Champernowne constant

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The Champernowne constant (named after D. G. Champernowne, and also called Mahler’s number, since Kurt Mahler proved that it is transcendental) is formed by concatenating the positive integers (in base 10) after the decimal point.

Decimal expansion of Champernowne constant

The decimal expansion of Champernowne constant is

C = 0.1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980...

giving the sequence of decimal digits (A033307)

{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, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, ...}

Continued fraction for Champernowne constant

The simple continued fraction for Champernowne constant is

     

C = 0 +  1 8 +  1 9 +  1 1 +  1 149083 +  1 ⋱

giving the sequence of integer part and partial quotients (A030167)

{0, 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 4575401113910310764836466282429561185996039397104575550006620043930902626592563149379532077471286563138641209375503552094607183089984575801469863148833592141783010987, 6, ...}

Base b Champernowne constant

The base b Champernowne constant is formed by concatenating the positive integers in base

b, b   ≥   2,

after the “binary point” (for base 2), or “ternary point” (for base 3), etc.

Base

b

Champernowne constant

b	Cb	Base b expansion Base 10 expansion Continued fraction (for base 10 expansion, integer part and partial quotients written in base 10)	A-numbers
2	C2	0.110111001011101111000100110101011110011011110111110000100011001010011101001010110110101111... 2 0.8622401258680545715577902832493945785657647427682990945160712145573067405905164580420384414386181334...10 {0, 1, 6, 3, 1, 6, 5, 3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, 7, 2, 4, 1, 2, 1, 2, 1, 1, 4534532, 1, 4, 5, 1, 2, 1, 7, 1, 16, 1, 4, 1, 5, 5, 1, 5, 1, 4, 1, 2, 1, 5, 3, 2, 38, 2, 12, 1, 15, 2, 6, 3, 30, 4682854730443938, 1, 1, 68, 1, ...}	A030190 A066716 A066717
3	C3	0.1210111220212210010110211011111212012112220020120221021121222022122210001001100210101011101210201021... 3 0.59895816753843399250017221792943659097820876867610593675478607547965184195280842055407211080527964157...10 {0, 1, 1, 2, 37, 1, 162, 1, 1, 1, 3, 1, 7, 1, 9, 2, 3, 1, 3068518062211324, 2, 1, 2, 6, 13, 1, 2, 1, 3, 1, 10, 1, 21, 1, 1, 4, 3, 577, 1, 1079268324684171943515797470873767312825026176345571319042096689270, ...}	A054635 A077771 A077772
4	C4	0.123101112132021222330313233100101102103110111112113120121122123130131132133200201202203210... 4 0.???...10 {?, ...}	A030373 A?????? A??????
5	C5	0.123410111213142021222324303132333440414243441001011021031041101111121131141201211221231241... 5 0.???...10 {?, ...}	A031219 A?????? A??????
6	C6	0.123451011121314152021222324253031323334354041424344455051525354551001011021031041051101111... 6 0.???...10 {?, ...}	A030548 A?????? A??????
7	C7	0.123456101112131415162021222324252630313233343536404142434445465051525354555660616263646566... 7 0.???...10 {?, ...}	A030998 A?????? A??????
8	C8	0.12345671011121314151617202122232425262730313233343536374041424344454647505152535455565760616263646 ... 8 0.???...10 {?, ...}	A054634 A?????? A??????
9	C9	0.123456781011121314151617182021222324252627283031323334353637384041424344454647485051525354... 9 0.???...10 {?, ...}	A031076 A?????? A??????
10	C10	0.123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475...10 0.123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475...10 {0, 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, ...}	A033307 A033307 A030167