Champernowne Number

Wikipedia states at https://en.wikipedia.org/wiki/Champernowne_constant
In mathematics, the Champernowne constant
C
10
is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933. The number is defined by concatenating the base 10 representations of the positive integers:
C
10
=123456789101112131415161718192021222324252627282930
Sequence A033307 in the OEIS at https://oeis.org/A033307.
Champernowne constants can also be constructed in other bases similarly; for example,
C
2
=
(0.11011100101110111...)
2
and
C
3
=
(0.12101112202122...)
3
We can create a function to do this.
Range[30]
Out[]=
{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}
IntegerDigits[Range[30]]
Out[]=
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{1,0},{1,1},{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{1,8},{1,9},{2,0},{2,1},{2,2},{2,3},{2,4},{2,5},{2,6},{2,7},{2,8},{2,9},{3,0}}
Catenate[IntegerDigits[Range[30]]]
Out[]=
{1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,1,3,1,4,1,5,1,6,1,7,1,8,1,9,2,0,2,1,2,2,2,3,2,4,2,5,2,6,2,7,2,8,2,9,3,0}
ToString/@Catenate[IntegerDigits[Range[30]]]
Out[]=
{1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,1,3,1,4,1,5,1,6,1,7,1,8,1,9,2,0,2,1,2,2,2,3,2,4,2,5,2,6,2,7,2,8,2,9,3,0}
Here the input form is different:
ToString/@Catenate[IntegerDigits[Range[30]]]//InputForm
Out[]//InputForm=
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1,
​
0, 1, 1, 1, 2, 1, 3, 1, 4, 1,
​
5, 1, 6, 1, 7, 1, 8, 1, 9, 2,
​
0, 2, 1, 2, 2, 2, 3, 2, 4, 2,
​
5, 2, 6, 2, 7, 2, 8, 2, 9, 3,
​
0}
StringJoin[ToString/@Catenate[IntegerDigits[Range[30]]]]
Out[]=
123456789101112131415161718192021222324252627282930
In[]:=
champernowneNumber[length_,base_:10]/;base>1&&IntegerQ[base]:=StringJoin[ToString/@Catenate[IntegerDigits[Range[length],base]]]
champernowneNumber[30]
Out[]=
123456789101112131415161718192021222324252627282930
champernowneNumber[30,2]
Out[]=
1101110010111011110001001101010111100110111101111100001000110010100111010010101101101011111000110011101011011111001110111110
The Champernowne word or Barbier word is the sequence of the digits of
C
10
obtained by writing it in base 10 and juxtaposing the digits:
champernowneNumber[30]
Out[]=
123456789101112131415161718192021222324252627282930
This is sequence A007376 in the OEIS.
This is what the function champernowneNumber generates.
More generally, a Champernowne sequence (sometimes also called a Champernowne word) is any sequence of digits obtained by concatenating all finite digit-strings (in any given base) in some recursive order. For instance, the binary Champernowne sequence in shortlex order is
0 1 00 01 10 11 000 001 ... (sequence A076478 in the OEIS)
where spaces (otherwise to be ignored) have been inserted just to show the strings being concatenated.
Tuples[{0,1},1]
Out[]=
{{0},{1}}
Tuples[{0,1},2]
Out[]=
{{0,0},{0,1},{1,0},{1,1}}
Table[Tuples[{0,1},length],{length,1,3}]
Out[]=
{{{0},{1}},{{0,0},{0,1},{1,0},{1,1}},{{0,0,0},{0,0,1},{0,1,0},{0,1,1},{1,0,0},{1,0,1},{1,1,0},{1,1,1}}}
Flatten[Table[Tuples[{0,1},length],{length,1,3}]]
Out[]=
{0,1,0,0,0,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,1,0,1,1,1,0,1,1,1}
ToString/@Flatten[Table[Tuples[{0,1},length],{length,1,3}]]
Out[]=
{0,1,0,0,0,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,1,0,1,1,1,0,1,1,1}
StringJoin[ToString/@Flatten[Table[Tuples[{0,1},length],{length,1,3}]]]
Out[]=
0100011011000001010011100101110111
StringJoin[ToString/@Flatten[Table[Tuples[Range[0,2-1],length],{length,1,3}]]]
Out[]=
0100011011000001010011100101110111
StringJoin[ToString/@Flatten[Table[Tuples[Range[0,3-1],length],{length,1,3}]]]
Out[]=
012000102101112202122000001002010011012020021022100101102110111112120121122200201202210211212220221222
In[]:=
ChampernowneSequenceInShortlexOrder[base_,lengthargument_]:=StringJoin[ToString/@Flatten[Table[Tuples[Range[0,base-1],length],{length,1,lengthargument}]]]
ChampernowneSequenceInShortlexOrder[2,3]
Out[]=
0100011011000001010011100101110111
You can see they are different:
The shortlex order and the Champernowne number are slightly different:
ChampernowneSequenceInShortlexOrder[10,2]
Out[]=
012345678900010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899
champernowneNumber[100]
Out[]=
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100
A real number x is said to be normal if its digits in every base follow a uniform distribution: all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc. A number x i said to be normal in base b if its digits in base b follow a uniform distribution.
Champernowne proved that
C
10
is normal in base 10, while Nakai and Shiokawa proved a more general theorem, a corollary of which is that
C
b
is normal in base
b
for any
b
. It is an open problem whether
C
k
is normal in bases
b!=k
. That is, it is not known, for example, if
C
10
expressed in base 9 is normal.
N[ChampernowneNumber[10],54]
Out[]=
0.123456789101112131415161718192021222324252627282930313
BaseForm[N[ChampernowneNumber[10],54],9]
Out[]//BaseForm=
0.108888888538230263265121113050277572014000015176608358870
9
Kurt Mahler showed that the constant is transcendental. The irrationality measure of
C
10
is
μ(
C
10
)=10
, and more generally
μ(
C
b
)=b
for any base
b>=2
.
The Champernowne word is a disjunctive sequence. A disjunctive sequence is an infinite sequence (over a finite alphabet of characters) in which every finite string appears as a substring.
The simple continued fraction of Champernowne’s constant does not terminate (because the constant is not rational) and is aperiodic (because it is not an irreducible quadratic). A simple continued fraction is a continued fraction where the denominator is 1. The simple continued fraction expansion of Champernowne’s constant exhibits extremely large terms appearing between many small ones.
Block[{$MaxExtraPrecision=20000},ContinuedFraction[ChampernowneNumber[10],40]]
Out[]=
{0,8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15,4575401113910310764836466282429561185996039397104575550006620043930902626592563149379532077471286563138641209375503552094607183089984575801469863148833592141783010987,6,1,1,21,1,9,1,1,2,3,1,7,2,1,83,1,156,4,58,8,54}
There is a related constant, the Copeland–Erdős constant. For more details, please visit https://en.wikipedia.org/wiki/Copeland%E2%80%93Erd%C5%91s_constant.
There is a built-in function to calculate the Champernowne number.
??ChampernowneNumber
Out[]=
Symbol
ChampernowneNumber[b] gives the base-b Champernowne number
C
b
.​ChampernowneNumber[] gives the base-10 Champernowne number.
Documentation
Web »
Attributes
{Constant,Listable,NHoldFirst,NumericFunction,Protected,ReadProtected}
Full Name
System`ChampernowneNumber
​
I handtyped the Champernowne number up to at least 2240. This was a way to have something to do when I was bored during a meeting.:
12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485868788899091929394959697989910010110210310410510610710810911011111211311411511611711811912012112212312412512612712812913013113213313413513613713813914014114214314414514614714814915015115215315415515615715815916016116216316416516616716816917017117217317417517617717817918018118218318418518618718818919019119219319419519619719819920020120220320420520620720820921021121221321421521621721821922022122222322422522622722822923023123223323423523623723823924024124224324424524624724824925025125225325425525625725825926026126226326426526626726826927027127227327427527627727827928028128228328428528628728828929029129229329429529629729829930030130230330430530630730830931031131231331431531631731831932032132232332432532632732832933033133233333433533633733833934034134234334434534634734834935035135235335435535635735835936036136236336436536636736836937037137237337437537637737837938038138238338438538638738838939039139239339439539639739839940040140240340440540640740840941041141241341441541641741841942042142242342442542642742842943043143243343443543643743843944044144244344444544644744844945045145245345445545645745845946046146246346446546646746846947047147247347447547647747847948048148248348448548648748848949049149249349449549649749849950050150250350450550650750850951051151251351451551651751851952052152252352452552652752852953053153253353453553653753853954054154254354454554654754854955055155255355455555655755855956056156256356456556656756856957057157257357457557657757857958058158258358458558658758858959059159259359459559659759859960060160260360460560660760860961061161261361461561661761861962062162262362462562662762862963063163263363463563663763863964064164264364464564664764864965065165265365465565665765865966066166266366466566666766866967067167267367467567667767867968068168268368468568668768868969069169269369469569669769869970070170270370470570670770870971071171271371471571671771871972072172272372472572672772872973073173273373473573673773873974074174274374474574674774874975075175275375475575675775875976076176276376476576676776876977077177277377477577677777877978078178278378478578678778878979079179279379479579679779879980080180280380480580680780880981081181281381481581681781881982082182282382482582682782882983083183283383483583683783883984084184284384484584684784884985085185285385485585685785885986086186286386486586686786886987087187287387487587687787887988088188288388488588688788888989089189289389489589689789889990090190290390490590690790890991091191291391491591691791891992092192292392492592692792892993093193293393493593693793893994094194294394494594694794894995095195295395495595695795895996096196296396496596696796896997097197297397497597697797897998098198298398498598698798898999099199299399499599699799899910001001100210031004100510061007100810091010101110121013101410151016101710181019102010211022102310241025102610271028102910301031103210331034103510361037103810391040104110421043104410451046104710481049105010511052105310541055105610571058105910601061106210631064106510661067106810691070107110721073107410751076107710781079108010811082108310841085108610871088108910901091109210931094109510961097109810991100110111021103110411051106110711081109111011111112111311141115111611171118111911201121112211231124112511261127112811291130113111321133113411351136113711381139114011411142114311441145114611471148114911501151115211531154115511561157115811591160116111621163116411651166116711681169117011711172117311741175117611771178117911801181118211831184118511861187118811891190119111921193119411951196119711981199120012011202120312041205120612071208120912101211121212131214121512161217121812191220122112221223122412251226122712281229123012311232123312341235123612371238123912401241124212431244124512461247124812491250125112521253125412551256125712581259126012611262126312641265126612671268126912701271127212731274127512761277127812791280128112821283128412851286128712881289129012911292129312941295129612971298129913001301130213031304130513061307130813091310131113121313131413151316131713181319132013211322132313241325132613271328132913301331133213331334133513361337133813391340134113421343134413451346134713481349135013511352135313541355135613571358135913601361136213631364136513661367136813691370137113721373137413751376137713781379138013811382138313841385138613871388138913901391139213931394139513961397139813991400140114021403140414051406140714081409141014111412141314141415141614171418141914201421142214231424142514261427142814291430143114321433143414351436143714381439144014411442144314441445144614471448144914501451145214531454145514561457145814591460146114621463146414651466146714681469147014711472147314741475147614771478147914801481148214831484148514861487148814891490149114921493149414951496149714981499150015011502150315041505150615071508150915101511151215131514151515161517151815191520152115221523152415251526152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In[]:=
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