1840:
1530:
1835:{\displaystyle {\begin{aligned}&\alpha =\prod _{m=2}^{\infty }\left({1-{\frac {1}{f\left(m\right)}}}\right)=\left(1-{\frac {1}{4}}\right)\left(1-{\frac {1}{9}}\right)\left(1-{\frac {1}{64}}\right)\left(1-{\frac {1}{152587890625}}\right)\left(1-{\frac {1}{6^{\left(5^{15}\right)}}}\right)\ldots =\\&=0.6562499999956991\underbrace {99999\ldots 99999} _{23,747,291,559}8528404201690728\ldots \end{aligned}}}
2862:, including possibly the empty string. (Since one digit is read from the input sequence for each state transition, it is necessary to be able to output the empty string in order to achieve any compression at all). An information lossless finite-state compressor is a finite-state compressor whose input can be uniquely recovered from its output and final state. In other words, for a finite-state compressor
1033:(as its Lebesgue measure as a subset of the real numbers is zero, so it essentially takes up no space within the real numbers). Also, the non-normal numbers (as well as the normal numbers) are dense in the reals: the set of non-normal numbers between two distinct real numbers is non-empty since it contains
99:
be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequence of equal length. No digit
86:
Intuitively, a number being simply normal means that no digit occurs more frequently than any other. If a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length. A normal number can be thought of as an infinite sequence
1056:
in order, is normal in base 10. Likewise, the different variants of
Champernowne's constant (done by performing the same concatenation in other bases) are normal in their respective bases (for example, the base-2 Champernowne constant is normal in base 2), but they have not been proven to be normal
2507:
is also normal. In other words, if one runs a finite-state machine on a normal sequence, where each of the finite-state machine's states are labeled either "output" or "no output", and the machine outputs the digit it reads next after entering an "output" state, but does not output the next digit
1524:
1334:
are strongly conjectured to be normal, it is still not known whether they are normal or not. It has not even been proven that all digits actually occur infinitely many times in the decimal expansions of those constants (for example, in the case of π, the popular claim "every string of numbers
3354:
1377:
3135:
These characterizations of normal sequences can be interpreted to mean that "normal" = "finite-state random"; i.e., the normal sequences are precisely those that appear random to any finite-state machine. Compare this with the
2204:(Copeland and Erdős 1946). From this it follows that Champernowne's number is normal in base 10 (since the set of all positive integers is obviously dense) and that the Copeland–Erdős constant is normal in base 10 (since the
3034:
536:
3236:
340:
2798:
1021:
absolutely normal number. Although this construction does not directly give the digits of the numbers constructed, it shows that it is possible in principle to enumerate each digit of a particular normal number.
1226:. (This result includes as special cases all of the above-mentioned results of Champernowne, Besicovitch, and Davenport & Erdős.) The authors also show that the same result holds even more generally when
2490:
number of digits in any normal sequence leaves it normal. Similarly, if a finite number of digits are added to, removed from, or changed in any simply normal sequence, the new sequence is still simply normal.
3418:
2074:
3221:
2924:
1535:
2408:
2473:
1893:
2133:
2194:
3079:
1973:
2833:
2303:
2602:
2697:
2003:
973:, a number can be simply normal (but not normal or rich), rich (but not simply normal or normal), normal (and thus simply normal and rich), or none of these. A number is
4668:
2727:
2651:
2960:
1938:
2564:
2544:
444:
1041:). For instance, there are uncountably many numbers whose decimal expansions (in base 3 or higher) do not contain the digit 1, and none of those numbers are normal.
265:
1860:
Every non-zero real number is the product of two normal numbers. This follows from the general fact that every number is the product of two numbers from a set
2743:
1519:{\displaystyle f\left(n\right)={\begin{cases}n^{\frac {f\left(n-1\right)}{n-1}},&n\in \mathbb {Z} \cap \left[3,\infty \right)\\4,&n=2\end{cases}}}
2699:
fraction of the gambler's money on the bit 1. The money bet on the digit that comes next in the input (total money times percent bet) is multiplied by
4397:
4393:
4292:
4638:
3140:, which are those infinite sequences that appear random to any algorithm (and in fact have similar gambling and compression characterizations with
4487:(1917), "Démonstration élémentaire d'un théorème de M. Borel sur les nombres absolutment normaux et détermination effective d'un tel nombre",
4428:
4189:
2729:, and the rest of the money is lost. After the bit is read, the FSG transitions to the next state according to the input it received. A FSG
1349:
is normal), and no counterexamples are known in any base. However, no irrational algebraic number has been proven to be normal in any base.
4040:
1169:
being any non-constant polynomial whose values on the positive integers are positive integers, expressed in base 10, is normal in base 10.
3806:
3349:{\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{k=0}^{n-1}e^{2\pi imb^{k}x}=0\quad {\text{ for all integers }}m\geq 1.}
883:
is a sequence in which every finite string appears. A normal sequence is disjunctive, but a disjunctive sequence need not be normal. A
3132:
compresses asymptotically as well as any ILFSC, this means that the LZ compression algorithm can compress any non-normal sequence.
4581:
4534:
3365:
4384:
4113:
3969:
3787:
2340:
A number is normal if and only if it is simply normal in every base. This follows from the previous characterization of base
2012:
3105:
Schnorr and Stimm showed that no FSG can succeed on any normal sequence, and Bourke, Hitchcock and
Vinodchandran showed the
4800:
4795:
Quéfflec, Martine (2006), "Old and new results on normality", in
Denteneer, Dee; den Hollander, F.; Verbitskiy, E. (eds.),
3981:(1935), "The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers",
3166:
2877:
4714:
2511:
A deeper connection exists with finite-state gamblers (FSGs) and information lossless finite-state compressors (ILFSCs).
2354:
4589:
4826:
4759:
4722:
4274:
4018:
3128:, a quantitative measure of its deviation from normality, which is 1 exactly when the sequence is normal). Since the
2417:
196:
4780:
4625:
4573:
4220:
3879:
3826:
3137:
2522:
1863:
4297:
3978:
2079:
1128:
2146:
1061:
4884:
4710:
4329:
4266:
3950:
3039:
1950:
3118:
A sequence is normal if and only if it is incompressible by any information lossless finite-state compressor
2803:
2273:
1313:
It has been an elusive goal to prove the normality of numbers that are not artificially constructed. While
4889:
4160:
1207:
796:
A given infinite sequence is either normal or not normal, whereas a real number, having a different base-
172:
4105:
4068:
Bourke, C.; Hitchcock, J. M.; Vinodchandran, N. V. (2005), "Entropy rates and finite-state dimension",
3887:
3779:
3101:
above) can always be made to equal 1 by the 1-state ILFSC that simply copies its input to the output.
2569:
1331:
155:
20:
4082:
1403:
1013:) showed that it is possible to specify a particular such number. Becher and Figueira (
807:, may be normal in one base but not in another (in which case it is not a normal number). For bases
3224:
3121:(they actually showed that the sequence's optimal compression ratio over all ILFSCs is exactly its
2656:
998:
1982:
4180:
3751:
2943:
1074:
3922:
4879:
4670:
Developments in
Language Theory: At the Crossroads of Mathematics, Computer Science and Biology
4484:
4254:
4077:
3770:
Adamczewski, Boris; Bugeaud, Yann (2010), "8. Transcendence and diophantine approximation", in
3094:
2859:
2702:
2611:
1327:
1006:
1917:
3959:
3741:
2549:
2529:
1044:
903:
131:
4629:
3756:
2500:
2205:
1299:
880:
4836:
4769:
4732:
4368:
4320:
4284:
4123:
3819:
3797:
2255:) and made explicit in the work of Bourke, Hitchcock, and Vinodchandran (
1335:
eventually occurs in π" is not known to be true). It has also been conjectured that every
8:
4633:
4424:
3771:
3228:
3123:
3112:
A sequence is normal if and only if there is no finite-state gambler that succeeds on it.
559:
258:
135:
49:
4804:
4776:
4703:
4682:
4660:
4614:
4606:
4473:
4356:
4338:
4239:
4155:
4057:
3998:
3910:
3868:
3834:
3746:
3721:
3719:
3106:
2935:
127:
3941:
2503:
and normal sequences: every infinite subsequence selected from a normal sequence by a
1085:). More generally, the latter authors proved that the real number represented in base
876:
irrational, there are uncountably many numbers normal in each base but not the other.
4852:
4822:
4755:
4718:
4509:
4380:
4270:
4109:
4061:
4024:
4014:
4002:
3965:
3783:
2546:, each of whose states is labelled with percentages of money to bet on each digit in
1362:
1336:
1018:
119:
4529:
4477:
4243:
4203:
3872:
3716:
4832:
4814:
4765:
4728:
4674:
4647:
4610:
4602:
4598:
4577:
4551:
4543:
4496:
4465:
4453:
4440:
4411:
4402:
4364:
4348:
4316:
4306:
4280:
4229:
4211:
4198:
4169:
4142:
4130:
4119:
4087:
4049:
3990:
3945:
3937:
3914:
3906:
3902:
3883:
3858:
3850:
3830:
3815:
3793:
2504:
1845:
1339:
1314:
1158:
139:
123:
4855:
4652:
562:
frequency. For example, in a normal binary sequence (a sequence over the alphabet
4743:
3854:
3593:
3029:{\displaystyle \liminf _{n\to \infty }{\frac {|C(S\upharpoonright n)|}{n}}<1,}
2740:
if, starting from $ 1, it makes unbounded money betting on the sequence; i.e., if
2508:
after entering a "no output state", then the sequence it outputs will be normal.
1358:
1307:
826:
673:
130:, and only a few specific numbers have been shown to be normal. For example, any
57:
4752:
Surveys in number theory: Papers from the millennial conference on number theory
3838:
2305:. This follows from the previous block characterization of normality: Since the
531:{\displaystyle \lim _{n\to \infty }{\frac {N_{S}(w,n)}{n}}={\frac {1}{b^{|w|}}}}
4818:
4525:
4262:
4250:
3474:
3437:
3160:
3141:
3098:
2939:
2848:
2212:
1053:
718:
4556:
4173:
4092:
4035:
3778:, Encyclopedia of Mathematics and its Applications, vol. 135, Cambridge:
990:
4873:
4547:
4038:(1909), "Les probabilités dénombrables et leurs applications arithmétiques",
4028:
1372:) gives an example of an irrational number that is absolutely abnormal. Let
1154:
947:
88:
4445:
4416:
4311:
4215:
4184:
1078:
4747:
4698:
4664:
4234:
1070:
1025:
The set of non-normal numbers, despite being "large" in the sense of being
4147:
335:{\displaystyle \lim _{n\to \infty }{\frac {N_{S}(a,n)}{n}}={\frac {1}{b}}}
4739:
3478:
1361:
is normal in any base, since the digit sequences of rational numbers are
1026:
885:
692:
661:
31:
27:
4809:
4663:(1994), "Borel normality and algorithmic randomness", in Rozenberg, G.;
4799:, IMS Lecture Notes – Monograph Series, vol. 48, Beachwood, Ohio:
4469:
4360:
4053:
3994:
2487:
1185:
1005:
real numbers are normal, establishing the existence of normal numbers.
1002:
115:
4501:
3863:
3632:
3362:
is normal in base β for any real number β if and only if the sequence
2251:, etc.) This was implicit in the work of Ziv and Lempel (
4860:
4521:
4343:
4249:
3725:
2793:{\displaystyle \limsup _{n\to \infty }d(S\upharpoonright n)=\infty ,}
943:
383:
4352:
4158:(1933), "The construction of decimals normal in the scale of ten",
1165:) proved that the number represented by the same expression, with
1038:
1030:
186:
111:) if it is normal in all integer bases greater than or equal to 2.
4742:(2002), "One hundred years of normal numbers", in Bennett, M. A.;
4685:; Zamfirescu, T. (1999), "Most numbers obey no probability laws",
1135:) proved that the number represented by the same expression, with
2219:
of equal length appears with equal frequency. (A block of length
897:
is disjunctive: one that is disjunctive to every base is called
684:
39:
3964:(illustrated ed.), American Mathematical Soc., p. 13,
3804:
Agafonov, V. N. (1968), "Normal sequences and finite automata",
3523:
1200:> 0, then the real number represented by the concatenation
672:
is precisely that expected if the sequence had been produced at
4797:
Dynamics & Stochastics: Festschrift in honor of M. S. Keane
4400:(1992), "Discrepancy estimates for a class of normal numbers",
2351:-normal if and only if there exists a set of positive integers
92:
4530:"Compression of individual sequences via variable-rate coding"
4516:, Ph.D. thesis, Berkeley, California: University of California
2227:
appearing at a position in the sequence that is a multiple of
4104:, Cambridge Tracts in Mathematics, vol. 193, Cambridge:
3837:; Dinneen, Michael J.; Dumitrescu, Monica; Yee, Alex (2012),
1052:
obtained by concatenating the decimal representations of the
42:
4582:"On the random character of fundamental constant expansions"
4456:; Stimm, H. (1972), "Endliche Automaten und Zufallsfolgen",
4067:
2858:
is a finite-state machine with output strings labelling its
2256:
3129:
1512:
4133:(1959), "On a problem of Steinhaus about normal numbers",
4013:(Anniversary ed.), Hoboken, N.J.: Wiley, p. 15,
3825:
3638:
3413:{\displaystyle \left({x\beta ^{k}}\right)_{k=0}^{\infty }}
3147:
558:
is normal if all strings of equal length occur with equal
3455:
3567:
3565:
3513:
3511:
4750:; Diamond, H.G.; Hildebrand, A.J.; Philipp, W. (eds.),
3620:
2653:
of the gambler's money on the bit 0, and the remaining
2069:{\displaystyle |A\cap \{1,\ldots ,n\}|\geq n^{\alpha }}
1323:
1049:
0.1234567891011121314151617181920212223242526272829...,
148:
138:). It is widely believed that the (computable) numbers
52:
is distributed uniformly in the sense that each of the
4850:
4261:, Mathematical Surveys and Monographs, vol. 104,
3644:
3552:
3550:
2494:
4102:
3923:"An example of a computable absolutely normal number"
3704:
3692:
3562:
3508:
3368:
3239:
3216:{\displaystyle {\left(b^{k}x\right)}_{k=0}^{\infty }}
3169:
3042:
2963:
2880:
2806:
2746:
2705:
2659:
2614:
2572:
2552:
2532:
2420:
2357:
2276:
2149:
2082:
2015:
1985:
1953:
1920:
1866:
1533:
1380:
447:
268:
4327:
Martin, Greg (2001), "Absolutely abnormal numbers",
3680:
3496:
3484:
2919:{\displaystyle f:\Sigma ^{*}\to \Sigma ^{*}\times Q}
2566:. For instance, for an FSG over the binary alphabet
3547:
2526:) is a finite-state machine over a finite alphabet
4702:
3769:
3608:
3535:
3529:
3412:
3348:
3215:
3073:
3028:
2918:
2827:
2792:
2721:
2691:
2645:
2596:
2558:
2538:
2475:No finite set suffices to show that the number is
2467:
2403:{\displaystyle m_{1}<m_{2}<m_{3}<\cdots }
2402:
2297:
2188:
2127:
2068:
1997:
1967:
1932:
1887:
1834:
1518:
1265:where the αs and βs are real numbers with β > β
1066:0.23571113171923293137414347535961677173798389...,
915:, but not necessarily conversely. The real number
530:
334:
4624:
3951:20.500.12110/paper_03043975_v270_n1-2_p947_Becher
1856:Additional properties of normal numbers include:
989:The concept of a normal number was introduced by
695:. Consider the infinite digit sequence expansion
19:For the floating-point meaning in computing, see
4871:
4681:
4639:Proceedings of the American Mathematical Society
3668:
3656:
3241:
2965:
2748:
957:if each individual digit appears with frequency
953:We defined a number to be simply normal in base
449:
270:
4697:
3443:
1073:in base 10, is normal in base 10, as proved by
4572:
4210:
3920:
3878:
3461:
3358:This connection leads to the terminology that
1295:
1162:
1037:(in fact, it is uncountably infinite and even
1014:
4788:Clay Mathematics Institute Annual Report 2006
4489:Bulletin de la Société Mathématique de France
4392:
4190:Bulletin of the American Mathematical Society
4179:
3961:Inevitable Randomness in Discrete Mathematics
3839:"An Empirical Approach to the Normality of π"
2468:{\displaystyle m\in \{m_{1},m_{2},\ldots \}.}
2325:expansion, a number is simply normal in base
1342:is absolutely normal (which would imply that
1173:
1082:
4775:
4452:
4154:
4041:Rendiconti del Circolo Matematico di Palermo
2591:
2579:
2499:Agafonov showed an early connection between
2459:
2427:
2045:
2027:
4008:
3977:
3626:
2410:where the number is simply normal in bases
2266:if and only if it is simply normal in base
1888:{\displaystyle X\subseteq \mathbb {R} ^{+}}
1132:
721:(we ignore the decimal point). We say that
4754:, Natick, MA: A K Peters, pp. 57–74,
4483:
4257:; Shparlinski, Igor; Ward, Thomas (2003),
4161:Journal of the London Mathematical Society
3776:Combinatorics, automata, and number theory
2196:, formed by concatenating the elements of
1010:
984:
189:that may be drawn from that alphabet, and
118:real numbers are normal (meaning that the
4808:
4651:
4555:
4520:
4500:
4444:
4415:
4342:
4310:
4233:
4202:
4146:
4091:
4081:
3949:
3862:
3698:
2946:can be implemented with ILFSCs. An ILFSC
2285:
2252:
2208:implies that the set of primes is dense).
2128:{\displaystyle a_{1},a_{2},a_{3},\ldots }
1961:
1875:
1462:
159:are normal, but a proof remains elusive.
4794:
3803:
3686:
2930:to the output string and final state of
2874:is information lossless if the function
2189:{\displaystyle 0.a_{1}a_{2}a_{3}\ldots }
981:if it is not simply normal in any base.
114:While a general proof can be given that
4535:IEEE Transactions on Information Theory
4423:
4129:
4099:
3710:
3650:
3571:
3556:
3541:
3517:
3502:
3490:
3148:Connection to equidistributed sequences
3074:{\displaystyle |C(S\upharpoonright n)|}
1968:{\displaystyle A\subseteq \mathbb {N} }
1034:
16:Number with all digits equally frequent
4872:
4738:
4659:
4326:
3888:"Random generators and normal numbers"
3614:
1369:
4851:
4374:
4034:
2828:{\displaystyle d(S\upharpoonright n)}
2298:{\displaystyle k\in \mathbb {Z} ^{+}}
1352:
994:
237:denote the number of times the digit
4801:Institute of Mathematical Statistics
4508:
4290:
3957:
3674:
3662:
3449:
4715:Mathematical Association of America
4687:Publicationes Mathematicae Debrecen
4218:(1952), "Note on normal decimals",
3436:The only bases considered here are
2835:is the amount of money the gambler
2495:Connection to finite-state machines
13:
4565:
4377:Problems in analytic number theory
4187:(1946), "Note on normal numbers",
3405:
3251:
3208:
3144:replacing finite-state machines).
3081:is the number of digits output by
2975:
2901:
2888:
2784:
2758:
2711:
2573:
2553:
2533:
2486:: adding, removing, or changing a
1561:
1480:
1294:Bailey and Crandall (
1157:in base 10, is normal in base 10.
459:
378:be the number of times the string
280:
14:
4901:
4844:
3921:Becher, V.; Figueira, S. (2002),
3227:modulo 1, or equivalently, using
2938:. Compression techniques such as
1188:with real coefficients such that
1172:Nakai and Shiokawa (
550:denotes the length of the string
4713:, vol. 29, Washington, DC:
3138:algorithmically random sequences
2329:if and only if blocks of length
195:the set of finite sequences, or
4221:Canadian Journal of Mathematics
4204:10.1090/S0002-9904-1946-08657-7
3577:
3331:
2597:{\displaystyle \Sigma =\{0,1\}}
2337:expansion with equal frequency.
2005:and for all sufficiently large
1914:≠ 0 is a rational number, then
893:is one whose expansion in base
71:if, for every positive integer
4603:10.1080/10586458.2001.10504441
4433:Pacific Journal of Mathematics
4298:Pacific Journal of Mathematics
3907:10.1080/10586458.2002.10504704
3530:Adamczewski & Bugeaud 2010
3467:
3430:
3248:
3067:
3063:
3057:
3051:
3044:
3007:
3003:
2997:
2991:
2984:
2972:
2926:, mapping the input string of
2897:
2822:
2816:
2810:
2778:
2772:
2766:
2755:
2715:
2707:
2640:
2628:
2484:closed under finite variations
2139:expansions of the elements of
2049:
2017:
1306:-normal numbers by perturbing
1153:obtained by concatenating the
1150:0.149162536496481100121144...,
1069:obtained by concatenating the
849:) every number normal in base
520:
512:
489:
477:
456:
310:
298:
277:
162:
79:digits long have density
1:
4711:Carus Mathematical Monographs
4677:, Singapore, pp. 113–119
4653:10.1090/S0002-9939-06-08551-0
4330:American Mathematical Monthly
4267:American Mathematical Society
4009:Billingsley, Patrick (2012),
3942:10.1016/S0304-3975(01)00170-0
3762:
3420:is equidistributed modulo 1.
2692:{\displaystyle q_{1}=1-q_{0}}
2317:expansion corresponds to the
1851:
660:; etc. Roughly speaking, the
209:be such a sequence. For each
4070:Theoretical Computer Science
3930:Theoretical Computer Science
3855:10.1080/10586458.2012.665333
3807:Soviet Mathematics - Doklady
3334: for all integers
2839:has after reading the first
1998:{\displaystyle \alpha <1}
1848:and is absolutely abnormal.
1230:is any function of the form
48:if its infinite sequence of
7:
4781:"Normal numbers are normal"
4634:"A strong hot spot theorem"
3735:
3473:ω is the smallest infinite
2262:A number is normal in base
800:expansion for each integer
429:if, for all finite strings
100:or sequence is "favored".
10:
4906:
4819:10.1214/074921706000000248
4375:Murty, Maruti Ram (2007),
4106:Cambridge University Press
3780:Cambridge University Press
3462:Bailey & Crandall 2002
907:. A number normal in base
785:) if it is normal in base
747:is simply normal and that
650:each occur with frequency
608:each occur with frequency
582:each occur with frequency
122:of non-normal numbers has
56:digit values has the same
18:
4790:: 15, continued pp. 27–31
4705:Ergodic theory of numbers
4701:; Kraaikamp, Cor (2002),
4093:10.1016/j.tcs.2005.09.040
3983:Mathematische Zeitschrift
2722:{\displaystyle |\Sigma |}
2646:{\displaystyle q_{0}\in }
2482:All normal sequences are
2223:is a substring of length
1161: and Erdős (
1017:) proved that there is a
668:in any given position in
126:zero), this proof is not
64:. A number is said to be
21:normal number (computing)
4590:Experimental Mathematics
4548:10.1109/TIT.1978.1055934
4379:(2 ed.), Springer,
4293:"Note on normal numbers"
3895:Experimental Mathematics
3843:Experimental Mathematics
3774:; Rigo, Michael (eds.),
3423:
3130:LZ compression algorithm
3085:after reading the first
2736:on an infinite sequence
2231:: e.g. the first length-
1933:{\displaystyle x\cdot a}
1119:prime expressed in base
783:absolutely normal number
719:positional number system
352:be any finite string in
185:the set of all infinite
4446:10.2140/pjm.1960.10.661
4417:10.4064/aa-62-3-271-284
4312:10.2140/pjm.1957.7.1163
4174:10.1112/jlms/s1-8.4.254
4135:Colloquium Mathematicum
4011:Probability and measure
3752:Infinite monkey theorem
3115:Ziv and Lempel showed:
2856:finite-state compressor
2559:{\displaystyle \Sigma }
2539:{\displaystyle \Sigma }
1940:is also normal in base
1062:Copeland–Erdős constant
1045:Champernowne's constant
985:Properties and examples
923:if and only if the set
390:digits of the sequence
245:digits of the sequence
103:A number is said to be
75:, all possible strings
4235:10.4153/CJM-1952-005-3
4100:Bugeaud, Yann (2012),
3414:
3350:
3292:
3217:
3075:
3030:
2920:
2829:
2794:
2723:
2693:
2647:
2598:
2560:
2540:
2469:
2404:
2299:
2190:
2129:
2070:
1999:
1969:
1934:
1889:
1836:
1565:
1520:
1184:) is any non-constant
1089:by the concatenation
899:absolutely disjunctive
773:is normal. The number
727:simply normal in base
664:of finding the string
532:
336:
4693:(Supplement): 619–623
4148:10.4064/cm-7-1-95-101
3958:Beck, József (2009),
3742:Champernowne constant
3699:Ziv & Lempel 1978
3415:
3351:
3266:
3218:
3076:
3031:
2953:an infinite sequence
2921:
2830:
2795:
2724:
2694:
2648:
2608:bets some percentage
2599:
2561:
2541:
2501:finite-state machines
2470:
2405:
2300:
2211:A sequence is normal
2191:
2130:
2071:
2000:
1970:
1935:
1895:if the complement of
1890:
1837:
1545:
1521:
1035:every rational number
1007:Wacław Sierpiński
975:absolutely non-normal
533:
337:
241:appears in the first
95:). Even though there
91:) or rolls of a die (
4885:Sets of real numbers
4803:, pp. 225–236,
4291:Long, C. T. (1957),
4259:Recurrence sequences
4255:van der Poorten, Alf
3831:Borwein, Jonathan M.
3782:, pp. 410–451,
3757:The Library of Babel
3366:
3237:
3167:
3040:
2961:
2878:
2804:
2744:
2703:
2657:
2612:
2604:, the current state
2570:
2550:
2530:
2517:finite-state gambler
2418:
2355:
2274:
2243:, the second length-
2206:prime number theorem
2200:, is normal in base
2147:
2080:
2013:
1983:
1951:
1918:
1864:
1531:
1378:
1300:uncountably infinite
1222:, is normal in base
1218:) expressed in base
1159:Harold Davenport
1123:, is normal in base
999:Borel–Cantelli lemma
881:disjunctive sequence
445:
394:. (For instance, if
266:
4777:Khoshnevisan, Davar
4429:"On normal numbers"
4156:Champernowne, D. G.
3835:Calude, Cristian S.
3726:Everest et al. 2003
3409:
3212:
2944:Shannon–Fano coding
2333:appear in its base
1363:eventually periodic
1298:) show an explicit
979:absolutely abnormal
969:. For a given base
901:or is said to be a
687:greater than 1 and
4890:Irrational numbers
4853:Weisstein, Eric W.
4557:10338.dmlcz/142945
4470:10.1007/BF00289514
4054:10.1007/BF03019651
3995:10.1007/BF01201350
3979:Besicovitch, A. S.
3747:De Bruijn sequence
3639:Bailey et al. 2012
3410:
3369:
3346:
3255:
3213:
3170:
3156:is normal in base
3071:
3026:
2979:
2916:
2825:
2790:
2762:
2719:
2689:
2643:
2594:
2556:
2536:
2465:
2400:
2321:digit in its base
2295:
2186:
2143:, then the number
2125:
2066:
1995:
1965:
1930:
1906:is normal in base
1885:
1832:
1830:
1821:
1796:
1777:0.6562499999956991
1516:
1511:
1353:Non-normal numbers
1075:A. H. Copeland
853:is normal in base
789:for every integer
554:. In other words,
528:
463:
332:
284:
132:Chaitin's constant
107:(sometimes called
4502:10.24033/bsmf.977
4386:978-0-387-72349-5
4131:Cassels, J. W. S.
4115:978-0-521-11169-0
3971:978-0-8218-4756-5
3789:978-0-521-51597-9
3335:
3264:
3240:
3231:, if and only if
3095:compression ratio
3015:
2964:
2860:state transitions
2747:
1844:Then α is a
1781:
1779:
1753:
1705:
1679:
1653:
1627:
1598:
1447:
1287:) > 0 for all
1196:) > 0 for all
1176:) proved that if
1001:, he proved that
781:(or sometimes an
679:Suppose now that
526:
496:
448:
330:
317:
269:
109:absolutely normal
4897:
4866:
4865:
4839:
4812:
4791:
4785:
4772:
4735:
4708:
4694:
4678:
4675:World Scientific
4656:
4655:
4646:(9): 2495–2501,
4621:
4619:
4613:, archived from
4586:
4560:
4559:
4517:
4505:
4504:
4480:
4458:Acta Informatica
4449:
4448:
4420:
4419:
4403:Acta Arithmetica
4389:
4371:
4346:
4323:
4314:
4305:(2): 1163–1165,
4287:
4246:
4237:
4207:
4206:
4176:
4151:
4150:
4126:
4096:
4095:
4085:
4064:
4031:
4005:
3974:
3954:
3953:
3936:(1–2): 947–958,
3927:
3917:
3892:
3875:
3866:
3827:Bailey, David H.
3822:
3800:
3729:
3723:
3714:
3708:
3702:
3696:
3690:
3684:
3678:
3672:
3666:
3660:
3654:
3648:
3642:
3636:
3630:
3627:Billingsley 2012
3624:
3618:
3612:
3606:
3604:
3591:
3581:
3575:
3569:
3560:
3554:
3545:
3539:
3533:
3527:
3521:
3515:
3506:
3500:
3494:
3488:
3482:
3471:
3465:
3459:
3453:
3447:
3441:
3434:
3419:
3417:
3416:
3411:
3408:
3403:
3392:
3388:
3387:
3386:
3355:
3353:
3352:
3347:
3336:
3333:
3324:
3323:
3319:
3318:
3291:
3280:
3265:
3257:
3254:
3229:Weyl's criterion
3222:
3220:
3219:
3214:
3211:
3206:
3195:
3194:
3190:
3186:
3185:
3080:
3078:
3077:
3072:
3070:
3047:
3035:
3033:
3032:
3027:
3016:
3011:
3010:
2987:
2981:
2978:
2925:
2923:
2922:
2917:
2909:
2908:
2896:
2895:
2834:
2832:
2831:
2826:
2799:
2797:
2796:
2791:
2761:
2728:
2726:
2725:
2720:
2718:
2710:
2698:
2696:
2695:
2690:
2688:
2687:
2669:
2668:
2652:
2650:
2649:
2644:
2624:
2623:
2603:
2601:
2600:
2595:
2565:
2563:
2562:
2557:
2545:
2543:
2542:
2537:
2505:regular language
2474:
2472:
2471:
2466:
2452:
2451:
2439:
2438:
2409:
2407:
2406:
2401:
2393:
2392:
2380:
2379:
2367:
2366:
2309:block of length
2304:
2302:
2301:
2296:
2294:
2293:
2288:
2195:
2193:
2192:
2187:
2182:
2181:
2172:
2171:
2162:
2161:
2134:
2132:
2131:
2126:
2118:
2117:
2105:
2104:
2092:
2091:
2075:
2073:
2072:
2067:
2065:
2064:
2052:
2020:
2004:
2002:
2001:
1996:
1974:
1972:
1971:
1966:
1964:
1939:
1937:
1936:
1931:
1894:
1892:
1891:
1886:
1884:
1883:
1878:
1846:Liouville number
1841:
1839:
1838:
1833:
1831:
1823:8528404201690728
1820:
1797:
1792:
1769:
1759:
1755:
1754:
1752:
1751:
1750:
1746:
1745:
1724:
1711:
1707:
1706:
1698:
1685:
1681:
1680:
1672:
1659:
1655:
1654:
1646:
1633:
1629:
1628:
1620:
1604:
1600:
1599:
1597:
1596:
1578:
1564:
1559:
1537:
1525:
1523:
1522:
1517:
1515:
1514:
1487:
1483:
1465:
1449:
1448:
1446:
1435:
1434:
1430:
1411:
1394:
1348:
1347:
1340:algebraic number
1320:
1319:
1308:Stoneham numbers
1057:in other bases.
972:
968:
967:
966:
961:
956:
941:
922:
919:is rich in base
918:
914:
911:is rich in base
910:
896:
892:
875:
864:
860:
856:
852:
848:
838:
825:
814:
810:
806:
799:
793:greater than 1.
792:
788:
776:
772:
758:if the sequence
756:
750:
746:
732:if the sequence
730:
724:
717:
713:
709:
690:
682:
671:
667:
659:
658:
654:
649:
645:
641:
637:
633:
629:
625:
621:
617:
616:
612:
607:
603:
599:
595:
591:
590:
586:
581:
577:
573:
571:
567:
557:
553:
549:
547:
537:
535:
534:
529:
527:
525:
524:
523:
515:
502:
497:
492:
476:
475:
465:
462:
438:
424:
420:
418:
404:
402:
393:
389:
381:
377:
357:
351:
347:
341:
339:
338:
333:
331:
323:
318:
313:
297:
296:
286:
283:
252:
248:
244:
240:
236:
216:
212:
208:
194:
184:
178:
170:
151:
145:
144:
124:Lebesgue measure
4905:
4904:
4900:
4899:
4898:
4896:
4895:
4894:
4870:
4869:
4856:"Normal number"
4847:
4842:
4829:
4810:math.DS/0608249
4783:
4762:
4725:
4630:Misiurewicz, M.
4617:
4584:
4578:Crandall, R. E.
4568:
4566:Further reading
4563:
4387:
4353:10.2307/2695618
4277:
4251:Everest, Graham
4197:(10): 857–860,
4181:Copeland, A. H.
4116:
4083:10.1.1.101.7244
4021:
3972:
3925:
3890:
3884:Crandall, R. E.
3790:
3772:Berthé, Valérie
3765:
3738:
3733:
3732:
3724:
3717:
3709:
3705:
3697:
3693:
3685:
3681:
3673:
3669:
3661:
3657:
3649:
3645:
3637:
3633:
3625:
3621:
3613:
3609:
3603:
3600:
3597:
3594:fractional part
3589:
3586:
3583:
3582:
3578:
3570:
3563:
3555:
3548:
3540:
3536:
3528:
3524:
3516:
3509:
3501:
3497:
3489:
3485:
3472:
3468:
3460:
3456:
3448:
3444:
3438:natural numbers
3435:
3431:
3426:
3404:
3393:
3382:
3378:
3374:
3370:
3367:
3364:
3363:
3332:
3314:
3310:
3297:
3293:
3281:
3270:
3256:
3244:
3238:
3235:
3234:
3225:equidistributed
3207:
3196:
3181:
3177:
3176:
3172:
3171:
3168:
3165:
3164:
3150:
3142:Turing machines
3119:
3113:
3066:
3043:
3041:
3038:
3037:
3006:
2983:
2982:
2980:
2968:
2962:
2959:
2958:
2904:
2900:
2891:
2887:
2879:
2876:
2875:
2866:with state set
2805:
2802:
2801:
2751:
2745:
2742:
2741:
2714:
2706:
2704:
2701:
2700:
2683:
2679:
2664:
2660:
2658:
2655:
2654:
2619:
2615:
2613:
2610:
2609:
2571:
2568:
2567:
2551:
2548:
2547:
2531:
2528:
2527:
2497:
2447:
2443:
2434:
2430:
2419:
2416:
2415:
2388:
2384:
2375:
2371:
2362:
2358:
2356:
2353:
2352:
2289:
2284:
2283:
2275:
2272:
2271:
2177:
2173:
2167:
2163:
2157:
2153:
2148:
2145:
2144:
2113:
2109:
2100:
2096:
2087:
2083:
2081:
2078:
2077:
2060:
2056:
2048:
2016:
2014:
2011:
2010:
1984:
1981:
1980:
1960:
1952:
1949:
1948:
1919:
1916:
1915:
1879:
1874:
1873:
1865:
1862:
1861:
1854:
1829:
1828:
1798:
1782:
1780:
1767:
1766:
1741:
1737:
1733:
1732:
1728:
1723:
1716:
1712:
1697:
1690:
1686:
1671:
1664:
1660:
1645:
1638:
1634:
1619:
1612:
1608:
1586:
1582:
1577:
1570:
1566:
1560:
1549:
1534:
1532:
1529:
1528:
1510:
1509:
1498:
1489:
1488:
1473:
1469:
1461:
1453:
1436:
1420:
1416:
1412:
1410:
1406:
1399:
1398:
1384:
1379:
1376:
1375:
1359:rational number
1355:
1345:
1343:
1317:
1315:
1278:
1273:> ... > β
1272:
1268:
1263:
1257:
1247:
1204:
1151:
1105:
1067:
1054:natural numbers
1050:
991:Émile Borel
987:
970:
964:
963:
959:
958:
954:
939:
935:
931:
928:
924:
920:
916:
912:
908:
894:
890:
874:
870:
866:
862:
858:
854:
850:
847:
843:
840:
837:
833:
830:
824:
820:
816:
812:
808:
804:
801:
797:
790:
786:
774:
771:
770:
766:
762:
759:
754:
753:normal in base
748:
745:
744:
740:
736:
733:
728:
722:
715:
711:
708:
707:
703:
699:
696:
688:
680:
669:
665:
656:
652:
651:
647:
643:
639:
635:
631:
627:
623:
619:
614:
610:
609:
605:
601:
597:
593:
588:
584:
583:
579:
575:
569:
565:
563:
555:
551:
546:
543:
541:
519:
511:
510:
506:
501:
471:
467:
466:
464:
452:
446:
443:
442:
437:
433:
430:
422:
416:
414:
413:
409:
406:
400:
398:
395:
391:
387:
379:
375:
371:
367:
366:
362:
359:
356:
353:
349:
345:
322:
292:
288:
287:
285:
273:
267:
264:
263:
250:
246:
242:
238:
234:
230:
226:
225:
221:
218:
214:
210:
207:
203:
200:
193:
190:
183:
180:
176:
168:
165:
149:
142:
140:
134:is normal (and
87:of coin flips (
82:
78:
74:
69:
66:normal in base
63:
58:natural density
55:
47:
24:
17:
12:
11:
5:
4903:
4893:
4892:
4887:
4882:
4868:
4867:
4846:
4845:External links
4843:
4841:
4840:
4827:
4792:
4773:
4760:
4736:
4723:
4695:
4679:
4657:
4622:
4597:(2): 175–190,
4569:
4567:
4564:
4562:
4561:
4542:(5): 530–536,
4518:
4514:Normal Numbers
4506:
4485:Sierpiński, W.
4481:
4464:(4): 345–359,
4454:Schnorr, C. P.
4450:
4439:(2): 661–672,
4421:
4410:(3): 271–284,
4390:
4385:
4372:
4337:(8): 746–754,
4324:
4288:
4275:
4263:Providence, RI
4247:
4208:
4177:
4168:(4): 254–260,
4152:
4127:
4114:
4097:
4076:(3): 392–406,
4065:
4032:
4019:
4006:
3975:
3970:
3955:
3918:
3901:(4): 527–546,
3876:
3849:(4): 375–384,
3823:
3801:
3788:
3766:
3764:
3761:
3760:
3759:
3754:
3749:
3744:
3737:
3734:
3731:
3730:
3728:, p. 127.
3715:
3703:
3691:
3679:
3667:
3655:
3653:, p. 113.
3643:
3631:
3619:
3607:
3601:
3598:
3587:
3584:
3576:
3561:
3546:
3534:
3532:, p. 413.
3522:
3520:, p. 102.
3507:
3495:
3483:
3475:ordinal number
3466:
3454:
3442:
3440:greater than 1
3428:
3427:
3425:
3422:
3407:
3402:
3399:
3396:
3391:
3385:
3381:
3377:
3373:
3345:
3342:
3339:
3330:
3327:
3322:
3317:
3313:
3309:
3306:
3303:
3300:
3296:
3290:
3287:
3284:
3279:
3276:
3273:
3269:
3263:
3260:
3253:
3250:
3247:
3243:
3210:
3205:
3202:
3199:
3193:
3189:
3184:
3180:
3175:
3161:if and only if
3149:
3146:
3117:
3111:
3103:
3102:
3099:limit inferior
3069:
3065:
3062:
3059:
3056:
3053:
3050:
3046:
3025:
3022:
3019:
3014:
3009:
3005:
3002:
2999:
2996:
2993:
2990:
2986:
2977:
2974:
2971:
2967:
2966:lim inf
2940:Huffman coding
2915:
2912:
2907:
2903:
2899:
2894:
2890:
2886:
2883:
2852:
2849:limit superior
2824:
2821:
2818:
2815:
2812:
2809:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2760:
2757:
2754:
2750:
2749:lim sup
2717:
2713:
2709:
2686:
2682:
2678:
2675:
2672:
2667:
2663:
2642:
2639:
2636:
2633:
2630:
2627:
2622:
2618:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2555:
2535:
2496:
2493:
2492:
2491:
2480:
2464:
2461:
2458:
2455:
2450:
2446:
2442:
2437:
2433:
2429:
2426:
2423:
2399:
2396:
2391:
2387:
2383:
2378:
2374:
2370:
2365:
2361:
2345:
2338:
2292:
2287:
2282:
2279:
2260:
2213:if and only if
2209:
2185:
2180:
2176:
2170:
2166:
2160:
2156:
2152:
2124:
2121:
2116:
2112:
2108:
2103:
2099:
2095:
2090:
2086:
2063:
2059:
2055:
2051:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2019:
1994:
1991:
1988:
1963:
1959:
1956:
1945:
1929:
1926:
1923:
1900:
1899:has measure 0.
1882:
1877:
1872:
1869:
1853:
1850:
1827:
1824:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1795:
1791:
1788:
1785:
1778:
1775:
1772:
1770:
1768:
1765:
1762:
1758:
1749:
1744:
1740:
1736:
1731:
1727:
1722:
1719:
1715:
1710:
1704:
1701:
1696:
1693:
1689:
1684:
1678:
1675:
1670:
1667:
1663:
1658:
1652:
1649:
1644:
1641:
1637:
1632:
1626:
1623:
1618:
1615:
1611:
1607:
1603:
1595:
1592:
1589:
1585:
1581:
1576:
1573:
1569:
1563:
1558:
1555:
1552:
1548:
1544:
1541:
1538:
1536:
1513:
1508:
1505:
1502:
1499:
1497:
1494:
1491:
1490:
1486:
1482:
1479:
1476:
1472:
1468:
1464:
1460:
1457:
1454:
1452:
1445:
1442:
1439:
1433:
1429:
1426:
1423:
1419:
1415:
1409:
1405:
1404:
1402:
1397:
1393:
1390:
1387:
1383:
1354:
1351:
1274:
1270:
1266:
1253:
1245:
1232:
1206:where is the
1202:
1155:square numbers
1149:
1091:
1065:
1048:
986:
983:
937:
933:
929:
926:
872:
868:
845:
841:
835:
831:
822:
818:
802:
768:
764:
763:
760:
742:
738:
737:
734:
705:
701:
700:
697:
544:
522:
518:
514:
509:
505:
500:
495:
491:
488:
485:
482:
479:
474:
470:
461:
458:
455:
451:
435:
431:
411:
410:
407:
396:
373:
369:
364:
363:
360:
354:
329:
326:
321:
316:
312:
309:
306:
303:
300:
295:
291:
282:
279:
276:
272:
249:. We say that
232:
228:
223:
222:
219:
205:
201:
191:
181:
164:
161:
80:
76:
72:
67:
61:
53:
45:
34:is said to be
15:
9:
6:
4:
3:
2:
4902:
4891:
4888:
4886:
4883:
4881:
4880:Number theory
4878:
4877:
4875:
4863:
4862:
4857:
4854:
4849:
4848:
4838:
4834:
4830:
4828:0-940600-64-1
4824:
4820:
4816:
4811:
4806:
4802:
4798:
4793:
4789:
4782:
4778:
4774:
4771:
4767:
4763:
4761:1-56881-162-4
4757:
4753:
4749:
4745:
4741:
4737:
4734:
4730:
4726:
4724:0-88385-034-6
4720:
4716:
4712:
4707:
4706:
4700:
4699:Dajani, Karma
4696:
4692:
4688:
4684:
4680:
4676:
4672:
4671:
4666:
4665:Salomaa, Arto
4662:
4658:
4654:
4649:
4645:
4641:
4640:
4635:
4631:
4627:
4626:Bailey, D. H.
4623:
4620:on 2008-11-23
4616:
4612:
4608:
4604:
4600:
4596:
4592:
4591:
4583:
4579:
4575:
4574:Bailey, D. H.
4571:
4570:
4558:
4553:
4549:
4545:
4541:
4537:
4536:
4531:
4527:
4523:
4519:
4515:
4511:
4507:
4503:
4498:
4494:
4490:
4486:
4482:
4479:
4475:
4471:
4467:
4463:
4459:
4455:
4451:
4447:
4442:
4438:
4434:
4430:
4426:
4422:
4418:
4413:
4409:
4405:
4404:
4399:
4395:
4391:
4388:
4382:
4378:
4373:
4370:
4366:
4362:
4358:
4354:
4350:
4345:
4340:
4336:
4332:
4331:
4325:
4322:
4318:
4313:
4308:
4304:
4300:
4299:
4294:
4289:
4286:
4282:
4278:
4276:0-8218-3387-1
4272:
4268:
4264:
4260:
4256:
4252:
4248:
4245:
4241:
4236:
4231:
4227:
4223:
4222:
4217:
4213:
4212:Davenport, H.
4209:
4205:
4200:
4196:
4192:
4191:
4186:
4182:
4178:
4175:
4171:
4167:
4163:
4162:
4157:
4153:
4149:
4144:
4140:
4136:
4132:
4128:
4125:
4121:
4117:
4111:
4107:
4103:
4098:
4094:
4089:
4084:
4079:
4075:
4071:
4066:
4063:
4059:
4055:
4051:
4047:
4043:
4042:
4037:
4033:
4030:
4026:
4022:
4020:9781118122372
4016:
4012:
4007:
4004:
4000:
3996:
3992:
3988:
3984:
3980:
3976:
3973:
3967:
3963:
3962:
3956:
3952:
3947:
3943:
3939:
3935:
3931:
3924:
3919:
3916:
3912:
3908:
3904:
3900:
3896:
3889:
3885:
3881:
3880:Bailey, D. H.
3877:
3874:
3870:
3865:
3860:
3856:
3852:
3848:
3844:
3840:
3836:
3832:
3828:
3824:
3821:
3817:
3813:
3809:
3808:
3802:
3799:
3795:
3791:
3785:
3781:
3777:
3773:
3768:
3767:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3739:
3727:
3722:
3720:
3713:, p. 89.
3712:
3707:
3700:
3695:
3688:
3687:Agafonov 1968
3683:
3676:
3671:
3664:
3659:
3652:
3647:
3640:
3635:
3628:
3623:
3616:
3611:
3595:
3580:
3574:, p. 92.
3573:
3568:
3566:
3558:
3553:
3551:
3543:
3538:
3531:
3526:
3519:
3514:
3512:
3505:, p. 79.
3504:
3499:
3493:, p. 78.
3492:
3487:
3480:
3476:
3470:
3463:
3458:
3451:
3446:
3439:
3433:
3429:
3421:
3400:
3397:
3394:
3389:
3383:
3379:
3375:
3371:
3361:
3356:
3343:
3340:
3337:
3328:
3325:
3320:
3315:
3311:
3307:
3304:
3301:
3298:
3294:
3288:
3285:
3282:
3277:
3274:
3271:
3267:
3261:
3258:
3245:
3232:
3230:
3226:
3203:
3200:
3197:
3191:
3187:
3182:
3178:
3173:
3163:the sequence
3162:
3159:
3155:
3145:
3143:
3139:
3133:
3131:
3127:
3125:
3116:
3110:
3109:. Therefore:
3108:
3100:
3096:
3092:
3088:
3084:
3060:
3054:
3048:
3023:
3020:
3017:
3012:
3000:
2994:
2988:
2969:
2956:
2952:
2949:
2945:
2941:
2937:
2933:
2929:
2913:
2910:
2905:
2892:
2884:
2881:
2873:
2869:
2865:
2861:
2857:
2853:
2850:
2846:
2842:
2838:
2819:
2813:
2807:
2787:
2781:
2775:
2769:
2763:
2752:
2739:
2735:
2732:
2684:
2680:
2676:
2673:
2670:
2665:
2661:
2637:
2634:
2631:
2625:
2620:
2616:
2607:
2588:
2585:
2582:
2576:
2525:
2524:
2521:finite-state
2518:
2514:
2513:
2512:
2509:
2506:
2502:
2489:
2485:
2481:
2478:
2462:
2456:
2453:
2448:
2444:
2440:
2435:
2431:
2424:
2421:
2413:
2397:
2394:
2389:
2385:
2381:
2376:
2372:
2368:
2363:
2359:
2350:
2346:
2343:
2339:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2308:
2290:
2280:
2277:
2269:
2265:
2261:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2222:
2218:
2214:
2210:
2207:
2203:
2199:
2183:
2178:
2174:
2168:
2164:
2158:
2154:
2150:
2142:
2138:
2135:are the base-
2122:
2119:
2114:
2110:
2106:
2101:
2097:
2093:
2088:
2084:
2061:
2057:
2053:
2042:
2039:
2036:
2033:
2030:
2024:
2021:
2008:
1992:
1989:
1986:
1978:
1957:
1954:
1946:
1943:
1927:
1924:
1921:
1913:
1909:
1905:
1901:
1898:
1880:
1870:
1867:
1859:
1858:
1857:
1849:
1847:
1842:
1825:
1822:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1793:
1789:
1786:
1783:
1776:
1773:
1771:
1763:
1760:
1756:
1747:
1742:
1738:
1734:
1729:
1725:
1720:
1717:
1713:
1708:
1702:
1699:
1694:
1691:
1687:
1682:
1676:
1673:
1668:
1665:
1661:
1656:
1650:
1647:
1642:
1639:
1635:
1630:
1624:
1621:
1616:
1613:
1609:
1605:
1601:
1593:
1590:
1587:
1583:
1579:
1574:
1571:
1567:
1556:
1553:
1550:
1546:
1542:
1539:
1526:
1506:
1503:
1500:
1495:
1492:
1484:
1477:
1474:
1470:
1466:
1458:
1455:
1450:
1443:
1440:
1437:
1431:
1427:
1424:
1421:
1417:
1413:
1407:
1400:
1395:
1391:
1388:
1385:
1381:
1373:
1371:
1368:Martin (
1366:
1364:
1360:
1350:
1341:
1338:
1333:
1329:
1325:
1321:
1311:
1309:
1305:
1301:
1297:
1292:
1290:
1286:
1282:
1277:
1261:
1256:
1251:
1243:
1239:
1235:
1231:
1229:
1225:
1221:
1217:
1213:
1209:
1201:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1170:
1168:
1164:
1160:
1156:
1148:
1146:
1142:
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1103:
1099:
1095:
1090:
1088:
1084:
1080:
1077: and
1076:
1072:
1071:prime numbers
1064:
1063:
1058:
1055:
1047:
1046:
1042:
1040:
1036:
1032:
1028:
1023:
1020:
1016:
1012:
1008:
1004:
1000:
997:). Using the
996:
992:
982:
980:
976:
951:
949:
948:unit interval
945:
932:mod 1 :
906:
905:
900:
888:
887:
882:
877:
828:
794:
784:
780:
779:normal number
757:
731:
720:
694:
686:
677:
675:
663:
561:
538:
516:
507:
503:
498:
493:
486:
483:
480:
472:
468:
453:
440:
428:
386:in the first
385:
382:appears as a
342:
327:
324:
319:
314:
307:
304:
301:
293:
289:
274:
261:
260:
256:
255:simply normal
198:
188:
174:
160:
158:
157:
152:
146:
137:
133:
129:
125:
121:
117:
112:
110:
106:
101:
98:
94:
90:
84:
70:
59:
51:
44:
41:
37:
36:simply normal
33:
29:
22:
4859:
4796:
4787:
4751:
4744:Berndt, B.C.
4740:Harman, Glyn
4704:
4690:
4686:
4683:Calude, C.S.
4669:
4643:
4637:
4615:the original
4594:
4588:
4539:
4533:
4513:
4492:
4488:
4461:
4457:
4436:
4432:
4407:
4401:
4398:Shiokawa, I.
4376:
4344:math/0006089
4334:
4328:
4302:
4296:
4258:
4225:
4219:
4194:
4188:
4165:
4159:
4138:
4134:
4101:
4073:
4069:
4045:
4039:
4010:
3986:
3982:
3960:
3933:
3929:
3898:
3894:
3846:
3842:
3811:
3805:
3775:
3711:Bugeaud 2012
3706:
3694:
3682:
3670:
3658:
3651:Bugeaud 2012
3646:
3634:
3622:
3610:
3592:denotes the
3579:
3572:Bugeaud 2012
3557:Schmidt 1960
3542:Cassels 1959
3537:
3525:
3518:Bugeaud 2012
3503:Bugeaud 2012
3498:
3491:Bugeaud 2012
3486:
3469:
3457:
3445:
3432:
3359:
3357:
3233:
3157:
3153:
3151:
3134:
3122:
3120:
3114:
3104:
3090:
3086:
3082:
2954:
2950:
2947:
2931:
2927:
2871:
2867:
2863:
2855:
2844:
2840:
2836:
2737:
2733:
2730:
2605:
2520:
2516:
2510:
2498:
2483:
2476:
2411:
2348:
2347:A number is
2341:
2334:
2330:
2326:
2322:
2318:
2314:
2313:in its base
2310:
2306:
2267:
2263:
2248:
2244:
2240:
2236:
2232:
2228:
2224:
2220:
2216:
2201:
2197:
2140:
2136:
2006:
1976:
1941:
1911:
1907:
1903:
1896:
1855:
1843:
1703:152587890625
1527:
1374:
1367:
1356:
1312:
1303:
1293:
1288:
1284:
1280:
1275:
1264:
1259:
1254:
1249:
1241:
1237:
1233:
1227:
1223:
1219:
1215:
1211:
1208:integer part
1205:
1197:
1193:
1189:
1181:
1177:
1171:
1166:
1152:
1144:
1140:
1136:
1124:
1120:
1116:
1112:
1108:
1106:
1101:
1097:
1093:
1086:
1068:
1059:
1051:
1043:
1029:, is also a
1024:
988:
978:
974:
952:
902:
898:
884:
878:
857:. For bases
795:
782:
778:
777:is called a
752:
726:
714:in the base
678:
539:
441:
426:
343:
262:
254:
171:be a finite
166:
154:
136:uncomputable
128:constructive
113:
108:
104:
102:
96:
85:
65:
35:
25:
4510:Wall, D. D.
4495:: 125–132,
4425:Schmidt, W.
4048:: 247–271,
3989:: 146–156,
3814:: 324–325,
3615:Martin 2001
3479:Kleene star
1979:(for every
1129:Besicovitch
1027:uncountable
886:rich number
693:real number
662:probability
163:Definitions
32:real number
28:mathematics
4874:Categories
4837:1130.11041
4770:1062.11052
4748:Boston, N.
4733:1033.11040
4661:Calude, C.
4526:Lempel, A.
4369:1036.11035
4321:0080.03604
4285:1033.11006
4141:: 95–101,
4124:1260.11001
3864:2292/10566
3820:0242.94040
3798:1271.11073
3763:References
3477:; is the
3089:digits of
2951:compresses
2843:digits of
2523:martingale
2344:normality.
1852:Properties
1337:irrational
1186:polynomial
1079:Paul Erdős
1019:computable
1003:almost all
560:asymptotic
348:. Now let
116:almost all
4861:MathWorld
4394:Nakai, Y.
4228:: 58–63,
4216:Erdős, P.
4185:Erdős, P.
4078:CiteSeerX
4062:184479669
4036:Borel, E.
4029:780289503
4003:123025145
3675:Long 1957
3663:Wall 1949
3450:Beck 2009
3406:∞
3380:β
3341:≥
3302:π
3286:−
3268:∑
3252:∞
3249:→
3209:∞
3152:A number
3058:↾
2998:↾
2976:∞
2973:→
2911:×
2906:∗
2902:Σ
2898:→
2893:∗
2889:Σ
2817:↾
2785:∞
2773:↾
2759:∞
2756:→
2712:Σ
2677:−
2626:∈
2574:Σ
2554:Σ
2534:Σ
2457:…
2425:∈
2398:⋯
2281:∈
2247:block is
2235:block in
2184:…
2123:…
2062:α
2054:≥
2037:…
2025:∩
1987:α
1958:⊆
1925:⋅
1871:⊆
1826:…
1794:⏟
1787:…
1761:…
1721:−
1695:−
1669:−
1643:−
1617:−
1575:−
1562:∞
1547:∏
1540:α
1481:∞
1467:∩
1459:∈
1441:−
1425:−
1302:class of
1279:≥ 0, and
1252:+ ... + α
1115:) is the
829:(so that
460:∞
457:→
384:substring
344:for each
281:∞
278:→
187:sequences
179:-digits,
4779:(2006),
4667:(eds.),
4632:(2006),
4580:(2001),
4528:(1978),
4512:(1949),
4478:31943843
4427:(1960),
4244:14621341
3886:(2002),
3873:17273684
3736:See also
3107:converse
2734:succeeds
2519:(a.k.a.
2479:-normal.
2414:for all
2270:for all
1291:> 0.
1039:comeagre
1031:null set
889:in base
827:rational
419:, 8) = 3
401:01010101
358:and let
173:alphabet
60: 1/
4611:2694690
4522:Ziv, J.
4361:2695618
3915:8944421
3124:entropy
1344:√
1316:√
1131: (
1104:(3)...,
1081: (
1009: (
993: (
962:⁄
946:in the
904:lexicon
685:integer
655:⁄
613:⁄
587:⁄
405:, then
257:if the
197:strings
141:√
40:integer
4835:
4825:
4768:
4758:
4731:
4721:
4609:
4476:
4383:
4367:
4359:
4319:
4283:
4273:
4242:
4122:
4112:
4080:
4060:
4027:
4017:
4001:
3968:
3913:
3871:
3818:
3796:
3786:
3093:. The
3036:where
2800:where
2488:finite
2215:every
2076:) and
1330:, and
1269:> β
1240:) = α·
1203:0....,
1107:where
871:/ log
821:/ log
683:is an
674:random
646:, and
604:, and
548:|
542:|
540:where
427:normal
199:. Let
153:, and
105:normal
93:base 6
89:binary
50:digits
38:in an
4805:arXiv
4784:(PDF)
4618:(PDF)
4607:S2CID
4585:(PDF)
4474:S2CID
4357:JSTOR
4339:arXiv
4240:S2CID
4058:S2CID
3999:S2CID
3926:(PDF)
3911:S2CID
3891:(PDF)
3869:S2CID
3590:mod 1
3424:Notes
3097:(the
2934:, is
2847:(see
2217:block
1977:dense
1790:99999
1784:99999
1328:ln(2)
944:dense
865:with
815:with
691:is a
259:limit
4823:ISBN
4756:ISBN
4719:ISBN
4381:ISBN
4271:ISBN
4110:ISBN
4025:OCLC
4015:ISBN
3966:ISBN
3784:ISBN
3126:rate
3018:<
2395:<
2382:<
2369:<
2257:2005
2253:1978
1990:<
1910:and
1370:2001
1296:2002
1174:1992
1163:1952
1143:) =
1133:1935
1083:1946
1060:The
1015:2002
1011:1917
995:1909
867:log
861:and
839:and
817:log
811:and
578:and
217:let
167:Let
97:will
43:base
30:, a
4833:Zbl
4815:doi
4766:Zbl
4729:Zbl
4648:doi
4644:134
4599:doi
4552:hdl
4544:doi
4497:doi
4466:doi
4441:doi
4412:doi
4365:Zbl
4349:doi
4335:108
4317:Zbl
4307:doi
4281:Zbl
4230:doi
4199:doi
4170:doi
4143:doi
4120:Zbl
4088:doi
4074:349
4050:doi
3991:doi
3946:hdl
3938:doi
3934:270
3903:doi
3859:hdl
3851:doi
3816:Zbl
3794:Zbl
3596:of
3242:lim
3223:is
2942:or
2936:1–1
2239:is
1975:is
1947:If
1902:If
1818:559
1812:291
1806:747
1357:No
1244:+ α
1210:of
1100:(2)
1096:(1)
977:or
942:is
805:≥ 2
751:is
725:is
710:of
648:111
644:110
640:101
636:100
632:011
628:010
624:001
620:000
574:),
450:lim
425:is
421:.)
417:010
403:...
271:lim
253:is
213:in
175:of
120:set
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4746:;
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4691:54
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4636:,
4628:;
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4265::
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3833:;
3829:;
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3792:,
3718:^
3564:^
3549:^
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2957:if
2870:,
2854:A
2851:).
2515:A
2259:).
2151:0.
2009:,
1800:23
1743:15
1677:64
1365:.
1326:,
1322:,
1310:.
1147:,
1127:.
1092:0.
950:.
936:∈
879:A
844:=
834:=
767:,
741:,
704:,
676:.
642:,
638:,
634:,
630:,
626:,
622:,
618:;
606:11
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600:,
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594:00
592:;
439:,
434:∈
399:=
372:,
231:,
204:∈
147:,
83:.
4864:.
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4807::
4650::
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4341::
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3701:.
3689:.
3677:.
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3641:.
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3605:.
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3481:.
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3321:x
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3299:2
3295:e
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3278:0
3275:=
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3262:n
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3198:k
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3188:x
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3174:(
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2995:S
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2841:n
2837:d
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2820:n
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2808:d
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2776:n
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2767:(
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2753:n
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2638:1
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2632:0
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2621:0
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2606:q
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2477:b
2463:.
2460:}
2454:,
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2436:1
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2428:{
2422:m
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2390:3
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2331:k
2327:b
2323:b
2319:n
2315:b
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2307:n
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2278:k
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2249:S
2245:k
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2233:k
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2221:k
2202:b
2198:A
2179:3
2175:a
2169:2
2165:a
2159:1
2155:a
2141:A
2137:b
2120:,
2115:3
2111:a
2107:,
2102:2
2098:a
2094:,
2089:1
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2058:n
2050:|
2046:}
2043:n
2040:,
2034:,
2031:1
2028:{
2022:A
2018:|
2007:n
1993:1
1962:N
1955:A
1944:.
1942:b
1928:a
1922:x
1912:a
1908:b
1904:x
1897:X
1881:+
1876:R
1868:X
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1809:,
1803:,
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1764:=
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1739:5
1735:(
1730:6
1726:1
1718:1
1714:(
1709:)
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1591:m
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1471:[
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1418:(
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1401:{
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1392:)
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1386:(
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1324:π
1318:2
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1289:x
1285:x
1283:(
1281:f
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1271:2
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1234:f
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1192:(
1190:f
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1180:(
1178:f
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1121:b
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1111:(
1109:f
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971:b
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960:1
955:b
940:}
938:N
934:n
930:b
927:x
925:{
921:b
917:x
913:b
909:b
895:b
891:b
873:s
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761:S
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702:x
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689:x
681:b
670:S
666:w
657:8
653:1
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580:1
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572:}
570:1
568:,
566:0
564:{
556:S
552:w
545:w
521:|
517:w
513:|
508:b
504:1
499:=
494:n
490:)
487:n
484:,
481:w
478:(
473:S
469:N
454:n
436:Σ
432:w
423:S
415:(
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392:S
388:n
380:w
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374:n
370:w
368:(
365:S
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350:w
346:a
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320:=
315:n
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294:S
290:N
275:n
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243:n
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233:n
229:a
227:(
224:S
220:N
215:Σ
211:a
206:Σ
202:S
192:Σ
182:Σ
177:b
169:Σ
156:e
150:π
143:2
81:b
77:n
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68:b
62:b
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23:.
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