582:
11529:
11448:
11419:10.1006/hmat.1998.2209
11380:The Square Root of Two
10913:
10606:
10571:
10496:
10366:
10314:
10267:
10244:
10216:
10175:
10150:
10126:
9828:
9795:
9578:, which follows from:
9545:
9274:
9138:
9116:
9005:
8855:
8828:
8669:
8612:
8324:
8246:
8168:
8102:trigonometric function
8087:
7892:
7843:
7624:
7575:
7434:
7298:
7271:
6168:
6141:
6012:
5842:
5816:
5710:
5606:
5580:
5453:
5420:
5332:
5270:
5249:
5208:
5102:This number satisfies
5093:
5016:
4992:
4903:
4799:multiplicative inverse
4793:Multiplicative inverse
4751:
4697:
4653:
4602:
4561:
4525:
4485:
4400:
4376:law of excluded middle
4253:
3890:
3595:
3527:
3509:argument showing that
3497:
3475:
3433:
3378:
3334:
3302:
3273:
3249:
3226:
3203:
3179:
3156:
3133:
3071:also follows from the
3065:
3025:
2974:
2948:
2918:
2888:
2861:
2831:
2751:
2750:{\displaystyle q>p}
2725:
2682:
2623:
2599:
2575:
2547:
2493:
2440:
2397:
2300:
2165:
2122:
2040:
2010:
1972:
1948:
1920:
1710:
1667:mathematical constants
1655:
1627:
1609:In 1997, the value of
1605:Records in computation
1434:
1182:
1146:
1008:
960:
928:Computation algorithms
853:
821:
794:
680:–200 BC), as follows:
656:
548:
512:
476:
359:rational approximation
311:, and therefore not a
301:
266:
225:
49:with legs of length 1.
11583:
11548:by Jerry Bonnell and
11530:
11449:
11175:Julian D. A. Wiseman
11138:Pythagorean Triangles
11036:Yanofsky, N. (2016).
10914:
10805:Annals of Mathematics
10780:"The Dangerous Ratio"
10584:
10572:
10497:
10367:
10315:
10268:
10245:
10217:
10194:of paper sizes under
10176:
10148:
10127:
9829:
9796:
9546:
9275:
9133:
9117:
8985:
8856:
8829:
8649:
8613:
8304:
8255:The Taylor series of
8247:
8148:
8088:
7872:
7844:
7604:
7576:
7414:
7299:
7272:
6169:
6142:
6013:
5843:
5817:
5711:
5607:
5581:
5474:arithmetic operations
5454:
5421:
5333:
5271:
5225:
5209:
5094:
5017:
4993:
4960:... (sequence
4904:
4752:
4698:
4654:
4603:
4562:
4526:
4486:
4401:
4254:
3891:
3725:is a right angle and
3596:
3528:
3487:
3476:
3434:
3366:
3335:
3303:
3274:
3250:
3248:{\displaystyle \pm 2}
3227:
3225:{\displaystyle \pm 1}
3204:
3180:
3178:{\displaystyle \pm 2}
3157:
3155:{\displaystyle \pm 1}
3134:
3073:rational root theorem
3066:
3047:The irrationality of
3026:
2975:
2949:
2919:
2889:
2862:
2832:
2752:
2726:
2683:
2624:
2600:
2576:
2548:
2494:
2441:
2398:
2301:
2166:
2123:
2041:
2016:can be written as an
2011:
1973:
1949:
1921:
1711:
1656:
1628:
1435:
1183:
1147:
1009:
981:First, pick a guess,
961:
932:Further information:
884:Pythagoras's constant
854:
822:
795:
657:
549:
513:
434:
313:transcendental number
302:
267:
226:
11681:Constructible number
11568:
11515:
11434:
11406:Historia Mathematica
11177:Sin and cos in surds
11008:edition (1883–1888).
10899:
10557:
10380:
10328:
10281:
10254:
10231:
10202:
10198:(A4, A0, etc.) is 1:
10161:
9842:
9814:
9585:
9531:
9152:
8949:
8841:
8636:
8291:
8130:
7859:
7591:
7396:
7284:
6185:
6154:
6028:
5863:
5828:
5744:
5696:
5592:
5499:
5439:
5348:
5283:
5256:
5109:
5045:
5002:
4978:
4828:
4708:
4664:
4616:
4610:Squaring both sides,
4571:
4537:
4498:
4471:
4386:
4047:
3876:
3568:
3513:
3506:reductio ad absurdum
3443:
3401:
3320:
3301:{\displaystyle p(x)}
3283:
3259:
3236:
3213:
3189:
3166:
3143:
3095:
3051:
2992:
2958:
2928:
2898:
2871:
2841:
2761:
2735:
2692:
2668:
2609:
2585:
2561:
2530:
2460:
2407:
2326:
2299:{\displaystyle a=2k}
2281:
2132:
2085:
2023:
2018:irreducible fraction
1996:
1958:
1934:
1906:
1696:
1641:
1613:
1195:
1159:
1025:
985:
946:
839:
807:
691:
561:
534:
498:
276:
252:
85:
12141:Pythagorean theorem
11807:Supersilver ratio (
11798:Supergolden ratio (
11382:, Springer-Verlag,
11120:2011arXiv1110.5456U
11090:Katz, Karin Usadi;
10855:The Book of Numbers
10250:shorter length and
9240:
9228:
9214:
9202:
9188:
9176:
8909:recurrence relation
6174:appears in various
5737:. Or symbolically:
5716:is the only number
4817:) is a widely used
4783:which implies that
4416:Pythagorean triples
4341:for the difference
3562:Pythagorean theorem
3089:leading coefficient
3033:prime factorization
2973:{\displaystyle q=p}
2216: ) (
1987:Euclidean algorithm
880:Pythagoras's number
331:Pythagorean theorem
189:
177:
163:
151:
137:
125:
111:
99:
28:
11876:Irrational numbers
11701:Eisenstein integer
11578:
11550:Robert J. Nemiroff
11525:
11444:
11182:2009-05-06 at the
11133:Sierpiński, Wacław
11023:2016-04-22 at the
10909:
10749:2012-08-13 at the
10607:
10585:Distances between
10567:
10492:
10362:
10310:
10266:{\displaystyle L=}
10263:
10243:{\displaystyle S=}
10240:
10212:
10171:
10151:
10122:
10120:
10071:
10049:
10028:
10004:
9940:
9918:
9897:
9873:
9824:
9791:
9551:by almost exactly
9541:
9270:
9263:
9258:
9253:
9235:
9209:
9183:
9143:continued fraction
9139:
9126:Continued fraction
9112:
8901:Egyptian fractions
8851:
8824:
8608:
8242:
8083:
7839:
7571:
7325:Series and product
7294:
7267:
7265:
7224:
7154:
7074:
7003:
6923:
6863:
6787:
6737:
6662:
6601:
6521:
6451:
6375:
6305:
6225:
6164:
6137:
6132:
6128: square roots
6120:
6052:
6008:
5838:
5812:
5706:
5602:
5576:
5449:
5416:
5328:
5266:
5250:
5204:
5145:
5122:
5089:
5012:
4988:
4899:
4747:
4693:
4649:
4598:
4557:
4521:
4481:
4406:and any rational.
4396:
4249:
3900:Constructive proof
3886:
3658:. It follows that
3613:are integers. Let
3591:
3523:
3498:
3471:
3429:
3383:Stanley Tennenbaum
3379:
3330:
3298:
3269:
3255:, it follows that
3245:
3222:
3199:
3175:
3152:
3129:
3061:
3021:
2970:
2944:
2914:
2884:
2857:
2837:, it follows that
2827:
2747:
2721:
2678:
2619:
2595:
2571:
2543:
2514:By steps 5 and 8,
2499:, it follows that
2489:
2436:
2393:
2296:
2171:. (
2161:
2118:
2036:
2006:
1968:
1944:
1916:
1706:
1651:
1623:
1430:
1428:
1391:
1336:
1281:
1229:
1178:
1142:
1092:
1004:
956:
942:for approximating
849:
833:continued fraction
817:
790:
652:
544:
508:
477:
297:
262:
221:
217:
212:
207:
202:
184:
158:
132:
106:
79:Continued fraction
26:
12118:
12117:
12019:Supersilver ratio
11984:Supergolden ratio
11944:Twelfth root of 2
11842:
11841:
11816:Twelfth root of 2
11696:Doubling the cube
11686:Conway's constant
11671:Algebraic integer
11660:Algebraic numbers
11576:
11564:"The Square Root
11523:
11496:978-0-88385-164-7
11442:
11370:, eBooks@Adelaide
11310:978-82-489-2018-2
11253:. OEIS Foundation
11148:978-0-486-43278-6
10907:
10597:of the first six
10565:
10551:cross-quarter day
10529:equal temperament
10513:physical sciences
10507:Physical sciences
10484:
10474:
10473:
10459:
10430:
10412:
10360:
10308:
10298:
10210:
10169:
10070:
10048:
10027:
10003:
9939:
9917:
9896:
9872:
9854:
9822:
9789:
9776:
9757:
9749:
9736:
9704:
9696:
9683:
9611:
9598:
9539:
9505:. The convergent
9265:
9260:
9255:
9239:
9227:
9213:
9201:
9187:
9175:
9160:
9099:
9086:
9073:
9058:
9045:
9032:
8983:
8970:
8957:
8849:
8813:
8800:
8787:
8774:
8761:
8748:
8735:
8644:
8597:
8584:
8571:
8558:
8545:
8520:
8473:
8438:
8417:
8398:
8299:
8237:
8204:
8143:
8142:
8070:
8043:
8016:
7989:
7960:
7923:
7867:
7852:or equivalently,
7830:
7795:
7760:
7725:
7692:
7599:
7561:
7534:
7507:
7478:
7409:
7408:
7292:
7261:
7259:
7257:
7255:
7223:
7205:
7181:
7179:
7177:
7153:
7135:
7111:
7109:
7107:
7105:
7073:
7055:
7030:
7028:
7026:
7002:
6984:
6960:
6958:
6956:
6954:
6922:
6904:
6880:
6878:
6862:
6844:
6824:
6822:
6820:
6818:
6786:
6768:
6744:
6736:
6718:
6699:
6697:
6695:
6693:
6661:
6643:
6618:
6616:
6600:
6582:
6558:
6556:
6554:
6552:
6520:
6502:
6478:
6476:
6474:
6450:
6432:
6412:
6410:
6408:
6406:
6374:
6356:
6332:
6330:
6328:
6304:
6286:
6262:
6260:
6258:
6256:
6224:
6206:
6162:
6129:
6116:
6114:
6112:
6110:
6108:
6065:
6063:
6037:
6000:
5998:
5996:
5995:
5984:
5971:
5959:
5946:
5931:
5929:
5928:
5917:
5904:
5889:
5888:
5874:
5836:
5792:
5784:
5776:
5771:
5762:
5753:
5704:
5600:
5574:
5563:
5547:
5534:
5529:
5523:
5510:
5447:
5385:
5361:
5320:
5310:
5301:
5289:
5264:
5161:
5160:
5146:
5144:
5129:
5121:
5078:
5074:
5063:
5059:
5034:that makes a 45°
5010:
4986:
4856:
4852:
4841:
4840:
4647:
4519:
4506:
4479:
4394:
4244:
4219:
4211:
4198:
4166:
4158:
4145:
4073:
4060:
3884:
3589:
3579:
3521:
3328:
3267:
3197:
3091:. In the case of
3059:
2936:
2906:
2894:. However, since
2882:
2849:
2791:
2772:
2719:
2700:
2676:
2617:
2593:
2569:
2541:
2110:
2034:
2004:
1966:
1942:
1914:
1882:
1881:
1736:Tizian Hanselmann
1728:Number of digits
1704:
1649:
1621:
1390:
1335:
1280:
1228:
1137:
1117:
1091:
1055:
972:Babylonian method
954:
847:
815:
785:
769:
756:
729:
708:
647:
631:
618:
598:
578:
542:
506:
378:decimal expansion
260:
234:
233:
219:
214:
209:
204:
188:
176:
162:
150:
136:
124:
110:
98:
21:Pythagoras number
12148:
12090:
12078:
12068:
12056:Square root of 7
12051:Square root of 6
12046:
12029:Square root of 5
12024:
12014:Square root of 3
12009:
11999:
11989:
11979:Square root of 2
11972:
11957:
11939:
11907:
11892:
11869:
11862:
11855:
11846:
11845:
11833:
11832:
11810:
11801:
11793:Square root of 7
11788:Square root of 6
11783:Square root of 5
11778:Square root of 3
11773:Square root of 2
11766:
11762:
11733:
11714:
11706:Gaussian integer
11691:Cyclotomic field
11653:
11646:
11639:
11630:
11629:
11625:
11621:
11617:
11616:
11608:
11607:
11599:
11587:
11585:
11584:
11579:
11577:
11572:
11541:
11534:
11532:
11531:
11526:
11524:
11519:
11499:
11478:
11453:
11451:
11450:
11445:
11443:
11438:
11422:
11421:
11392:
11371:
11367:Analytica priora
11356:
11315:
11314:
11296:
11290:
11289:
11271:
11262:
11261:
11259:
11258:
11236:
11230:
11229:
11227:
11226:
11220:
11213:
11204:
11198:
11192:
11186:
11173:
11167:
11166:
11158:
11152:
11151:
11129:
11123:
11122:
11113:
11092:Katz, Mikhail G.
11086:
11080:
11079:
11052:
11046:
11045:
11033:
11027:
11015:
11009:
10994:
10988:
10981:
10975:
10974:
10972:
10971:
10956:
10947:
10946:
10944:
10943:
10928:
10922:
10918:
10916:
10915:
10910:
10908:
10903:
10893:
10887:
10886:
10865:
10859:
10858:
10843:
10837:
10836:
10800:
10794:
10793:
10791:
10790:
10776:
10770:
10767:
10761:
10735:
10729:
10728:
10726:
10725:
10715:"A002193 - OEIS"
10711:
10705:
10704:
10689:
10675:
10674:
10673:
10660:
10651:
10650:
10649:
10641:Square root of 5
10637:
10636:
10635:
10627:Square root of 3
10576:
10574:
10573:
10568:
10566:
10561:
10501:
10499:
10498:
10493:
10485:
10480:
10475:
10469:
10465:
10460:
10458:
10451:
10436:
10431:
10426:
10418:
10413:
10411:
10407:
10395:
10390:
10371:
10369:
10368:
10363:
10361:
10359:
10351:
10343:
10338:
10319:
10317:
10316:
10311:
10309:
10304:
10299:
10291:
10272:
10270:
10269:
10264:
10249:
10247:
10246:
10241:
10221:
10219:
10218:
10213:
10211:
10206:
10180:
10178:
10177:
10172:
10170:
10165:
10131:
10129:
10128:
10123:
10121:
10114:
10113:
10108:
10104:
10103:
10102:
10097:
10093:
10092:
10091:
10086:
10085:
10072:
10063:
10060:
10059:
10050:
10041:
10029:
10020:
10005:
9996:
9987:
9983:
9982:
9977:
9973:
9972:
9971:
9966:
9962:
9961:
9960:
9955:
9954:
9941:
9932:
9929:
9928:
9919:
9910:
9898:
9889:
9874:
9865:
9855:
9850:
9835:
9833:
9831:
9830:
9825:
9823:
9818:
9800:
9798:
9797:
9792:
9790:
9788:
9787:
9786:
9777:
9772:
9763:
9758:
9756:
9755:
9751:
9750:
9742:
9737:
9732:
9724:
9723:
9710:
9705:
9703:
9702:
9698:
9697:
9689:
9684:
9679:
9671:
9670:
9660:
9659:
9654:
9653:
9641:
9640:
9628:
9622:
9617:
9613:
9612:
9604:
9599:
9594:
9577:
9576:
9574:
9573:
9569:
9568:
9561:
9558:
9550:
9548:
9547:
9542:
9540:
9535:
9526:
9525:
9523:
9522:
9517:
9514:
9504:
9503:
9501:
9500:
9491:
9488:
9474:
9473:
9471:
9470:
9465:
9462:
9452:
9451:
9449:
9448:
9445:
9442:
9435:
9433:
9432:
9429:
9426:
9419:
9417:
9416:
9413:
9410:
9403:
9401:
9400:
9397:
9394:
9387:
9385:
9384:
9381:
9378:
9371:
9369:
9368:
9365:
9362:
9355:
9353:
9352:
9349:
9346:
9339:
9337:
9336:
9333:
9330:
9322:
9307:
9306:
9304:
9303:
9298:
9295:
9279:
9277:
9276:
9271:
9266:
9264:
9262:
9261:
9259:
9257:
9256:
9254:
9252:
9236:
9234:
9224:
9210:
9208:
9198:
9184:
9182:
9172:
9161:
9156:
9145:representation:
9121:
9119:
9118:
9113:
9111:
9107:
9100:
9092:
9087:
9079:
9074:
9066:
9059:
9051:
9046:
9038:
9033:
9031:
9027:
9026:
9007:
9004:
8999:
8984:
8976:
8971:
8963:
8958:
8953:
8895:
8893:
8892:
8882:
8881:
8874:
8873:
8872:
8863:BBP-type formula
8860:
8858:
8857:
8852:
8850:
8845:
8833:
8831:
8830:
8825:
8814:
8806:
8801:
8793:
8788:
8780:
8775:
8767:
8762:
8754:
8749:
8741:
8736:
8734:
8733:
8732:
8714:
8713:
8694:
8671:
8668:
8663:
8645:
8640:
8617:
8615:
8614:
8609:
8598:
8590:
8585:
8577:
8572:
8564:
8559:
8551:
8546:
8538:
8521:
8519:
8496:
8479:
8474:
8472:
8455:
8444:
8439:
8437:
8423:
8418:
8410:
8399:
8397:
8377:
8351:
8349:
8348:
8323:
8318:
8300:
8295:
8283:
8276:double factorial
8273:
8266:
8265:
8264:
8251:
8249:
8248:
8243:
8238:
8236:
8219:
8218:
8217:
8209:
8205:
8197:
8190:
8189:
8170:
8167:
8162:
8144:
8138:
8134:
8122:
8121:
8119:
8118:
8115:
8112:
8092:
8090:
8089:
8084:
8076:
8072:
8071:
8063:
8049:
8045:
8044:
8036:
8022:
8018:
8017:
8009:
7995:
7991:
7990:
7982:
7966:
7962:
7961:
7959:
7942:
7929:
7925:
7924:
7922:
7905:
7891:
7886:
7868:
7863:
7848:
7846:
7845:
7840:
7835:
7831:
7829:
7818:
7807:
7800:
7796:
7794:
7783:
7772:
7765:
7761:
7759:
7748:
7737:
7730:
7726:
7724:
7713:
7702:
7693:
7691:
7653:
7652:
7651:
7626:
7623:
7618:
7600:
7595:
7580:
7578:
7577:
7572:
7567:
7563:
7562:
7554:
7540:
7536:
7535:
7527:
7513:
7509:
7508:
7500:
7484:
7480:
7479:
7477:
7476:
7475:
7447:
7433:
7428:
7410:
7404:
7400:
7384:
7383:
7381:
7380:
7379:
7378:
7372:
7369:
7362:
7360:
7359:
7356:
7353:
7346:
7344:
7343:
7340:
7337:
7310:binary expansion
7303:
7301:
7300:
7295:
7293:
7288:
7276:
7274:
7273:
7268:
7266:
7262:
7260:
7258:
7256:
7251:
7243:
7235:
7227:
7225:
7216:
7206:
7201:
7193:
7182:
7180:
7178:
7173:
7165:
7157:
7155:
7146:
7136:
7131:
7123:
7112:
7110:
7108:
7106:
7101:
7093:
7085:
7077:
7075:
7066:
7056:
7051:
7043:
7031:
7029:
7027:
7022:
7014:
7006:
7004:
6995:
6985:
6980:
6972:
6961:
6959:
6957:
6955:
6950:
6942:
6934:
6926:
6924:
6915:
6905:
6900:
6892:
6881:
6879:
6874:
6866:
6864:
6855:
6845:
6837:
6825:
6823:
6821:
6819:
6814:
6806:
6798:
6790:
6788:
6779:
6769:
6764:
6756:
6745:
6740:
6738:
6729:
6719:
6711:
6700:
6698:
6696:
6694:
6689:
6681:
6673:
6665:
6663:
6654:
6644:
6639:
6631:
6619:
6617:
6612:
6604:
6602:
6593:
6583:
6578:
6570:
6559:
6557:
6555:
6553:
6548:
6540:
6532:
6524:
6522:
6513:
6503:
6498:
6490:
6479:
6477:
6475:
6470:
6462:
6454:
6452:
6443:
6433:
6425:
6413:
6411:
6409:
6407:
6402:
6394:
6386:
6378:
6376:
6367:
6357:
6352:
6344:
6333:
6331:
6329:
6324:
6316:
6308:
6306:
6297:
6287:
6282:
6274:
6263:
6261:
6259:
6257:
6252:
6244:
6236:
6228:
6226:
6217:
6207:
6199:
6173:
6171:
6170:
6165:
6163:
6158:
6146:
6144:
6143:
6138:
6131:
6130:
6127:
6121:
6115:
6113:
6111:
6109:
6104:
6090:
6082:
6074:
6066:
6062:
6061:
6051:
6017:
6015:
6014:
6009:
6001:
5999:
5997:
5988:
5987:
5985:
5977:
5972:
5964:
5962:
5960:
5952:
5947:
5939:
5937:
5932:
5930:
5921:
5920:
5918:
5910:
5905:
5897:
5895:
5890:
5881:
5880:
5875:
5867:
5855:
5847:
5845:
5844:
5839:
5837:
5832:
5821:
5819:
5818:
5813:
5805:
5804:
5803:
5802:
5801:
5800:
5799:
5798:
5797:
5796:
5790:
5782:
5774:
5772:
5767:
5763:
5758:
5754:
5749:
5736:
5722:
5715:
5713:
5712:
5707:
5705:
5700:
5691:
5680:
5673:
5658:
5651:
5635:
5622:
5611:
5609:
5608:
5603:
5601:
5596:
5585:
5583:
5582:
5577:
5575:
5573:
5565:
5564:
5556:
5548:
5540:
5537:
5535:
5532:
5530:
5525:
5524:
5519:
5511:
5506:
5503:
5491:
5484:
5467:
5458:
5456:
5455:
5450:
5448:
5443:
5425:
5423:
5422:
5417:
5397:
5393:
5386:
5381:
5373:
5369:
5362:
5357:
5337:
5335:
5334:
5329:
5321:
5316:
5311:
5309:
5302:
5297:
5291:
5287:
5275:
5273:
5272:
5267:
5265:
5260:
5247:
5241:
5240:
5239:
5229:size and sector
5213:
5211:
5210:
5205:
5200:
5199:
5181:
5180:
5162:
5156:
5152:
5147:
5137:
5135:
5130:
5125:
5123:
5114:
5098:
5096:
5095:
5090:
5084:
5080:
5079:
5070:
5069:
5064:
5055:
5054:
5021:
5019:
5018:
5013:
5011:
5006:
4997:
4995:
4994:
4989:
4987:
4982:
4965:
4959:
4958:
4955:
4952:
4949:
4946:
4943:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4913:
4908:
4906:
4905:
4900:
4895:
4894:
4876:
4875:
4857:
4848:
4847:
4842:
4836:
4832:
4816:
4814:
4813:
4810:
4807:
4788:
4782:
4771:
4765:
4756:
4754:
4753:
4748:
4746:
4745:
4733:
4732:
4720:
4719:
4702:
4700:
4699:
4694:
4692:
4691:
4679:
4678:
4658:
4656:
4655:
4650:
4648:
4646:
4645:
4636:
4635:
4626:
4607:
4605:
4604:
4599:
4566:
4564:
4563:
4558:
4556:
4530:
4528:
4527:
4522:
4520:
4512:
4507:
4502:
4490:
4488:
4487:
4482:
4480:
4475:
4459:
4453:
4439:
4433:
4427:
4405:
4403:
4402:
4397:
4395:
4390:
4373:
4371:
4370:
4368:
4367:
4362:
4359:
4350:
4349:
4340:
4339:
4337:
4336:
4330:
4327:
4319:
4317:
4316:
4310:
4308:
4307:
4302:
4299:
4289:
4287:
4285:
4284:
4279:
4276:
4258:
4256:
4255:
4250:
4245:
4243:
4242:
4241:
4225:
4220:
4218:
4217:
4213:
4212:
4204:
4199:
4194:
4186:
4185:
4172:
4167:
4165:
4164:
4160:
4159:
4151:
4146:
4141:
4133:
4132:
4122:
4121:
4116:
4115:
4103:
4102:
4090:
4084:
4079:
4075:
4074:
4066:
4061:
4056:
4039:
4037:
4035:
4034:
4029:
4026:
4017:
4016:
4005:
4003:
4002:
4000:
3999:
3994:
3991:
3982:
3981:
3972:
3970:
3957:
3951:
3944:
3940:
3938:
3936:
3935:
3930:
3927:
3916:
3910:
3895:
3893:
3892:
3887:
3885:
3880:
3871:
3865:
3859:
3849:
3843:
3837:
3827:
3813:
3787:
3780:
3766:
3752:
3738:
3731:
3724:
3706:
3700:
3691:
3684:
3677:
3667:
3657:
3651:
3645:
3639:
3622:
3612:
3606:
3600:
3598:
3597:
3592:
3590:
3585:
3580:
3572:
3559:
3553:
3547:
3532:
3530:
3529:
3524:
3522:
3517:
3496:
3495:
3494:
3480:
3478:
3477:
3472:
3470:
3469:
3438:
3436:
3435:
3430:
3428:
3427:
3377:
3376:
3353:Infinite descent
3339:
3337:
3336:
3331:
3329:
3324:
3310:monic polynomial
3307:
3305:
3304:
3299:
3278:
3276:
3275:
3270:
3268:
3263:
3254:
3252:
3251:
3246:
3231:
3229:
3228:
3223:
3209:is not equal to
3208:
3206:
3205:
3200:
3198:
3193:
3184:
3182:
3181:
3176:
3161:
3159:
3158:
3153:
3138:
3136:
3135:
3130:
3122:
3121:
3070:
3068:
3067:
3062:
3060:
3055:
3030:
3028:
3027:
3022:
3020:
3019:
3004:
3003:
2979:
2977:
2976:
2971:
2953:
2951:
2950:
2945:
2937:
2932:
2923:
2921:
2920:
2915:
2907:
2902:
2893:
2891:
2890:
2885:
2883:
2875:
2866:
2864:
2863:
2858:
2850:
2845:
2836:
2834:
2833:
2828:
2820:
2819:
2792:
2787:
2773:
2768:
2756:
2754:
2753:
2748:
2730:
2728:
2727:
2722:
2720:
2712:
2701:
2696:
2687:
2685:
2684:
2679:
2677:
2672:
2639:Analytica Priora
2628:
2626:
2625:
2620:
2618:
2613:
2604:
2602:
2601:
2596:
2594:
2589:
2580:
2578:
2577:
2572:
2570:
2565:
2553:is irreducible).
2552:
2550:
2549:
2544:
2542:
2534:
2525:
2519:
2510:
2504:
2498:
2496:
2495:
2490:
2488:
2487:
2475:
2474:
2455:
2445:
2443:
2442:
2437:
2435:
2434:
2419:
2418:
2402:
2400:
2399:
2394:
2392:
2391:
2376:
2375:
2354:
2353:
2341:
2340:
2321:
2316:from step 7 for
2315:
2305:
2303:
2302:
2297:
2276:
2270:
2261:
2256:It follows that
2252:
2245:
2234:
2225:
2215:
2213:
2211:
2210:
2205:
2202:
2193:
2191:
2190:
2185:
2182:
2170:
2168:
2167:
2162:
2160:
2159:
2144:
2143:
2127:
2125:
2124:
2119:
2111:
2109:
2108:
2099:
2098:
2089:
2081:It follows that
2073:
2067:
2060:coprime integers
2057:
2051:
2045:
2043:
2042:
2037:
2035:
2027:
2015:
2013:
2012:
2007:
2005:
2000:
1977:
1975:
1974:
1969:
1967:
1962:
1953:
1951:
1950:
1945:
1943:
1938:
1925:
1923:
1922:
1917:
1915:
1910:
1878:
1877:
1874:
1871:
1868:
1853:
1852:
1849:
1846:
1843:
1833:February 9, 2012
1828:
1827:
1824:
1821:
1818:
1808:January 20, 2016
1803:
1802:
1799:
1796:
1793:
1778:
1777:
1774:
1771:
1768:
1753:
1752:
1749:
1746:
1743:
1719:
1718:
1715:
1713:
1712:
1707:
1705:
1700:
1679:
1673:
1660:
1658:
1657:
1652:
1650:
1645:
1632:
1630:
1629:
1624:
1622:
1617:
1600:
1599:
1596:
1593:
1587:
1585:
1579:
1577:
1576:
1573:
1570:
1563:
1550:
1548:
1538:
1536:
1535:
1532:
1529:
1522:
1520:
1510:
1508:
1507:
1504:
1501:
1491:
1489:
1483:
1481:
1480:
1477:
1474:
1463:
1461:
1460:
1457:
1454:
1439:
1437:
1436:
1431:
1429:
1418:
1405:
1394:
1392:
1383:
1373:
1372:
1350:
1339:
1337:
1328:
1318:
1317:
1295:
1284:
1282:
1273:
1263:
1262:
1243:
1232:
1230:
1221:
1211:
1210:
1187:
1185:
1184:
1179:
1171:
1170:
1151:
1149:
1148:
1143:
1138:
1136:
1135:
1123:
1118:
1113:
1112:
1103:
1098:
1094:
1093:
1090:
1089:
1077:
1071:
1070:
1056:
1048:
1043:
1042:
1013:
1011:
1010:
1005:
997:
996:
965:
963:
962:
957:
955:
950:
858:
856:
855:
850:
848:
843:
826:
824:
823:
818:
816:
811:
799:
797:
796:
791:
786:
778:
770:
762:
757:
755:
735:
730:
728:
714:
709:
701:
679:
676:
661:
659:
658:
653:
648:
640:
632:
624:
619:
617:
616:
604:
599:
597:
596:
584:
579:
571:
553:
551:
550:
545:
543:
538:
525:
517:
515:
514:
509:
507:
502:
493:
490:
474:
471:, in which case
470:
468:
467:
464:
461:
454:
450:
446:
422:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
352:
350:
349:
346:
343:
309:algebraic number
306:
304:
303:
298:
296:
295:
291:
271:
269:
268:
263:
261:
256:
238:square root of 2
230:
228:
227:
222:
220:
218:
216:
215:
213:
211:
210:
208:
206:
205:
203:
201:
185:
183:
173:
159:
157:
147:
133:
131:
121:
107:
105:
95:
74:
73:
70:
67:
36:
29:
27:Square root of 2
25:
12156:
12155:
12151:
12150:
12149:
12147:
12146:
12145:
12121:
12120:
12119:
12114:
12091:
12082:
12076:
12066:
12045:
12037:
12022:
12007:
11997:
11987:
11970:
11952:
11937:
11905:
11890:
11878:
11873:
11843:
11838:
11827:
11820:
11808:
11799:
11767:
11764:
11760:
11744:Rational number
11731:
11730:Plastic ratio (
11712:
11676:Chebyshev nodes
11662:
11657:
11623:
11619:
11614:
11612:
11605:
11603:
11571:
11569:
11566:
11565:
11518:
11516:
11513:
11512:
11507:
11497:
11468:10.2307/2344040
11437:
11435:
11432:
11431:
11402:Robson, Eleanor
11390:
11346:10.2307/2695741
11328:Apostol, Tom M.
11323:
11318:
11311:
11297:
11293:
11286:
11272:
11265:
11256:
11254:
11237:
11233:
11224:
11222:
11218:
11211:
11205:
11201:
11193:
11189:
11184:Wayback Machine
11174:
11170:
11159:
11155:
11149:
11130:
11126:
11087:
11083:
11069:10.2307/2695741
11053:
11049:
11034:
11030:
11025:Wayback Machine
11016:
11012:
11006:J. L. Heiberg's
10995:
10991:
10982:
10978:
10969:
10967:
10958:
10957:
10950:
10941:
10939:
10930:
10929:
10925:
10920:
10902:
10900:
10897:
10896:
10894:
10890:
10883:
10866:
10862:
10851:Guy, Richard K.
10847:Conway, John H.
10844:
10840:
10817:10.2307/1969021
10801:
10797:
10788:
10786:
10784:nrich.maths.org
10778:
10777:
10773:
10768:
10764:
10753:
10751:Wayback Machine
10738:
10736:
10732:
10723:
10721:
10713:
10712:
10708:
10690:
10686:
10682:
10671:
10669:
10667:
10658:
10647:
10645:
10644:
10633:
10631:
10630:
10618:
10599:natural numbers
10560:
10558:
10555:
10554:
10521:frequency ratio
10509:
10479:
10464:
10447:
10440:
10435:
10419:
10417:
10403:
10399:
10394:
10383:
10381:
10378:
10377:
10373:
10352:
10344:
10342:
10331:
10329:
10326:
10325:
10303:
10290:
10282:
10279:
10278:
10274:
10255:
10252:
10251:
10232:
10229:
10228:
10226:
10205:
10203:
10200:
10199:
10164:
10162:
10159:
10158:
10143:
10138:
10119:
10118:
10109:
10098:
10087:
10081:
10080:
10079:
10061:
10055:
10054:
10039:
10038:
10034:
10033:
10018:
10017:
10013:
10012:
9994:
9985:
9984:
9978:
9967:
9956:
9950:
9949:
9948:
9930:
9924:
9923:
9908:
9907:
9903:
9902:
9887:
9886:
9882:
9881:
9863:
9856:
9849:
9845:
9843:
9840:
9839:
9817:
9815:
9812:
9811:
9810:
9807:
9782:
9778:
9771:
9767:
9762:
9741:
9731:
9730:
9726:
9719:
9715:
9714:
9709:
9688:
9678:
9677:
9673:
9666:
9662:
9661:
9655:
9649:
9645:
9636:
9632:
9624:
9623:
9621:
9603:
9593:
9592:
9588:
9586:
9583:
9582:
9566:
9564:
9562:
9559:
9556:
9555:
9553:
9552:
9534:
9532:
9529:
9528:
9518:
9515:
9510:
9509:
9507:
9506:
9492:
9489:
9480:
9479:
9477:
9476:
9466:
9463:
9458:
9457:
9455:
9454:
9446:
9443:
9440:
9439:
9437:
9430:
9427:
9424:
9423:
9421:
9414:
9411:
9408:
9407:
9405:
9398:
9395:
9392:
9391:
9389:
9382:
9379:
9376:
9375:
9373:
9366:
9363:
9360:
9359:
9357:
9350:
9347:
9344:
9343:
9341:
9334:
9331:
9328:
9327:
9325:
9324:
9313:
9299:
9296:
9291:
9290:
9288:
9287:
9242:
9237:
9230:
9225:
9223:
9216:
9211:
9204:
9199:
9197:
9190:
9185:
9178:
9173:
9171:
9155:
9153:
9150:
9149:
9128:
9091:
9078:
9065:
9064:
9060:
9050:
9037:
9022:
9018:
9011:
9006:
9000:
8989:
8975:
8962:
8952:
8950:
8947:
8946:
8890:
8888:
8879:
8877:
8876:
8870:
8868:
8866:
8844:
8842:
8839:
8838:
8805:
8792:
8779:
8766:
8753:
8740:
8728:
8724:
8700:
8696:
8695:
8672:
8670:
8664:
8653:
8639:
8637:
8634:
8633:
8627:Euler transform
8589:
8576:
8563:
8550:
8537:
8497:
8480:
8478:
8456:
8445:
8443:
8427:
8422:
8409:
8378:
8352:
8350:
8338:
8334:
8319:
8308:
8294:
8292:
8289:
8288:
8278:
8268:
8259:
8257:
8256:
8220:
8210:
8196:
8192:
8191:
8185:
8181:
8171:
8169:
8163:
8152:
8133:
8131:
8128:
8127:
8116:
8113:
8110:
8109:
8107:
8105:
8062:
8055:
8051:
8035:
8028:
8024:
8008:
8001:
7997:
7981:
7974:
7970:
7946:
7941:
7934:
7930:
7909:
7904:
7897:
7893:
7887:
7876:
7862:
7860:
7857:
7856:
7819:
7808:
7806:
7802:
7784:
7773:
7771:
7767:
7749:
7738:
7736:
7732:
7714:
7703:
7701:
7697:
7654:
7647:
7643:
7627:
7625:
7619:
7608:
7594:
7592:
7589:
7588:
7553:
7546:
7542:
7526:
7519:
7515:
7499:
7492:
7488:
7471:
7467:
7451:
7446:
7439:
7435:
7429:
7418:
7399:
7397:
7394:
7393:
7387:sine and cosine
7376:
7374:
7373:
7370:
7367:
7366:
7364:
7357:
7354:
7351:
7350:
7348:
7341:
7338:
7335:
7334:
7332:
7330:
7327:
7322:
7320:Representations
7287:
7285:
7282:
7281:
7264:
7263:
7250:
7242:
7234:
7226:
7214:
7207:
7194:
7192:
7183:
7172:
7164:
7156:
7144:
7137:
7124:
7122:
7113:
7100:
7092:
7084:
7076:
7064:
7057:
7044:
7042:
7033:
7032:
7021:
7013:
7005:
6993:
6986:
6973:
6971:
6962:
6949:
6941:
6933:
6925:
6913:
6906:
6893:
6891:
6882:
6873:
6865:
6853:
6846:
6836:
6827:
6826:
6813:
6805:
6797:
6789:
6777:
6770:
6757:
6755:
6746:
6739:
6727:
6720:
6710:
6701:
6688:
6680:
6672:
6664:
6652:
6645:
6632:
6630:
6621:
6620:
6611:
6603:
6591:
6584:
6571:
6569:
6560:
6547:
6539:
6531:
6523:
6511:
6504:
6491:
6489:
6480:
6469:
6461:
6453:
6441:
6434:
6424:
6415:
6414:
6401:
6393:
6385:
6377:
6365:
6358:
6345:
6343:
6334:
6323:
6315:
6307:
6295:
6288:
6275:
6273:
6264:
6251:
6243:
6235:
6227:
6215:
6208:
6198:
6188:
6186:
6183:
6182:
6157:
6155:
6152:
6151:
6126:
6122:
6103:
6089:
6081:
6073:
6064:
6057:
6053:
6041:
6029:
6026:
6025:
5986:
5976:
5963:
5961:
5951:
5938:
5936:
5919:
5909:
5896:
5894:
5879:
5866:
5864:
5861:
5860:
5853:
5850:Viète's formula
5831:
5829:
5826:
5825:
5789:
5785:
5781:
5777:
5773:
5766:
5765:
5764:
5757:
5756:
5755:
5748:
5747:
5745:
5742:
5741:
5724:
5717:
5699:
5697:
5694:
5693:
5682:
5675:
5672:
5664:
5653:
5646:
5637:
5630:
5624:
5617:
5595:
5593:
5590:
5589:
5566:
5555:
5539:
5538:
5536:
5533: and
5531:
5518:
5505:
5504:
5502:
5500:
5497:
5496:
5486:
5480:
5478:complex numbers
5468:using only the
5463:
5442:
5440:
5437:
5436:
5380:
5379:
5375:
5356:
5355:
5351:
5349:
5346:
5345:
5315:
5296:
5295:
5290:
5284:
5281:
5280:
5259:
5257:
5254:
5253:
5243:
5237:
5235:
5234:
5220:
5195:
5191:
5176:
5172:
5151:
5134:
5124:
5112:
5110:
5107:
5106:
5068:
5053:
5052:
5048:
5046:
5043:
5042:
5005:
5003:
5000:
4999:
4981:
4979:
4976:
4975:
4961:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4909:
4890:
4886:
4871:
4867:
4846:
4831:
4829:
4826:
4825:
4811:
4808:
4805:
4804:
4802:
4795:
4784:
4773:
4767:
4761:
4741:
4737:
4728:
4724:
4715:
4711:
4709:
4706:
4705:
4687:
4683:
4674:
4670:
4665:
4662:
4661:
4641:
4637:
4631:
4627:
4625:
4617:
4614:
4613:
4572:
4569:
4568:
4552:
4538:
4535:
4534:
4511:
4501:
4499:
4496:
4495:
4474:
4472:
4469:
4468:
4455:
4441:
4435:
4429:
4423:
4412:
4389:
4387:
4384:
4383:
4363:
4360:
4355:
4354:
4352:
4347:
4345:
4344:
4342:
4331:
4328:
4325:
4324:
4322:
4321:
4314:
4312:
4303:
4300:
4295:
4294:
4292:
4291:
4280:
4277:
4272:
4271:
4269:
4267:
4237:
4233:
4229:
4224:
4203:
4193:
4192:
4188:
4181:
4177:
4176:
4171:
4150:
4140:
4139:
4135:
4128:
4124:
4123:
4117:
4111:
4107:
4098:
4094:
4086:
4085:
4083:
4065:
4055:
4054:
4050:
4048:
4045:
4044:
4030:
4027:
4022:
4021:
4019:
4014:
4012:
4007:
3995:
3992:
3987:
3986:
3984:
3979:
3977:
3976:
3974:
3961:
3959:
3953:
3946:
3942:
3931:
3928:
3923:
3922:
3920:
3918:
3912:
3906:
3902:
3896:is irrational.
3879:
3877:
3874:
3873:
3867:
3861:
3851:
3845:
3839:
3829:
3818:
3789:
3782:
3768:
3767:. By symmetry,
3754:
3740:
3733:
3726:
3719:
3702:
3696:
3686:
3679:
3669:
3659:
3653:
3647:
3641:
3635:
3614:
3608:
3602:
3584:
3571:
3569:
3566:
3565:
3555:
3549:
3542:
3516:
3514:
3511:
3510:
3492:
3490:
3489:
3465:
3461:
3444:
3441:
3440:
3423:
3419:
3402:
3399:
3398:
3374:
3372:
3361:
3359:Geometric proof
3323:
3321:
3318:
3317:
3284:
3281:
3280:
3262:
3260:
3257:
3256:
3237:
3234:
3233:
3214:
3211:
3210:
3192:
3190:
3187:
3186:
3167:
3164:
3163:
3144:
3141:
3140:
3117:
3113:
3096:
3093:
3092:
3054:
3052:
3049:
3048:
3045:
3015:
3011:
2999:
2995:
2993:
2990:
2989:
2986:
2959:
2956:
2955:
2931:
2929:
2926:
2925:
2901:
2899:
2896:
2895:
2874:
2872:
2869:
2868:
2844:
2842:
2839:
2838:
2815:
2811:
2786:
2767:
2762:
2759:
2758:
2736:
2733:
2732:
2711:
2695:
2693:
2690:
2689:
2671:
2669:
2666:
2665:
2662:
2629:is irrational.
2612:
2610:
2607:
2606:
2588:
2586:
2583:
2582:
2564:
2562:
2559:
2558:
2533:
2531:
2528:
2527:
2521:
2515:
2506:
2500:
2483:
2479:
2470:
2466:
2461:
2458:
2457:
2450:
2430:
2426:
2414:
2410:
2408:
2405:
2404:
2387:
2383:
2371:
2367:
2349:
2345:
2336:
2332:
2327:
2324:
2323:
2317:
2310:
2282:
2279:
2278:
2272:
2266:
2257:
2247:
2240:
2230:
2217:
2206:
2203:
2198:
2197:
2195:
2186:
2183:
2178:
2177:
2175:
2172:
2155:
2151:
2139:
2135:
2133:
2130:
2129:
2104:
2100:
2094:
2090:
2088:
2086:
2083:
2082:
2069:
2063:
2053:
2047:
2026:
2024:
2021:
2020:
1999:
1997:
1994:
1993:
1961:
1959:
1956:
1955:
1937:
1935:
1932:
1931:
1909:
1907:
1904:
1903:
1898:. It is also a
1892:
1887:
1875:
1872:
1869:
1866:
1864:
1850:
1847:
1844:
1841:
1839:
1825:
1822:
1819:
1816:
1814:
1800:
1797:
1794:
1791:
1789:
1775:
1772:
1769:
1766:
1764:
1750:
1747:
1744:
1741:
1739:
1733:January 5, 2022
1699:
1697:
1694:
1693:
1677:
1671:
1644:
1642:
1639:
1638:
1635:Yasumasa Kanada
1616:
1614:
1611:
1610:
1607:
1597:
1594:
1591:
1589:
1583:
1581:
1574:
1571:
1568:
1567:
1565:
1561:
1555:
1546:
1544:
1533:
1530:
1527:
1526:
1524:
1518:
1516:
1505:
1502:
1499:
1498:
1496:
1487:
1485:
1478:
1475:
1472:
1471:
1469:
1458:
1455:
1452:
1451:
1449:
1446:
1427:
1426:
1416:
1415:
1401:
1393:
1381:
1374:
1368:
1364:
1361:
1360:
1346:
1338:
1326:
1319:
1313:
1309:
1306:
1305:
1291:
1283:
1271:
1264:
1258:
1254:
1251:
1250:
1239:
1231:
1219:
1212:
1206:
1202:
1198:
1196:
1193:
1192:
1166:
1162:
1160:
1157:
1156:
1131:
1127:
1122:
1108:
1104:
1102:
1085:
1081:
1075:
1066:
1062:
1061:
1057:
1047:
1032:
1028:
1026:
1023:
1022:
992:
988:
986:
983:
982:
976:Newton's method
949:
947:
944:
943:
938:There are many
936:
930:
925:
892:
842:
840:
837:
836:
810:
808:
805:
804:
777:
761:
739:
734:
718:
713:
700:
692:
689:
688:
677:
639:
623:
612:
608:
603:
592:
588:
583:
570:
562:
559:
558:
537:
535:
532:
531:
523:
501:
499:
496:
495:
491:
472:
465:
462:
459:
458:
456:
452:
448:
444:
429:
419:
416:
413:
410:
407:
404:
401:
398:
395:
392:
389:
386:
384:
347:
344:
341:
340:
338:
337:. The fraction
287:
283:
279:
277:
274:
273:
255:
253:
250:
249:
191:
186:
179:
174:
172:
165:
160:
153:
148:
146:
139:
134:
127:
122:
120:
113:
108:
101:
96:
94:
86:
83:
82:
71:
68:
65:
63:
55:Representations
50:
24:
17:
12:
11:
5:
12154:
12144:
12143:
12138:
12133:
12116:
12115:
12113:
12112:
12107:
12105:Transcendental
12102:
12096:
12093:
12092:
12085:
12083:
12081:
12080:
12070:
12059:
12058:
12053:
12048:
12041:
12031:
12026:
12016:
12011:
12001:
11991:
11981:
11975:
11974:
11964:
11962:Cube root of 2
11959:
11946:
11941:
11931:
11926:
11924:Logarithm of 2
11920:
11919:
11914:
11909:
11899:
11894:
11883:
11880:
11879:
11872:
11871:
11864:
11857:
11849:
11840:
11839:
11825:
11822:
11821:
11819:
11818:
11813:
11804:
11795:
11790:
11785:
11780:
11775:
11770:
11763:
11759:Silver ratio (
11756:
11751:
11746:
11741:
11736:
11727:
11722:
11717:
11711:Golden ratio (
11708:
11703:
11698:
11693:
11688:
11683:
11678:
11673:
11667:
11664:
11663:
11656:
11655:
11648:
11641:
11633:
11627:
11626:
11600:
11575:
11559:
11553:
11543:
11522:
11506:
11505:External links
11503:
11502:
11501:
11495:
11480:
11462:(1): 102–107,
11441:
11424:
11412:(4): 366–378,
11394:
11388:
11375:
11372:
11358:
11340:(9): 841–842,
11322:
11319:
11317:
11316:
11309:
11291:
11285:978-0393244809
11284:
11263:
11231:
11199:
11187:
11168:
11153:
11147:
11124:
11081:
11063:(9): 841–842,
11047:
11028:
11010:
10989:
10976:
10948:
10923:
10906:
10888:
10881:
10860:
10838:
10811:(2): 242–264.
10795:
10771:
10762:
10730:
10706:
10683:
10681:
10678:
10677:
10676:
10661:
10652:
10638:
10624:
10617:
10614:
10613:
10612:
10579:
10578:
10564:
10547:
10532:
10508:
10505:
10504:
10503:
10491:
10488:
10483:
10478:
10472:
10468:
10463:
10457:
10454:
10450:
10446:
10443:
10439:
10434:
10429:
10425:
10422:
10416:
10410:
10406:
10402:
10398:
10393:
10389:
10386:
10358:
10355:
10350:
10347:
10341:
10337:
10334:
10322:
10321:
10307:
10302:
10297:
10294:
10289:
10286:
10262:
10259:
10239:
10236:
10209:
10168:
10142:
10139:
10137:
10134:
10133:
10132:
10117:
10112:
10107:
10101:
10096:
10090:
10084:
10078:
10075:
10069:
10066:
10058:
10053:
10047:
10044:
10037:
10032:
10026:
10023:
10016:
10011:
10008:
10002:
9999:
9993:
9990:
9988:
9986:
9981:
9976:
9970:
9965:
9959:
9953:
9947:
9944:
9938:
9935:
9927:
9922:
9916:
9913:
9906:
9901:
9895:
9892:
9885:
9880:
9877:
9871:
9868:
9862:
9859:
9857:
9853:
9848:
9847:
9821:
9806:
9803:
9802:
9801:
9785:
9781:
9775:
9770:
9766:
9761:
9754:
9748:
9745:
9740:
9735:
9729:
9722:
9718:
9713:
9708:
9701:
9695:
9692:
9687:
9682:
9676:
9669:
9665:
9658:
9652:
9648:
9644:
9639:
9635:
9631:
9627:
9620:
9616:
9610:
9607:
9602:
9597:
9591:
9538:
9281:
9280:
9269:
9251:
9248:
9245:
9233:
9222:
9219:
9207:
9196:
9193:
9181:
9170:
9167:
9164:
9159:
9127:
9124:
9123:
9122:
9110:
9106:
9103:
9098:
9095:
9090:
9085:
9082:
9077:
9072:
9069:
9063:
9057:
9054:
9049:
9044:
9041:
9036:
9030:
9025:
9021:
9017:
9014:
9010:
9003:
8998:
8995:
8992:
8988:
8982:
8979:
8974:
8969:
8966:
8961:
8956:
8848:
8835:
8834:
8823:
8820:
8817:
8812:
8809:
8804:
8799:
8796:
8791:
8786:
8783:
8778:
8773:
8770:
8765:
8760:
8757:
8752:
8747:
8744:
8739:
8731:
8727:
8723:
8720:
8717:
8712:
8709:
8706:
8703:
8699:
8693:
8690:
8687:
8684:
8681:
8678:
8675:
8667:
8662:
8659:
8656:
8652:
8648:
8643:
8619:
8618:
8607:
8604:
8601:
8596:
8593:
8588:
8583:
8580:
8575:
8570:
8567:
8562:
8557:
8554:
8549:
8544:
8541:
8536:
8533:
8530:
8527:
8524:
8518:
8515:
8512:
8509:
8506:
8503:
8500:
8495:
8492:
8489:
8486:
8483:
8477:
8471:
8468:
8465:
8462:
8459:
8454:
8451:
8448:
8442:
8436:
8433:
8430:
8426:
8421:
8416:
8413:
8408:
8405:
8402:
8396:
8393:
8390:
8387:
8384:
8381:
8376:
8373:
8370:
8367:
8364:
8361:
8358:
8355:
8347:
8344:
8341:
8337:
8333:
8330:
8327:
8322:
8317:
8314:
8311:
8307:
8303:
8298:
8274:and using the
8253:
8252:
8241:
8235:
8232:
8229:
8226:
8223:
8216:
8213:
8208:
8203:
8200:
8195:
8188:
8184:
8180:
8177:
8174:
8166:
8161:
8158:
8155:
8151:
8147:
8141:
8137:
8094:
8093:
8082:
8079:
8075:
8069:
8066:
8061:
8058:
8054:
8048:
8042:
8039:
8034:
8031:
8027:
8021:
8015:
8012:
8007:
8004:
8000:
7994:
7988:
7985:
7980:
7977:
7973:
7969:
7965:
7958:
7955:
7952:
7949:
7945:
7940:
7937:
7933:
7928:
7921:
7918:
7915:
7912:
7908:
7903:
7900:
7896:
7890:
7885:
7882:
7879:
7875:
7871:
7866:
7850:
7849:
7838:
7834:
7828:
7825:
7822:
7817:
7814:
7811:
7805:
7799:
7793:
7790:
7787:
7782:
7779:
7776:
7770:
7764:
7758:
7755:
7752:
7747:
7744:
7741:
7735:
7729:
7723:
7720:
7717:
7712:
7709:
7706:
7700:
7696:
7690:
7687:
7684:
7681:
7678:
7675:
7672:
7669:
7666:
7663:
7660:
7657:
7650:
7646:
7642:
7639:
7636:
7633:
7630:
7622:
7617:
7614:
7611:
7607:
7603:
7598:
7582:
7581:
7570:
7566:
7560:
7557:
7552:
7549:
7545:
7539:
7533:
7530:
7525:
7522:
7518:
7512:
7506:
7503:
7498:
7495:
7491:
7487:
7483:
7474:
7470:
7466:
7463:
7460:
7457:
7454:
7450:
7445:
7442:
7438:
7432:
7427:
7424:
7421:
7417:
7413:
7407:
7403:
7326:
7323:
7321:
7318:
7291:
7278:
7277:
7254:
7249:
7246:
7241:
7238:
7233:
7230:
7222:
7219:
7213:
7210:
7208:
7204:
7200:
7197:
7191:
7188:
7184:
7176:
7171:
7168:
7163:
7160:
7152:
7149:
7143:
7140:
7138:
7134:
7130:
7127:
7121:
7118:
7114:
7104:
7099:
7096:
7091:
7088:
7083:
7080:
7072:
7069:
7063:
7060:
7058:
7054:
7050:
7047:
7041:
7038:
7035:
7034:
7025:
7020:
7017:
7012:
7009:
7001:
6998:
6992:
6989:
6987:
6983:
6979:
6976:
6970:
6967:
6963:
6953:
6948:
6945:
6940:
6937:
6932:
6929:
6921:
6918:
6912:
6909:
6907:
6903:
6899:
6896:
6890:
6887:
6883:
6877:
6872:
6869:
6861:
6858:
6852:
6849:
6847:
6843:
6840:
6835:
6832:
6829:
6828:
6817:
6812:
6809:
6804:
6801:
6796:
6793:
6785:
6782:
6776:
6773:
6771:
6767:
6763:
6760:
6754:
6751:
6747:
6743:
6735:
6732:
6726:
6723:
6721:
6717:
6714:
6709:
6706:
6702:
6692:
6687:
6684:
6679:
6676:
6671:
6668:
6660:
6657:
6651:
6648:
6646:
6642:
6638:
6635:
6629:
6626:
6623:
6622:
6615:
6610:
6607:
6599:
6596:
6590:
6587:
6585:
6581:
6577:
6574:
6568:
6565:
6561:
6551:
6546:
6543:
6538:
6535:
6530:
6527:
6519:
6516:
6510:
6507:
6505:
6501:
6497:
6494:
6488:
6485:
6481:
6473:
6468:
6465:
6460:
6457:
6449:
6446:
6440:
6437:
6435:
6431:
6428:
6423:
6420:
6417:
6416:
6405:
6400:
6397:
6392:
6389:
6384:
6381:
6373:
6370:
6364:
6361:
6359:
6355:
6351:
6348:
6342:
6339:
6335:
6327:
6322:
6319:
6314:
6311:
6303:
6300:
6294:
6291:
6289:
6285:
6281:
6278:
6272:
6269:
6265:
6255:
6250:
6247:
6242:
6239:
6234:
6231:
6223:
6220:
6214:
6211:
6209:
6205:
6202:
6197:
6194:
6191:
6190:
6161:
6148:
6147:
6136:
6125:
6119:
6107:
6102:
6099:
6096:
6093:
6088:
6085:
6080:
6077:
6072:
6069:
6060:
6056:
6050:
6047:
6044:
6040:
6036:
6033:
6019:
6018:
6007:
6004:
5994:
5991:
5983:
5980:
5975:
5970:
5967:
5958:
5955:
5950:
5945:
5942:
5935:
5927:
5924:
5916:
5913:
5908:
5903:
5900:
5893:
5887:
5884:
5878:
5873:
5870:
5835:
5823:
5822:
5811:
5808:
5795:
5788:
5780:
5770:
5761:
5752:
5703:
5668:
5641:
5628:
5599:
5587:
5586:
5572:
5569:
5562:
5559:
5554:
5551:
5546:
5543:
5528:
5522:
5517:
5514:
5509:
5461:imaginary unit
5446:
5427:
5426:
5415:
5412:
5409:
5406:
5403:
5400:
5396:
5392:
5389:
5384:
5378:
5372:
5368:
5365:
5360:
5354:
5339:
5338:
5327:
5324:
5319:
5314:
5308:
5305:
5300:
5294:
5263:
5219:
5216:
5215:
5214:
5203:
5198:
5194:
5190:
5187:
5184:
5179:
5175:
5171:
5168:
5165:
5159:
5155:
5150:
5143:
5140:
5133:
5128:
5120:
5117:
5100:
5099:
5088:
5083:
5077:
5073:
5067:
5062:
5058:
5051:
5009:
4985:
4972:
4971:
4898:
4893:
4889:
4885:
4882:
4879:
4874:
4870:
4866:
4863:
4860:
4855:
4851:
4845:
4839:
4835:
4794:
4791:
4789:must be even.
4758:
4757:
4744:
4740:
4736:
4731:
4727:
4723:
4718:
4714:
4703:
4690:
4686:
4682:
4677:
4673:
4669:
4659:
4644:
4640:
4634:
4630:
4624:
4621:
4611:
4608:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4576:
4555:
4551:
4548:
4545:
4542:
4531:
4518:
4515:
4510:
4505:
4478:
4462:
4461:
4460:is never even.
4411:
4408:
4393:
4260:
4259:
4248:
4240:
4236:
4232:
4228:
4223:
4216:
4210:
4207:
4202:
4197:
4191:
4184:
4180:
4175:
4170:
4163:
4157:
4154:
4149:
4144:
4138:
4131:
4127:
4120:
4114:
4110:
4106:
4101:
4097:
4093:
4089:
4082:
4078:
4072:
4069:
4064:
4059:
4053:
3943:1<2< 9/4
3901:
3898:
3883:
3634:Draw the arcs
3588:
3583:
3578:
3575:
3520:
3501:Tom M. Apostol
3468:
3464:
3460:
3457:
3454:
3451:
3448:
3426:
3422:
3418:
3415:
3412:
3409:
3406:
3360:
3357:
3345:natural number
3327:
3297:
3294:
3291:
3288:
3266:
3244:
3241:
3221:
3218:
3196:
3174:
3171:
3151:
3148:
3128:
3125:
3120:
3116:
3112:
3109:
3106:
3103:
3100:
3058:
3044:
3041:
3018:
3014:
3010:
3007:
3002:
2998:
2985:
2982:
2969:
2966:
2963:
2943:
2940:
2935:
2913:
2910:
2905:
2881:
2878:
2856:
2853:
2848:
2826:
2823:
2818:
2814:
2810:
2807:
2804:
2801:
2798:
2795:
2790:
2785:
2782:
2779:
2776:
2771:
2766:
2746:
2743:
2740:
2718:
2715:
2710:
2707:
2704:
2699:
2675:
2661:
2658:
2616:
2592:
2568:
2555:
2554:
2540:
2537:
2512:
2486:
2482:
2478:
2473:
2469:
2465:
2447:
2433:
2429:
2425:
2422:
2417:
2413:
2390:
2386:
2382:
2379:
2374:
2370:
2366:
2363:
2360:
2357:
2352:
2348:
2344:
2339:
2335:
2331:
2307:
2295:
2292:
2289:
2286:
2277:that fulfills
2263:
2254:
2227:
2158:
2154:
2150:
2147:
2142:
2138:
2117:
2114:
2107:
2103:
2097:
2093:
2079:
2033:
2030:
2003:
1990:
1979:
1965:
1941:
1913:
1891:
1888:
1886:
1883:
1880:
1879:
1862:
1859:
1858:March 22, 2010
1855:
1854:
1837:
1834:
1830:
1829:
1812:
1809:
1805:
1804:
1787:
1784:
1780:
1779:
1762:
1759:
1755:
1754:
1737:
1734:
1730:
1729:
1726:
1723:
1703:
1648:
1620:
1606:
1603:
1559:
1445:
1442:
1441:
1440:
1425:
1421:
1419:
1417:
1414:
1411:
1408:
1404:
1400:
1397:
1395:
1389:
1386:
1380:
1377:
1375:
1371:
1367:
1363:
1362:
1359:
1356:
1353:
1349:
1345:
1342:
1340:
1334:
1331:
1325:
1322:
1320:
1316:
1312:
1308:
1307:
1304:
1301:
1298:
1294:
1290:
1287:
1285:
1279:
1276:
1270:
1267:
1265:
1261:
1257:
1253:
1252:
1249:
1246:
1242:
1238:
1235:
1233:
1227:
1224:
1218:
1215:
1213:
1209:
1205:
1201:
1200:
1177:
1174:
1169:
1165:
1153:
1152:
1141:
1134:
1130:
1126:
1121:
1116:
1111:
1107:
1101:
1097:
1088:
1084:
1080:
1074:
1069:
1065:
1060:
1054:
1051:
1046:
1041:
1038:
1035:
1031:
1003:
1000:
995:
991:
966:as a ratio of
953:
929:
926:
924:
921:
891:
888:
846:
814:
801:
800:
789:
784:
781:
780:56862745098039
776:
773:
768:
765:
760:
754:
751:
748:
745:
742:
738:
733:
727:
724:
721:
717:
712:
707:
704:
699:
696:
667:ancient Indian
663:
662:
651:
646:
643:
638:
635:
630:
627:
622:
615:
611:
607:
602:
595:
591:
587:
582:
577:
574:
569:
566:
541:
505:
428:
425:
424:
423:
294:
290:
286:
282:
259:
232:
231:
200:
197:
194:
182:
171:
168:
156:
145:
142:
130:
119:
116:
104:
93:
90:
80:
76:
75:
61:
57:
56:
52:
51:
47:right triangle
37:
15:
9:
6:
4:
3:
2:
12153:
12142:
12139:
12137:
12134:
12132:
12129:
12128:
12126:
12111:
12110:Trigonometric
12108:
12106:
12103:
12101:
12100:Schizophrenic
12098:
12097:
12094:
12089:
12074:
12071:
12064:
12061:
12060:
12057:
12054:
12052:
12049:
12044:
12040:
12035:
12032:
12030:
12027:
12020:
12017:
12015:
12012:
12005:
12002:
11995:
11994:Erdős–Borwein
11992:
11985:
11982:
11980:
11977:
11976:
11968:
11967:Plastic ratio
11965:
11963:
11960:
11955:
11950:
11947:
11945:
11942:
11935:
11932:
11930:
11927:
11925:
11922:
11921:
11918:
11915:
11913:
11910:
11903:
11900:
11898:
11895:
11888:
11885:
11884:
11881:
11877:
11870:
11865:
11863:
11858:
11856:
11851:
11850:
11847:
11837:
11836:
11831:
11823:
11817:
11814:
11812:
11805:
11803:
11796:
11794:
11791:
11789:
11786:
11784:
11781:
11779:
11776:
11774:
11771:
11769:
11757:
11755:
11752:
11750:
11749:Root of unity
11747:
11745:
11742:
11740:
11737:
11735:
11728:
11726:
11723:
11721:
11720:Perron number
11718:
11716:
11709:
11707:
11704:
11702:
11699:
11697:
11694:
11692:
11689:
11687:
11684:
11682:
11679:
11677:
11674:
11672:
11669:
11668:
11665:
11661:
11654:
11649:
11647:
11642:
11640:
11635:
11634:
11631:
11610:
11609:Search Engine
11601:
11597:
11593:
11589:
11573:
11560:
11557:
11554:
11551:
11547:
11544:
11540:
11539:
11520:
11509:
11508:
11498:
11492:
11488:
11487:
11481:
11477:
11473:
11469:
11465:
11461:
11457:
11439:
11429:
11425:
11420:
11415:
11411:
11407:
11403:
11399:
11398:Fowler, David
11395:
11391:
11389:0-387-20220-X
11385:
11381:
11376:
11373:
11369:
11368:
11363:
11359:
11355:
11351:
11347:
11343:
11339:
11335:
11334:
11329:
11325:
11324:
11312:
11306:
11302:
11295:
11287:
11281:
11277:
11270:
11268:
11252:
11251:
11245:
11241:
11235:
11217:
11210:
11203:
11196:
11191:
11185:
11181:
11178:
11172:
11164:
11157:
11150:
11144:
11140:
11139:
11134:
11128:
11121:
11117:
11112:
11107:
11103:
11099:
11098:
11093:
11085:
11078:
11074:
11070:
11066:
11062:
11058:
11051:
11043:
11039:
11032:
11026:
11022:
11019:
11014:
11007:
11003:
10999:
10993:
10986:
10980:
10965:
10961:
10955:
10953:
10937:
10933:
10927:
10904:
10892:
10884:
10882:9783319001371
10878:
10874:
10870:
10869:Williams, Kim
10864:
10856:
10852:
10848:
10842:
10834:
10830:
10826:
10822:
10818:
10814:
10810:
10806:
10799:
10785:
10781:
10775:
10766:
10760:
10758:
10752:
10748:
10745:
10743:
10734:
10720:
10716:
10710:
10703:
10699:
10696:(10): 45–61,
10695:
10688:
10684:
10665:
10662:
10656:
10653:
10642:
10639:
10628:
10625:
10623:
10620:
10619:
10609:
10608:
10604:
10600:
10596:
10592:
10588:
10583:
10562:
10552:
10548:
10545:
10541:
10537:
10533:
10530:
10526:
10522:
10518:
10517:
10516:
10514:
10489:
10486:
10481:
10476:
10470:
10466:
10461:
10452:
10448:
10444:
10437:
10432:
10427:
10423:
10420:
10414:
10408:
10404:
10400:
10396:
10391:
10387:
10384:
10376:
10375:
10374:
10356:
10353:
10348:
10345:
10339:
10335:
10332:
10305:
10300:
10295:
10292:
10287:
10284:
10277:
10276:
10275:
10260:
10257:
10237:
10234:
10223:
10207:
10197:
10193:
10189:
10185:
10166:
10156:
10147:
10115:
10110:
10105:
10099:
10094:
10088:
10076:
10073:
10067:
10064:
10051:
10045:
10042:
10035:
10030:
10024:
10021:
10014:
10009:
10006:
10000:
9997:
9991:
9989:
9979:
9974:
9968:
9963:
9957:
9945:
9942:
9936:
9933:
9920:
9914:
9911:
9904:
9899:
9893:
9890:
9883:
9878:
9875:
9869:
9866:
9860:
9858:
9851:
9838:
9837:
9836:
9819:
9805:Nested square
9783:
9779:
9773:
9768:
9764:
9752:
9746:
9743:
9738:
9733:
9727:
9720:
9716:
9711:
9706:
9699:
9693:
9690:
9685:
9680:
9674:
9667:
9663:
9650:
9646:
9642:
9637:
9633:
9629:
9618:
9614:
9608:
9605:
9600:
9595:
9589:
9581:
9580:
9579:
9572:
9536:
9527:differs from
9521:
9513:
9499:
9495:
9487:
9483:
9469:
9461:
9320:
9316:
9311:
9302:
9294:
9286:
9267:
9249:
9246:
9243:
9231:
9220:
9217:
9205:
9194:
9191:
9179:
9168:
9165:
9162:
9157:
9148:
9147:
9146:
9144:
9137:
9132:
9108:
9104:
9101:
9096:
9093:
9088:
9083:
9080:
9075:
9070:
9067:
9061:
9055:
9052:
9047:
9042:
9039:
9034:
9023:
9019:
9012:
9008:
8996:
8993:
8990:
8986:
8980:
8977:
8972:
8967:
8964:
8959:
8954:
8945:
8944:
8943:
8941:
8937:
8933:
8929:
8925:
8921:
8917:
8913:
8910:
8906:
8902:
8897:
8886:
8864:
8846:
8821:
8818:
8815:
8810:
8807:
8802:
8797:
8794:
8789:
8784:
8781:
8776:
8771:
8768:
8763:
8758:
8755:
8750:
8745:
8742:
8737:
8729:
8721:
8718:
8710:
8707:
8704:
8701:
8697:
8691:
8685:
8682:
8679:
8676:
8660:
8657:
8654:
8650:
8646:
8641:
8632:
8631:
8630:
8628:
8624:
8605:
8602:
8599:
8594:
8591:
8586:
8581:
8578:
8573:
8568:
8565:
8560:
8555:
8552:
8547:
8542:
8539:
8534:
8531:
8528:
8525:
8522:
8516:
8513:
8510:
8507:
8504:
8501:
8498:
8493:
8490:
8487:
8484:
8481:
8475:
8469:
8466:
8463:
8460:
8457:
8452:
8449:
8446:
8440:
8434:
8431:
8428:
8424:
8419:
8414:
8411:
8406:
8403:
8400:
8394:
8391:
8385:
8382:
8374:
8371:
8365:
8362:
8359:
8356:
8345:
8342:
8339:
8331:
8328:
8315:
8312:
8309:
8305:
8301:
8296:
8287:
8286:
8285:
8281:
8277:
8271:
8263:
8239:
8233:
8227:
8224:
8214:
8211:
8206:
8201:
8198:
8193:
8186:
8178:
8175:
8159:
8156:
8153:
8149:
8145:
8139:
8135:
8126:
8125:
8124:
8103:
8099:
8098:Taylor series
8080:
8077:
8073:
8067:
8064:
8059:
8056:
8052:
8046:
8040:
8037:
8032:
8029:
8025:
8019:
8013:
8010:
8005:
8002:
7998:
7992:
7986:
7983:
7978:
7975:
7971:
7967:
7963:
7956:
7953:
7950:
7947:
7943:
7938:
7935:
7931:
7926:
7919:
7916:
7913:
7910:
7906:
7901:
7898:
7894:
7883:
7880:
7877:
7873:
7869:
7864:
7855:
7854:
7853:
7836:
7832:
7826:
7823:
7820:
7815:
7812:
7809:
7803:
7797:
7791:
7788:
7785:
7780:
7777:
7774:
7768:
7762:
7756:
7753:
7750:
7745:
7742:
7739:
7733:
7727:
7721:
7718:
7715:
7710:
7707:
7704:
7698:
7694:
7685:
7682:
7679:
7676:
7667:
7664:
7661:
7658:
7648:
7640:
7637:
7634:
7631:
7615:
7612:
7609:
7605:
7601:
7596:
7587:
7586:
7585:
7568:
7564:
7558:
7555:
7550:
7547:
7543:
7537:
7531:
7528:
7523:
7520:
7516:
7510:
7504:
7501:
7496:
7493:
7489:
7485:
7481:
7472:
7464:
7461:
7458:
7455:
7448:
7443:
7440:
7436:
7425:
7422:
7419:
7415:
7411:
7405:
7401:
7392:
7391:
7390:
7388:
7329:The identity
7317:
7315:
7311:
7307:
7306:normal number
7289:
7252:
7247:
7244:
7239:
7236:
7231:
7228:
7220:
7217:
7211:
7209:
7202:
7198:
7195:
7189:
7186:
7174:
7169:
7166:
7161:
7158:
7150:
7147:
7141:
7139:
7132:
7128:
7125:
7119:
7116:
7102:
7097:
7094:
7089:
7086:
7081:
7078:
7070:
7067:
7061:
7059:
7052:
7048:
7045:
7039:
7036:
7023:
7018:
7015:
7010:
7007:
6999:
6996:
6990:
6988:
6981:
6977:
6974:
6968:
6965:
6951:
6946:
6943:
6938:
6935:
6930:
6927:
6919:
6916:
6910:
6908:
6901:
6897:
6894:
6888:
6885:
6875:
6870:
6867:
6859:
6856:
6850:
6848:
6841:
6838:
6833:
6830:
6815:
6810:
6807:
6802:
6799:
6794:
6791:
6783:
6780:
6774:
6772:
6765:
6761:
6758:
6752:
6749:
6741:
6733:
6730:
6724:
6722:
6715:
6712:
6707:
6704:
6690:
6685:
6682:
6677:
6674:
6669:
6666:
6658:
6655:
6649:
6647:
6640:
6636:
6633:
6627:
6624:
6613:
6608:
6605:
6597:
6594:
6588:
6586:
6579:
6575:
6572:
6566:
6563:
6549:
6544:
6541:
6536:
6533:
6528:
6525:
6517:
6514:
6508:
6506:
6499:
6495:
6492:
6486:
6483:
6471:
6466:
6463:
6458:
6455:
6447:
6444:
6438:
6436:
6429:
6426:
6421:
6418:
6403:
6398:
6395:
6390:
6387:
6382:
6379:
6371:
6368:
6362:
6360:
6353:
6349:
6346:
6340:
6337:
6325:
6320:
6317:
6312:
6309:
6301:
6298:
6292:
6290:
6283:
6279:
6276:
6270:
6267:
6253:
6248:
6245:
6240:
6237:
6232:
6229:
6221:
6218:
6212:
6210:
6203:
6200:
6195:
6192:
6181:
6180:
6179:
6177:
6159:
6134:
6123:
6117:
6105:
6100:
6097:
6094:
6091:
6086:
6083:
6078:
6075:
6070:
6067:
6058:
6054:
6042:
6034:
6031:
6024:
6023:
6022:
6005:
6002:
5992:
5989:
5981:
5978:
5973:
5968:
5965:
5956:
5953:
5948:
5943:
5940:
5933:
5925:
5922:
5914:
5911:
5906:
5901:
5898:
5891:
5885:
5882:
5876:
5871:
5868:
5859:
5858:
5857:
5851:
5833:
5809:
5806:
5793:
5786:
5778:
5768:
5759:
5750:
5740:
5739:
5738:
5735:
5731:
5727:
5720:
5701:
5689:
5685:
5678:
5671:
5667:
5662:
5656:
5650:
5644:
5640:
5634:
5627:
5620:
5615:
5597:
5570:
5567:
5560:
5557:
5552:
5549:
5544:
5541:
5526:
5520:
5515:
5512:
5507:
5495:
5494:
5493:
5490:
5483:
5479:
5475:
5471:
5466:
5462:
5444:
5434:
5432:
5431:silver ratios
5413:
5410:
5407:
5404:
5401:
5398:
5394:
5390:
5387:
5382:
5376:
5370:
5366:
5363:
5358:
5352:
5344:
5343:
5342:
5325:
5322:
5317:
5312:
5306:
5303:
5298:
5292:
5279:
5278:
5277:
5261:
5246:
5232:
5228:
5224:
5201:
5196:
5192:
5188:
5185:
5182:
5177:
5173:
5169:
5166:
5163:
5157:
5153:
5148:
5141:
5138:
5131:
5126:
5118:
5115:
5105:
5104:
5103:
5086:
5081:
5075:
5071:
5065:
5060:
5056:
5049:
5041:
5040:
5039:
5037:
5033:
5029:
5025:
5007:
4983:
4969:
4964:
4896:
4891:
4887:
4883:
4880:
4877:
4872:
4868:
4864:
4861:
4858:
4853:
4849:
4843:
4837:
4833:
4824:
4823:
4822:
4820:
4800:
4790:
4787:
4781:
4777:
4770:
4764:
4742:
4738:
4734:
4729:
4725:
4721:
4716:
4712:
4704:
4688:
4684:
4680:
4675:
4671:
4667:
4660:
4642:
4638:
4632:
4628:
4622:
4619:
4612:
4609:
4595:
4592:
4586:
4583:
4580:
4549:
4546:
4543:
4540:
4532:
4516:
4513:
4508:
4503:
4494:
4493:
4492:
4476:
4465:
4458:
4452:
4448:
4444:
4438:
4432:
4426:
4421:
4420:
4419:
4417:
4407:
4391:
4381:
4380:Errett Bishop
4377:
4366:
4358:
4335:
4306:
4298:
4283:
4275:
4265:
4246:
4238:
4234:
4230:
4226:
4221:
4214:
4208:
4205:
4200:
4195:
4189:
4182:
4178:
4173:
4168:
4161:
4155:
4152:
4147:
4142:
4136:
4129:
4125:
4112:
4108:
4104:
4099:
4095:
4091:
4080:
4076:
4070:
4067:
4062:
4057:
4051:
4043:
4042:
4041:
4033:
4025:
4010:
3998:
3990:
3969:
3965:
3956:
3950:
3934:
3926:
3915:
3909:
3897:
3881:
3870:
3864:
3858:
3854:
3848:
3842:
3836:
3832:
3826:
3822:
3815:
3812:
3808:
3804:
3800:
3796:
3792:
3786:
3779:
3775:
3771:
3765:
3761:
3757:
3751:
3747:
3743:
3737:
3730:
3723:
3716:
3714:
3710:
3705:
3699:
3695:
3690:
3683:
3676:
3672:
3666:
3662:
3656:
3650:
3644:
3638:
3632:
3630:
3627:given in its
3626:
3621:
3617:
3611:
3605:
3586:
3581:
3576:
3573:
3563:
3558:
3552:
3546:
3539:
3536:
3518:
3508:
3507:
3502:
3486:
3482:
3466:
3458:
3455:
3452:
3446:
3424:
3416:
3413:
3410:
3407:
3396:
3392:
3388:
3384:
3370:
3369:irrationality
3365:
3356:
3354:
3350:
3346:
3341:
3325:
3315:
3312:with integer
3311:
3292:
3286:
3264:
3242:
3239:
3219:
3216:
3194:
3172:
3169:
3149:
3146:
3126:
3123:
3118:
3114:
3110:
3104:
3098:
3090:
3086:
3082:
3078:
3074:
3056:
3040:
3038:
3034:
3016:
3012:
3008:
3005:
3000:
2996:
2981:
2967:
2964:
2961:
2941:
2938:
2933:
2911:
2908:
2903:
2879:
2876:
2854:
2851:
2846:
2824:
2821:
2816:
2812:
2808:
2805:
2802:
2796:
2793:
2788:
2777:
2774:
2769:
2744:
2741:
2738:
2716:
2713:
2708:
2705:
2702:
2697:
2673:
2657:
2655:
2654:interpolation
2651:
2650:
2645:
2641:
2640:
2635:
2630:
2614:
2590:
2566:
2538:
2535:
2524:
2518:
2513:
2509:
2503:
2484:
2480:
2476:
2471:
2467:
2463:
2454:
2448:
2431:
2427:
2423:
2420:
2415:
2411:
2388:
2384:
2380:
2377:
2372:
2364:
2361:
2355:
2350:
2346:
2342:
2337:
2333:
2329:
2320:
2314:
2309:Substituting
2308:
2293:
2290:
2287:
2284:
2275:
2269:
2264:
2260:
2255:
2251:
2244:
2238:
2233:
2228:
2226:are integers)
2224:
2220:
2214:
2209:
2201:
2189:
2181:
2156:
2152:
2148:
2145:
2140:
2136:
2115:
2112:
2105:
2101:
2095:
2091:
2080:
2077:
2072:
2066:
2061:
2056:
2050:
2031:
2028:
2019:
2001:
1991:
1988:
1984:
1980:
1963:
1939:
1929:
1928:
1927:
1911:
1901:
1897:
1863:
1861:Shigeru Kondo
1857:
1856:
1838:
1836:Alexander Yee
1832:
1831:
1813:
1807:
1806:
1788:
1783:April 3, 2016
1782:
1781:
1763:
1758:June 28, 2016
1757:
1756:
1738:
1732:
1731:
1724:
1720:
1717:
1701:
1690:
1688:
1684:
1680:
1674:
1668:
1664:
1646:
1636:
1618:
1602:
1558:
1552:
1542:
1514:
1493:
1467:
1423:
1420:
1412:
1409:
1406:
1403:1.41421356237
1398:
1396:
1387:
1384:
1378:
1376:
1369:
1365:
1357:
1354:
1351:
1343:
1341:
1332:
1329:
1323:
1321:
1314:
1310:
1302:
1299:
1296:
1288:
1286:
1277:
1274:
1268:
1266:
1259:
1255:
1247:
1244:
1236:
1234:
1225:
1222:
1216:
1214:
1207:
1203:
1191:
1190:
1189:
1175:
1172:
1167:
1163:
1139:
1132:
1128:
1124:
1119:
1114:
1109:
1105:
1099:
1095:
1086:
1082:
1078:
1072:
1067:
1063:
1058:
1052:
1049:
1044:
1039:
1036:
1033:
1029:
1021:
1020:
1019:
1018:computation:
1017:
1001:
998:
993:
989:
979:
977:
973:
969:
951:
941:
935:
923:Decimal value
920:
918:
914:
910:
905:
901:
897:
887:
885:
881:
876:
872:
868:
864:
860:
844:
835:expansion of
834:
830:
812:
787:
779:
774:
771:
766:
763:
758:
752:
749:
746:
743:
740:
736:
731:
725:
722:
719:
715:
710:
705:
702:
697:
694:
687:
686:
685:
683:
672:
668:
649:
641:
636:
633:
628:
625:
620:
613:
609:
605:
600:
593:
589:
585:
580:
575:
572:
567:
564:
557:
556:
555:
539:
529:
521:
503:
486:
482:
442:
438:
433:
383:
382:
381:
379:
375:
371:
366:
364:
360:
356:
336:
332:
328:
323:
321:
318:
314:
310:
292:
288:
284:
280:
257:
247:
243:
239:
198:
195:
192:
180:
169:
166:
154:
143:
140:
128:
117:
114:
102:
91:
88:
81:
77:
62:
58:
53:
48:
45:
41:
35:
30:
22:
12042:
12038:
12034:Silver ratio
12004:Golden ratio
11978:
11953:
11826:
11772:
11754:Salem number
11591:
11552:. May, 1994.
11537:
11485:
11459:
11455:
11409:
11405:
11379:
11365:
11337:
11331:
11300:
11294:
11275:
11255:. Retrieved
11247:
11234:
11223:. Retrieved
11202:
11190:
11171:
11162:
11156:
11137:
11127:
11101:
11097:Intellectica
11095:
11084:
11060:
11056:
11050:
11031:
11013:
10997:
10992:
10979:
10968:. Retrieved
10940:. Retrieved
10926:
10891:
10872:
10863:
10854:
10841:
10808:
10804:
10798:
10787:. Retrieved
10783:
10774:
10765:
10756:
10741:
10733:
10722:. Retrieved
10718:
10709:
10693:
10687:
10664:Silver ratio
10595:square roots
10589:of a double
10539:
10510:
10323:
10224:
10192:aspect ratio
10152:
10136:Applications
9808:
9570:
9519:
9511:
9497:
9493:
9485:
9481:
9467:
9459:
9318:
9314:
9310:Pell numbers
9300:
9292:
9282:
9140:
8939:
8935:
8931:
8927:
8923:
8919:
8915:
8911:
8898:
8836:
8629:, producing
8620:
8279:
8269:
8261:
8254:
8095:
7851:
7583:
7347:= sin
7328:
7279:
6149:
6020:
5824:
5733:
5729:
5725:
5718:
5687:
5683:
5676:
5669:
5665:
5654:
5648:
5642:
5638:
5632:
5625:
5618:
5588:
5488:
5481:
5464:
5435:
5428:
5340:
5251:
5244:
5101:
5030:because the
5028:trigonometry
4974:One-half of
4973:
4796:
4785:
4779:
4775:
4768:
4762:
4759:
4466:
4463:
4456:
4450:
4446:
4442:
4436:
4430:
4424:
4413:
4364:
4356:
4333:
4304:
4296:
4281:
4273:
4261:
4031:
4023:
4008:
3996:
3988:
3967:
3963:
3954:
3948:
3932:
3924:
3913:
3907:
3903:
3868:
3862:
3856:
3852:
3846:
3840:
3834:
3830:
3824:
3820:
3816:
3810:
3806:
3802:
3798:
3794:
3790:
3784:
3777:
3773:
3769:
3763:
3759:
3755:
3749:
3745:
3741:
3735:
3728:
3721:
3717:
3703:
3697:
3688:
3681:
3674:
3670:
3664:
3660:
3654:
3648:
3646:with centre
3642:
3636:
3633:
3629:lowest terms
3619:
3615:
3609:
3603:
3556:
3550:
3544:
3540:
3504:
3499:
3390:
3386:
3380:
3342:
3314:coefficients
3046:
2987:
2663:
2647:
2637:
2631:
2556:
2522:
2516:
2507:
2501:
2452:
2318:
2312:
2273:
2267:
2258:
2249:
2242:
2231:
2222:
2218:
2207:
2199:
2187:
2179:
2070:
2064:
2054:
2048:
1930:Assume that
1893:
1691:
1683:golden ratio
1608:
1556:
1553:
1540:
1512:
1494:
1465:
1447:
1154:
980:
937:
904:ad quadratum
903:
893:
883:
879:
863:Pythagoreans
861:
829:Pell numbers
802:
681:
664:
483:clay tablet
478:
367:
354:
324:
316:
237:
235:
11596:Brady Haran
11592:Numberphile
11428:Good, I. J.
10184:paper sizes
9285:convergents
8896:, however.
8623:convergence
5848:appears in
5470:square root
5032:unit vector
4262:the latter
2229:Therefore,
1811:Ron Watkins
1786:Ron Watkins
1761:Ron Watkins
671:Sulbasutras
520:sexagesimal
492: 1800
441:sexagesimal
363:denominator
320:square root
307:. It is an
242:real number
12125:Categories
11934:Lemniscate
11321:References
11257:2016-09-05
11225:2010-04-30
10970:2022-04-07
10942:2012-09-07
10789:2023-09-18
10769:Henderson.
10724:2020-08-10
10188:"A" series
10141:Paper size
8106:cos
7331:cos
5723:for which
5218:Properties
4264:inequality
3971:| ≥ 1
3601:. Suppose
3081:polynomial
2046:such that
1681:, and the
940:algorithms
871:irrational
678: 800
524:1 24 51 10
481:Babylonian
473:0 42 25 35
445:1 24 51 10
335:irrational
40:hypotenuse
11897:Liouville
11887:Chaitin's
11362:Aristotle
11111:1110.5456
10825:0003-486X
10591:unit cube
10544:apertures
10077:⋯
10007:−
9946:⋯
9943:−
9921:−
9900:−
9876:−
9760:≈
9643:−
9601:−
9250:⋱
9105:…
9048:−
9002:∞
8987:∑
8973:−
8942:(1) = 6.
8938:(0) = 0,
8905:Fibonacci
8819:⋯
8666:∞
8651:∑
8603:⋯
8574:−
8548:−
8526:⋯
8514:⋅
8508:⋅
8502:⋅
8491:⋅
8485:⋅
8476:−
8467:⋅
8461:⋅
8450:⋅
8432:⋅
8420:−
8363:−
8329:−
8321:∞
8306:∑
8199:π
8176:−
8165:∞
8150:∑
8078:⋯
8060:−
8006:−
7939:−
7889:∞
7874:∏
7837:⋯
7824:⋅
7813:⋅
7789:⋅
7778:⋅
7754:⋅
7743:⋅
7719:⋅
7708:⋅
7621:∞
7606:∏
7569:⋯
7551:−
7524:−
7497:−
7444:−
7431:∞
7416:∏
7199:π
7190:
7170:−
7129:π
7120:
7098:−
7090:−
7082:−
7049:π
7040:
6978:π
6969:
6939:−
6898:π
6889:
6871:−
6839:π
6834:
6811:−
6762:π
6753:
6713:π
6708:
6686:−
6670:−
6637:π
6628:
6576:π
6567:
6537:−
6529:−
6496:π
6487:
6459:−
6427:π
6422:
6399:−
6391:−
6350:π
6341:
6321:−
6313:−
6280:π
6271:
6233:−
6201:π
6196:
6118:⏟
6098:⋯
6071:−
6049:∞
6046:→
6032:π
6003:⋯
5934:⋅
5892:⋅
5872:π
5794:⋅
5787:⋅
5779:⋅
5568:−
5558:−
5550:−
5542:−
5405:−
5388:−
5304:−
5197:∘
5189:
5178:∘
5170:
4892:∘
4884:
4873:∘
4865:
4763:(b, b, a)
4550:∈
4290:, giving
4222:≥
4169:≥
4105:−
4063:−
3828:and legs
3709:congruent
3694:triangles
3554:and legs
3456:−
3414:−
3240:±
3217:±
3170:±
3147:±
3124:−
2909:−
2852:−
2809:−
2775:−
2636:, in his
2634:Aristotle
1484:(approx.
1424:⋮
1410:…
1355:…
1300:…
1016:recursive
900:Vitruvius
783:¯
750:×
744:×
732:−
723:×
684:That is,
645:¯
522:figures,
368:Sequence
317:principal
199:⋱
44:isosceles
11364:(2007),
11216:Archived
11180:Archived
11135:(2003),
11042:Archived
11021:Archived
11018:Proof 8‴
10998:Elements
10964:Archived
10936:Archived
10853:(1996),
10747:Archived
10719:oeis.org
10616:See also
10587:vertices
10388:′
10357:′
10349:′
10336:′
7314:base two
5024:geometry
4819:constant
4288:< 3/2
3939:< 3/2
3783:△
3753:implies
3734:△
3718:Because
3543:△
3085:quotient
2757:. Since
2649:Elements
2511:is even.
2449:Because
2265:Because
2074:must be
1663:trillion
968:integers
875:Hippasus
518:in four
485:YBC 7289
449:42 25 35
437:YBC 7289
246:number 2
12063:Euler's
11949:Apéry's
11613:√
11604:√
11588:of Two"
11476:2344040
11354:2695741
11242:(ed.).
11116:Bibcode
11077:2695741
10833:1969021
10757:root(2)
10742:root(2)
10702:1891736
10670:√
10646:√
10632:√
10536:f-stops
10525:tritone
10196:ISO 216
9575:
9565:√
9554:
9524:
9508:
9502:
9478:
9472:
9456:
9450:
9438:
9434:
9422:
9418:
9406:
9402:
9390:
9386:
9374:
9370:
9358:
9354:
9342:
9338:
9326:
9312:(i.e.,
9305:
9289:
8889:√
8883:
8878:√
8869:√
8258:√
8120:
8108:
7382:
7375:√
7365:
7361:
7349:
7345:
7333:
5692:. Then
5614:tetrate
5236:√
4966:in the
4963:A010503
4815:
4803:
4454:, then
4369:
4353:
4346:√
4338:
4323:
4313:√
4309:
4293:
4286:
4270:
4036:
4020:
4013:√
4001:
3985:
3978:√
3937:
3921:
3652:. Join
3491:√
3373:√
3035:by the
2212:
2196:
2192:
2176:
1578:
1575:470,832
1569:665,857
1566:
1537:
1525:
1523:), and
1509:
1497:
1482:
1470:
1462:
1450:
1348:1.41421
913:tangent
775:1.41421
637:1.41421
528:decimal
469:
457:
427:History
385:1.41421
372:in the
370:A002193
351:
339:
72:0488...
64:1.41421
60:Decimal
11929:Dottie
11493:
11474:
11386:
11352:
11307:
11282:
11145:
11075:
11002:Berlin
10879:
10831:
10823:
10700:
10694:Neusis
10531:music.
10225:Proof:
9097:235416
8926:−1) −
8918:) = 34
8907:-like
8284:gives
8123:gives
5791:
5783:
5775:
5721:> 1
5659:, the
5657:> 1
5621:> 1
5341:since
5288:
4760:Here,
4533:where
4434:, and
4378:; see
4372:|
4343:|
4004:|
3975:|
3960:|
3781:, and
2644:Euclid
1983:factor
1687:normal
1541:1.4142
1479:10,000
1466:1.4142
1388:470832
1385:665857
867:square
629:216000
626:305470
355:1.4142
42:of an
11917:Cahen
11912:Omega
11902:Prime
11472:JSTOR
11350:JSTOR
11219:(PDF)
11212:(PDF)
11106:arXiv
11073:JSTOR
10829:JSTOR
10680:Notes
10546:is 2.
10540:areas
10523:of a
8934:−2),
8811:16384
8267:with
8100:of a
7304:is a
5661:limit
5227:Angle
5036:angle
4910:0.707
4268:1<
3919:1<
3805:) = 2
3625:ratio
3623:be a
3308:is a
3185:. As
3079:of a
1992:Then
1590:2.000
1545:−0.12
1517:−0.72
1513:1.414
1486:+0.72
917:atria
909:Plato
420:73799
417:76679
414:80731
411:37694
408:71875
405:85696
402:69807
399:24209
396:16887
393:04880
390:73095
387:35623
69:73095
66:35623
11622:and
11491:ISBN
11384:ISBN
11305:ISBN
11280:ISBN
11248:The
11143:ISBN
11088:See
10877:ISBN
10821:ISSN
10668:1 +
10593:are
10324:Let
10227:Let
9321:= ±1
9283:The
8875:and
8798:4096
8621:The
8260:1 +
7584:and
5852:for
5732:) =
5652:for
5636:and
5485:and
5472:and
5231:area
5026:and
4968:OEIS
4797:The
4567:and
4318:≤ 3
3952:and
3941:(as
3911:and
3866:and
3844:and
3707:are
3701:and
3685:and
3678:and
3640:and
3607:and
3541:Let
3395:area
3389:and
3162:and
3077:root
2924:and
2742:>
2520:and
2237:even
2221:and
2194:) =
2128:and
2058:are
2052:and
1725:Name
1722:Date
1598:0045
1492:).
1293:1.41
999:>
479:The
453:0 30
236:The
11956:(3)
11535:",
11464:doi
11460:130
11454:",
11414:doi
11342:doi
11338:107
11065:doi
11061:107
10813:doi
9484:+ 2
9475:is
9447:408
9441:577
9431:169
9425:239
9317:− 2
9084:204
8887:(1+
8808:693
8795:315
8785:256
8595:256
8582:128
8272:= 1
7559:100
7187:sin
7117:sin
7037:sin
6966:sin
6886:sin
6831:sin
6750:sin
6705:sin
6625:sin
6564:sin
6484:sin
6419:sin
6338:sin
6268:sin
6193:sin
6039:lim
5679:→ ∞
5674:as
5663:of
5276:is
5186:sin
5167:cos
4954:688
4951:937
4948:835
4945:284
4942:039
4939:849
4936:104
4933:362
4930:844
4927:400
4924:524
4921:547
4918:186
4915:781
4912:106
4881:cos
4862:sin
4575:gcd
4422:If
4006:by
3797:− (
3785:FDC
3736:BEF
3729:BEF
3722:EBF
3713:SAS
3711:by
3704:ADE
3698:ABC
3689:DAE
3682:BAC
3545:ABC
3371:of
3351:or
3232:or
2646:'s
2246:. (
2235:is
2076:odd
2068:or
1876:000
1873:000
1870:000
1867:000
1851:050
1848:000
1845:000
1842:000
1826:100
1823:000
1820:000
1817:000
1801:000
1798:000
1795:000
1792:000
1776:000
1773:000
1770:000
1767:000
1751:000
1748:001
1745:000
1742:000
1595:000
1592:000
1582:1.6
1562:= 1
1539:(≈
1534:169
1528:239
1511:(≈
1500:140
1464:(≈
1333:408
1330:577
894:In
882:or
767:408
764:577
642:296
353:(≈
272:or
12127::
12073:Pi
11618:,
11594:.
11590:.
11470:,
11458:,
11410:25
11408:,
11400:;
11348:,
11336:,
11266:^
11246:.
11214:.
11114:,
11102:56
11100:,
11071:,
11059:,
11040:.
10962:.
10951:^
10849:;
10827:.
10819:.
10809:46
10807:.
10782:.
10717:.
10698:MR
10666:,
10657:,
10643:,
10629:,
10515::
10222:.
9496:+
9436:,
9420:,
9415:70
9409:99
9404:,
9399:29
9393:41
9388:,
9383:12
9377:17
9372:,
9356:,
9340:,
8885:ln
8782:35
8772:64
8769:15
8569:16
8282:!!
7827:15
7821:13
7816:14
7810:14
7792:11
7781:10
7775:10
7532:36
7363:=
7316:.
7203:32
7196:15
7133:16
7053:32
6982:16
6902:32
6766:32
6759:13
6641:32
6500:32
6430:16
6354:32
6347:11
6284:16
6204:32
6178::
5856:,
5810:2.
5647:=
5645:+1
5631:=
5623:,
5492::
5433:.
5414:1.
5193:45
5174:45
4957:47
4888:45
4869:45
4821:.
4778:=
4449:=
4445:+
4428:,
4418::
4351:−
4311:+
4018:+
3983:−
3966:−
3833:−
3823:−
3814:.
3809:−
3801:−
3793:=
3791:FC
3776:−
3772:=
3770:DF
3762:−
3758:=
3756:BF
3748:−
3744:=
3742:BE
3715:.
3675:AE
3673:=
3671:AC
3668:,
3665:AD
3663:=
3661:AB
3655:DE
3643:CE
3637:BD
3631:.
3564:,
3355:.
1765:10
1740:10
1716:.
1689:.
1675:,
1601:.
1586:10
1551:.
1549:10
1521:10
1506:99
1490:10
1459:70
1453:99
1407:46
1278:12
1275:17
1245:.5
898:,
886:.
753:34
675:c.
610:60
606:10
590:60
586:51
576:60
573:24
489:c.
455:=
365:.
348:70
342:99
12079:)
12077:π
12075:(
12069:)
12067:e
12065:(
12047:)
12043:S
12039:δ
12036:(
12025:)
12023:ς
12021:(
12010:)
12008:φ
12006:(
12000:)
11998:E
11996:(
11990:)
11988:ψ
11986:(
11973:)
11971:ρ
11969:(
11958:)
11954:ζ
11951:(
11940:)
11938:ϖ
11936:(
11908:)
11906:ρ
11904:(
11893:)
11891:Ω
11889:(
11868:e
11861:t
11854:v
11811:)
11809:ς
11802:)
11800:ψ
11768:)
11765:S
11761:δ
11734:)
11732:ρ
11715:)
11713:φ
11652:e
11645:t
11638:v
11624:e
11620:π
11615:2
11606:2
11598:.
11574:2
11542:.
11521:2
11500:.
11479:.
11466::
11440:2
11423:.
11416::
11393:.
11357:.
11344::
11313:.
11288:.
11260:.
11228:.
11197:.
11118::
11108::
11067::
10973:.
10945:.
10905:2
10885:.
10835:.
10815::
10792:.
10727:.
10672:2
10659:2
10648:5
10634:3
10605:)
10577:.
10563:2
10502:.
10490:R
10487:=
10482:2
10477:=
10471:2
10467:2
10462:=
10456:)
10453:S
10449:/
10445:L
10442:(
10438:2
10433:=
10428:L
10424:S
10421:2
10415:=
10409:2
10405:/
10401:L
10397:S
10392:=
10385:R
10354:S
10346:L
10340:=
10333:R
10306:2
10301:=
10296:S
10293:L
10288:=
10285:R
10261:=
10258:L
10238:=
10235:S
10208:2
10167:2
10116:.
10111:2
10106:)
10100:2
10095:)
10089:2
10083:)
10074:+
10068:8
10065:1
10057:(
10052:+
10046:8
10043:1
10036:(
10031:+
10025:8
10022:1
10015:(
10010:4
10001:2
9998:3
9992:=
9980:2
9975:)
9969:2
9964:)
9958:2
9952:)
9937:4
9934:1
9926:(
9915:4
9912:1
9905:(
9894:4
9891:1
9884:(
9879:2
9870:2
9867:3
9861:=
9852:2
9834::
9820:2
9784:2
9780:q
9774:2
9769:2
9765:1
9753:)
9747:q
9744:p
9739:+
9734:2
9728:(
9721:2
9717:q
9712:1
9707:=
9700:)
9694:q
9691:p
9686:+
9681:2
9675:(
9668:2
9664:q
9657:|
9651:2
9647:p
9638:2
9634:q
9630:2
9626:|
9619:=
9615:|
9609:q
9606:p
9596:2
9590:|
9571:q
9567:2
9563:2
9560:/
9557:1
9537:2
9520:q
9516:/
9512:p
9498:q
9494:p
9490:/
9486:q
9482:p
9468:q
9464:/
9460:p
9444:/
9428:/
9412:/
9396:/
9380:/
9367:5
9364:/
9361:7
9351:2
9348:/
9345:3
9335:1
9332:/
9329:1
9319:q
9315:p
9301:q
9297:/
9293:p
9268:.
9247:+
9244:2
9232:1
9221:+
9218:2
9206:1
9195:+
9192:2
9180:1
9169:+
9166:1
9163:=
9158:2
9109:)
9102:+
9094:1
9089:+
9081:1
9076:+
9071:6
9068:1
9062:(
9056:2
9053:1
9043:2
9040:3
9035:=
9029:)
9024:n
9020:2
9016:(
9013:a
9009:1
8997:0
8994:=
8991:n
8981:2
8978:1
8968:2
8965:3
8960:=
8955:2
8940:a
8936:a
8932:n
8930:(
8928:a
8924:n
8922:(
8920:a
8916:n
8914:(
8912:a
8894:)
8891:2
8880:2
8871:2
8867:π
8847:2
8822:.
8816:+
8803:+
8790:+
8777:+
8764:+
8759:8
8756:3
8751:+
8746:2
8743:1
8738:=
8730:2
8726:)
8722:!
8719:k
8716:(
8711:1
8708:+
8705:k
8702:3
8698:2
8692:!
8689:)
8686:1
8683:+
8680:k
8677:2
8674:(
8661:0
8658:=
8655:k
8647:=
8642:2
8606:.
8600:+
8592:7
8587:+
8579:5
8566:1
8561:+
8556:8
8553:1
8543:2
8540:1
8535:+
8532:1
8529:=
8523:+
8517:8
8511:6
8505:4
8499:2
8494:5
8488:3
8482:1
8470:6
8464:4
8458:2
8453:3
8447:1
8441:+
8435:4
8429:2
8425:1
8415:2
8412:1
8407:+
8404:1
8401:=
8395:!
8392:!
8389:)
8386:k
8383:2
8380:(
8375:!
8372:!
8369:)
8366:3
8360:k
8357:2
8354:(
8346:1
8343:+
8340:k
8336:)
8332:1
8326:(
8316:0
8313:=
8310:k
8302:=
8297:2
8280:n
8270:x
8262:x
8240:.
8234:!
8231:)
8228:k
8225:2
8222:(
8215:k
8212:2
8207:)
8202:4
8194:(
8187:k
8183:)
8179:1
8173:(
8160:0
8157:=
8154:k
8146:=
8140:2
8136:1
8117:4
8114:/
8111:π
8081:.
8074:)
8068:7
8065:1
8057:1
8053:(
8047:)
8041:5
8038:1
8033:+
8030:1
8026:(
8020:)
8014:3
8011:1
8003:1
7999:(
7993:)
7987:1
7984:1
7979:+
7976:1
7972:(
7968:=
7964:)
7957:3
7954:+
7951:k
7948:4
7944:1
7936:1
7932:(
7927:)
7920:1
7917:+
7914:k
7911:4
7907:1
7902:+
7899:1
7895:(
7884:0
7881:=
7878:k
7870:=
7865:2
7833:)
7804:(
7798:)
7786:9
7769:(
7763:)
7757:7
7751:5
7746:6
7740:6
7734:(
7728:)
7722:3
7716:1
7711:2
7705:2
7699:(
7695:=
7689:)
7686:3
7683:+
7680:k
7677:4
7674:(
7671:)
7668:1
7665:+
7662:k
7659:4
7656:(
7649:2
7645:)
7641:2
7638:+
7635:k
7632:4
7629:(
7616:0
7613:=
7610:k
7602:=
7597:2
7565:)
7556:1
7548:1
7544:(
7538:)
7529:1
7521:1
7517:(
7511:)
7505:4
7502:1
7494:1
7490:(
7486:=
7482:)
7473:2
7469:)
7465:2
7462:+
7459:k
7456:4
7453:(
7449:1
7441:1
7437:(
7426:0
7423:=
7420:k
7412:=
7406:2
7402:1
7377:2
7371:/
7368:1
7358:4
7355:/
7352:π
7342:4
7339:/
7336:π
7290:2
7253:2
7248:+
7245:2
7240:+
7237:2
7232:+
7229:2
7221:2
7218:1
7212:=
7175:2
7167:2
7162:+
7159:2
7151:2
7148:1
7142:=
7126:5
7103:2
7095:2
7087:2
7079:2
7071:2
7068:1
7062:=
7046:5
7024:2
7019:+
7016:2
7011:+
7008:2
7000:2
6997:1
6991:=
6975:7
6952:2
6947:+
6944:2
6936:2
6931:+
6928:2
6920:2
6917:1
6911:=
6895:9
6876:2
6868:2
6860:2
6857:1
6851:=
6842:8
6816:2
6808:2
6803:+
6800:2
6795:+
6792:2
6784:2
6781:1
6775:=
6742:2
6734:2
6731:1
6725:=
6716:4
6691:2
6683:2
6678:+
6675:2
6667:2
6659:2
6656:1
6650:=
6634:3
6614:2
6609:+
6606:2
6598:2
6595:1
6589:=
6580:8
6573:3
6550:2
6545:+
6542:2
6534:2
6526:2
6518:2
6515:1
6509:=
6493:7
6472:2
6467:+
6464:2
6456:2
6448:2
6445:1
6439:=
6404:2
6396:2
6388:2
6383:+
6380:2
6372:2
6369:1
6363:=
6326:2
6318:2
6310:2
6302:2
6299:1
6293:=
6277:3
6254:2
6249:+
6246:2
6241:+
6238:2
6230:2
6222:2
6219:1
6213:=
6160:2
6135:.
6124:m
6106:2
6101:+
6095:+
6092:2
6087:+
6084:2
6079:+
6076:2
6068:2
6059:m
6055:2
6043:m
6035:=
6006:,
5993:2
5990:1
5982:2
5979:1
5974:+
5969:2
5966:1
5957:2
5954:1
5949:+
5944:2
5941:1
5926:2
5923:1
5915:2
5912:1
5907:+
5902:2
5899:1
5886:2
5883:1
5877:=
5869:2
5854:π
5834:2
5807:=
5769:2
5760:2
5751:2
5734:c
5730:c
5728:(
5726:f
5719:c
5702:2
5690:)
5688:c
5686:(
5684:f
5677:n
5670:n
5666:x
5655:n
5649:c
5643:n
5639:x
5633:c
5629:1
5626:x
5619:c
5598:2
5571:i
5561:i
5553:i
5545:i
5527:i
5521:i
5516:i
5513:+
5508:i
5489:i
5487:−
5482:i
5465:i
5445:2
5411:=
5408:1
5402:2
5399:=
5395:)
5391:1
5383:2
5377:(
5371:)
5367:1
5364:+
5359:2
5353:(
5326:1
5323:+
5318:2
5313:=
5307:1
5299:2
5293:1
5262:2
5248:.
5245:u
5238:2
5202:.
5183:=
5164:=
5158:2
5154:1
5149:=
5142:2
5139:1
5132:=
5127:2
5119:2
5116:1
5087:.
5082:)
5076:2
5072:2
5066:,
5061:2
5057:2
5050:(
5008:2
4984:2
4970:)
4897:=
4878:=
4859:=
4854:2
4850:2
4844:=
4838:2
4834:1
4812:2
4809:/
4806:1
4786:a
4780:a
4776:b
4774:2
4769:a
4743:2
4739:a
4735:=
4730:2
4726:b
4722:+
4717:2
4713:b
4689:2
4685:a
4681:=
4676:2
4672:b
4668:2
4643:2
4639:b
4633:2
4629:a
4623:=
4620:2
4596:1
4593:=
4590:)
4587:b
4584:,
4581:a
4578:(
4554:Z
4547:b
4544:,
4541:a
4517:b
4514:a
4509:=
4504:2
4477:2
4457:c
4451:c
4447:b
4443:a
4437:c
4431:b
4425:a
4392:2
4365:b
4361:/
4357:a
4348:2
4334:b
4332:3
4329:/
4326:1
4315:2
4305:b
4301:/
4297:a
4282:b
4278:/
4274:a
4247:,
4239:2
4235:b
4231:3
4227:1
4215:)
4209:b
4206:a
4201:+
4196:2
4190:(
4183:2
4179:b
4174:1
4162:)
4156:b
4153:a
4148:+
4143:2
4137:(
4130:2
4126:b
4119:|
4113:2
4109:a
4100:2
4096:b
4092:2
4088:|
4081:=
4077:|
4071:b
4068:a
4058:2
4052:|
4038:)
4032:b
4028:/
4024:a
4015:2
4011:(
4009:b
3997:b
3993:/
3989:a
3980:2
3968:a
3964:b
3962:2
3955:a
3949:b
3947:2
3933:b
3929:/
3925:a
3914:b
3908:a
3882:2
3869:n
3863:m
3857:n
3855::
3853:m
3847:n
3841:m
3835:n
3831:m
3825:m
3821:n
3819:2
3811:m
3807:n
3803:n
3799:m
3795:n
3778:n
3774:m
3764:n
3760:m
3750:n
3746:m
3727:∠
3720:∠
3687:∠
3680:∠
3649:A
3620:n
3618::
3616:m
3610:n
3604:m
3587:2
3582:=
3577:n
3574:m
3557:n
3551:m
3519:2
3493:2
3467:2
3463:)
3459:b
3453:a
3450:(
3447:2
3425:2
3421:)
3417:a
3411:b
3408:2
3405:(
3391:b
3387:a
3375:2
3326:2
3296:)
3293:x
3290:(
3287:p
3265:2
3243:2
3220:1
3195:2
3173:2
3150:1
3127:2
3119:2
3115:x
3111:=
3108:)
3105:x
3102:(
3099:p
3057:2
3017:2
3013:b
3009:2
3006:=
3001:2
2997:a
2968:p
2965:=
2962:q
2942:1
2939:+
2934:2
2912:1
2904:2
2880:q
2877:p
2855:1
2847:2
2825:1
2822:=
2817:2
2813:1
2806:2
2803:=
2800:)
2797:1
2794:+
2789:2
2784:(
2781:)
2778:1
2770:2
2765:(
2745:p
2739:q
2717:p
2714:q
2709:=
2706:1
2703:+
2698:2
2674:2
2615:2
2591:2
2567:2
2539:b
2536:a
2523:b
2517:a
2508:b
2502:b
2485:2
2481:b
2477:=
2472:2
2468:k
2464:2
2453:k
2451:2
2446:.
2432:2
2428:k
2424:2
2421:=
2416:2
2412:b
2389:2
2385:k
2381:4
2378:=
2373:2
2369:)
2365:k
2362:2
2359:(
2356:=
2351:2
2347:a
2343:=
2338:2
2334:b
2330:2
2319:a
2313:k
2311:2
2306:.
2294:k
2291:2
2288:=
2285:a
2274:k
2268:a
2259:a
2250:b
2248:2
2243:b
2241:2
2232:a
2223:b
2219:a
2208:b
2204:/
2200:a
2188:b
2184:/
2180:a
2174:(
2157:2
2153:b
2149:2
2146:=
2141:2
2137:a
2116:2
2113:=
2106:2
2102:b
2096:2
2092:a
2078:.
2071:b
2065:a
2055:b
2049:a
2032:b
2029:a
2002:2
1989:.
1978:.
1964:2
1940:2
1912:2
1865:1
1840:2
1815:2
1790:5
1702:2
1678:e
1672:π
1647:2
1619:2
1584:×
1572:/
1564:(
1560:0
1557:a
1547:×
1531:/
1519:×
1503:/
1488:×
1476:/
1473:1
1456:/
1413:,
1399:=
1379:=
1370:4
1366:a
1358:,
1352:5
1344:=
1324:=
1315:3
1311:a
1303:,
1297:6
1289:=
1269:=
1260:2
1256:a
1248:,
1241:1
1237:=
1226:2
1223:3
1217:=
1208:1
1204:a
1176:1
1173:=
1168:0
1164:a
1140:.
1133:n
1129:a
1125:1
1120:+
1115:2
1110:n
1106:a
1100:=
1096:)
1087:n
1083:a
1079:2
1073:+
1068:n
1064:a
1059:(
1053:2
1050:1
1045:=
1040:1
1037:+
1034:n
1030:a
1002:0
994:0
990:a
952:2
845:2
813:2
788:.
772:=
759:=
747:4
741:3
737:1
726:4
720:3
716:1
711:+
706:3
703:1
698:+
695:1
673:(
650:.
634:=
621:=
614:3
601:+
594:2
581:+
568:+
565:1
540:2
504:2
487:(
466:2
463:/
460:1
443:(
345:/
293:2
289:/
285:1
281:2
258:2
196:+
193:2
181:1
170:+
167:2
155:1
144:+
141:2
129:1
118:+
115:2
103:1
92:+
89:1
23:.
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