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Topic review (newest first)
visit http://www.geocities.com/chengyuanwu/baseconverter.html for a base converter. removing the decimal point, i get A4AJ8AI
- John E. Franklin
- 2005-08-31 11:01:53
I have been noticing something logarithmic that someone taught me years ago. The number of digits in the base-26 number to be similar accuracy as the base-10 number is approximately the length of the base-10 number divided by the common log of 26. In other words, the base-26 number is shorter in digits by a factor of log(26) or 1.41. My latest run of the program ran slower because the whole numbers were over 2400 digits long. I haven't read the documentation of Just BASIC enough to know what the limit is, but I am happy it seems to be working. Here is pi in base-26 to over 1700 digits.
pi in base-26 where Z=0, A=1, B=2, C=3, ... X=24, Y=25.
C. CQRPKNKXQSQNC MKGMPSFJTCPFS THQWMDPABJARY HUPPUFCLDKLTD WQNHPHXZKUTYU DALHIOPPWKJOK QMBEVIOAXLFFN GILLPHRLRRBHD JGUCTSBWSIORA VGTENLPIZNRXF ONVTOXLKHEREH HYQNCOKXWODCN RWLEPSPGLEWEO UYDYQJEBVJWGS GTGBOKDLKMTCS LROVAAIEFRIGM BNWYMCFGJUNYQ MJVACLETZXIEN YWXCJZXLMPTVK XJZOKXAHYTXAQ NTIYMCCILNIXN YRBJRVOJOZCXK OBSKICHKJTTVJ PJVIJSYLDKFBN GQAQIDMQPUGIS GCKDDIUHEZEPH BPRLSIEOOYWYN GXPKVDCECPIQM TGQKLBMJVPIOZ LUMNSFUXIPMYL TBTLXUMCAOFLY UZLKTEAQYZOLT JSRJATOEZUKRV SVLZDSLUDCBHT ISWLJMUWJCSEF EGPAZMJNQMOEA FMBCTJVYOJKSN ADLNBNIFFWXAU NZDSLGBSSLZIC WZTVVOXULTERT CIUNBLZGLHHGM BKXVMOHNIDFPV YLVQTXPDVIXUA TGNNVZLBSTWQH HHQURKKSZUTSV AFWLDFFEIVPLR UMWHODZYKACKM GRWYDCPPCNKZO DYGJVLNZDQKRT IWUUGJQEJEDYO BGKLOCVQUDQNB JVGOPECNVNXUI VOWTNFXGSHCTZ QPYGDPPUNMKLU YRMNOZWMKDJEM DVEBDTIKDWUDC LMLGTXNWEZMTI BELURXMVSTGOV KPPFMUQANBIGW DHUKNXWWXVUZR YGORDOMKVDYFR NVODVVURUXSSS WKRVBVBDGDGBV CELWMLLGPRTUX HXVIKFGHIBKGX YSRAJOKGJPMBC UQVQRHAJRZNHS USZWMCXJMGLLQ OUHIXIKWMGPST YPPBSBJCKCVAQ APYLUFUGTAYDE JGRKCHLEKQOZC MSIABBCTHKNHJ ILPEAUECDPNDN RMWQECLKNOBRQ DIESFQPDAOOXK THXRKAANEMXYP XMQLYYSDGCXTP XQMYWHRJBCCSA SKVFWXGLRZEAK ASWMHQNPLJTJT SUDOMPWMUYYVS XLYEBOURQXFBX FRUPTERACZQTT VIHPVNHXHICFV TZPKVIRPVGHHY NZGTRCKBLETTK HJTPOGVQTTKDL OBUNCOBVXYQCI HYHLYTYCEIYZZ KIRIQUCNVGLBI CQLJURMGFFLRC AEBKMBPGGSCZM QFFPKBFSHGJEP GWYCFLCRKONWR HVLCFROEBXXKQ DKDKKFMYPJPHR IGGTYHDUVTYUK XLWGCNOBHKEQK DAUIXQNQGGGJF VYZRRVCACQLKQ WOCEPPBJJNHPS RYNRMXWQRHMPI GTWMZQSHCJBEZ QBJBOZZPZBERO IWNOZFJTQQRYA JPINCLZSXIMZB DSUVXKYBVFLIV LDTFRSKACJOWL VMWHKGDGSEEMM UXLENEISPQTUT OLJDRSWZPHIXH JOWGZNALVKHCW DCGCQCDCSCPHE EVWNQRZYAXHKB WCVBNPJXAJBVS JKOEEHFIAJASW ODJVMDTCNIOLP WAMRFXYRJDDBR DYZKGQVENKHYM FVVQNZSMZLCQI SXFHAVAOXPFGX NIIWVUESTQFFT WKICWRZDFGRNB DRRSTJMGMWSPU GWWRCCUFFZSKJ ZDGYPLZILDFJU EJDRFLQXMFHZM VSIZATHPVHKTK UTVSCQSGAVIBC IIZJFCSAMMHDI MGVQODXTJNIEU IUJFGK Started with a base-10 accuracy of 2438 decimal places. # of digits found is: 1722

- John E. Franklin
- 2005-08-31 08:22:09
That would be cool to find words in pi! Here are the results a little neater:
pi in base-26 where Z=0, A=1, B=2, C=3, ... X=24, Y=25.
C. CQRPKNKXQSQNC MKGMPSFJTCPFS THQWMDPABJARY HUPPUFCLDKLTD WQNHPHXZKUTYU DALHIOPPWKJOK QMBEVIOAXLFFN GILLPHRLRRBHD JGUCTSBWSIORA VGTENLPIZNRXF ONVTOXLKHE
- MathsIsFun
- 2005-08-31 07:12:52
Very well done.
So now we have to know - are there any hidden messages in PI ???
- John E. Franklin
- 2005-08-31 07:07:56
Hi again...
The BASIC program was a preliminary version. I added these lines so the letter Z could be displayed not as an @ (at sign). The @ sign precedes the capital letters in the ASCII chart, #64. if "@"= nextMinChar$ then nextMinChar$= "Z" if "@"= nextMaxChar$ then nextMaxChar$= "Z" Also the loopAround number of times I changed as needed for different tests. Also the last and longest one that min and max digits diverge at the 140th decimal place in base-26, that run of the program started off with 199 decimal places of pi in base-10. I noticed it went off the screen to the right and couldn't be viewed even with the horizontal scroll bar. Anyway, it was kind of messy above, but I didn't want to edit the file or I might have accidentally deleted the correct data. Finally, the left column is the minimum possible digit, and the right column is the maximum possible digit value.
- John E. Franklin
- 2005-08-31 06:13:42
I wrote a "Just BASIC" program to compute the digits. I used a max and min value for pi so I could see where the results diverged. See results below.
Code:C.CQRPK
piMax in decimal started out as: 14159265358979323847, This is: 20 digits.
piMin in decimal started out as: 14159265358979323846, This is: 20 digits.
(C,C)
(Q,Q)
(R,R)
(P,P)
(K,K)
(N,N)
(K,K)
(X,X)
(Q,Q)
(S,S)
(Q,Q)
(N,N)
(C,C)
(M,M) fourteen digits! Tuesday, August 30, 2005
(F,W) This is where min and max diverge.
(V,P)
(F,H)
(M,T)
(B,E)
(L,C)
(U,T)
(Q,J)
(I,Y)
(F,F)
(R,V)
________________________________________
see code below by John E. Franklin...
'program name: base26pi_BBK.bas (using Just BASIC v1.0 from internet)
piMax= 14159265358979323847''6'264338327950288419716939937511
piMin= 14159265358979323846 '264338327950288419716939937510
piMaxString$= STR$(piMax)
piMinString$= STR$(piMin)
piMaxLen= LEN(piMaxString$)
piMinLen= LEN(piMinString$)
print "piMax in decimal started out as: ";piMax;", This is: ";piMaxLen;" digits."
print "piMin in decimal started out as: ";piMin;", This is: ";piMinLen;" digits."
nextMax= piMax
nextMin= piMin
'commence loop
For loopAround= 1 to 25
nextMax= nextMax * 26
nextMin= nextMin * 26
nextMaxString$= STR$(nextMax)
nextMinString$= STR$(nextMin)
rightMaxString$= RIGHT$(nextMaxString$,piMaxLen)
rightMinString$= RIGHT$(nextMinString$,piMinLen)
leftMaxString$= LEFT$(nextMaxString$,LEN(nextMaxString$) - piMaxLen)
leftMinString$= LEFT$(nextMinString$,LEN(nextMinString$) - piMinLen)
'print "@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@"
'print leftMaxString$;"%";rightMaxString$
'print leftMinString$;"%";rightMinString$
leftMax= val(leftMaxString$)
leftMin= val(leftMinString$)
'print "Next Max Digit is: ";CHR$(leftMax + 64);" ";leftMax
'print "Next Min Digit is: ";CHR$(leftMin + 64);" ";leftMin
nextMaxChar$=CHR$(leftMax + 64)
nextMinChar$=CHR$(leftMin + 64)
print "(";nextMinChar$;",";nextMaxChar$;")"
nextMax= val(rightMaxString$)
nextMin= val(rightMinString$)
next loopAround
_______________________________________________________
piMax in decimal started out as: 14160, This is: 5 digits.
piMin in decimal started out as: 14159, This is: 5 digits.
(C,C)
(Q,Q)
(R,R)
(O,S) Diverges after 3rd digit.
(F,T)
(Z,U)
(R,V)
_______________________________________________
piMax in decimal started out as: 14159265358979323846264338327950288419716939937511, This is: 50 digits.
piMin in decimal started out as: 14159265358979323846264338327950288419716939937510, This is: 50 digits.
(C,C)
(Q,Q)
(R,R)
(P,P)
(K,K)
(N,N)
(K,K)
(X,X)
(Q,Q)
(S,S)
(Q,Q)
(N,N)
(C,C)
(M,M)
(K,K)
(G,G)
(M,M)
(P,P)
(S,S)
(F,F)
(J,J)
(T,T)
(C,C)
(P,P)
(F,F)
(S,S)
(T,T)
(H,H)
(Q,Q)
(W,W)
(M,M)
(D,D)
(P,P)
(A,A)
(B,B)
(E,M) Diverges after 35 good digits
(Z,R)
(E,D)
(H,Z)
(D,W)
(N,R)
(B,W)
(R,Z)
(D,H)
(I,N)
(A,U)
(F,Y)
(O,S)
(L,C)
(Y,X)
__________________________________
piMax in decimal started out as: 1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303820, This is: 199 digits.
piMin in decimal started out as: 1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819, This is: 199 digits.
(min,max)
(C,C)
(Q,Q)
(R,R)
(P,P)
(K,K)
(N,N)
(K,K)
(X,X)
(Q,Q)
(S,S)
(Q,Q)
(N,N)
(C,C)
(M,M)
(K,K)
(G,G)
(M,M)
(P,P)
(S,S)
(F,F)
(J,J)
(T,T)
(C,C)
(P,P)
(F,F)
(S,S)
(T,T)
(H,H)
(Q,Q)
(W,W)
(M,M)
(D,D)
(P,P)
(A,A)
(B,B)
(J,J)
(A,A)
(R,R)
(Y,Y)
(H,H)
(U,U)
(P,P)
(P,P)
(U,U)
(F,F)
(C,C)
(L,L)
(D,D)
(K,K)
(L,L)
(T,T)
(D,D)
(W,W)
(Q,Q)
(N,N)
(H,H)
(P,P)
(H,H)
(X,X)
(Z,Z)
(K,K)
(U,U)
(T,T)
(Y,Y)
(U,U)
(D,D)
(A,A)
(L,L)
(H,H)
(I,I)
(O,O)
(P,P)
(P,P)
(W,W)
(K,K)
(J,J)
(O,O)
(K,K)
(Q,Q)
(M,M)
(B,B)
(E,E)
(V,V)
(I,I)
(O,O)
(A,A)
(X,X)
(L,L)
(F,F)
(F,F)
(N,N)
(G,G)
(I,I)
(L,L)
(L,L)
(P,P)
(H,H)
(R,R)
(L,L)
(R,R)
(R,R)
(B,B)
(H,H)
(D,D)
(J,J)
(G,G)
(U,U)
(C,C)
(T,T)
(S,S)
(B,B)
(W,W)
(S,S)
(I,I)
(O,O)
(R,R)
(A,A)
(V,V)
(G,G)
(T,T)
(E,E)
(N,N)
(L,L)
(P,P)
(I,I)
(Z,Z)
(N,N)
(R,R)
(X,X)
(F,F)
(O,O)
(N,N)
(V,V)
(T,T)
(O,O)
(X,X)
(L,L)
(K,K)
(H,H)
(E,E) This is the 140th decimal place.
(P,S) min and max diverge here; digits are good above this point.
(B,I)
(X,H)
(X,Q)
(F,Y)
(X,L)
(J,S)
(Z,J)
(J,E)
(M,Z)
(E,D)
(P,J)
(W,Z)
(U,W)
(T,W)
(G,A)
(L,T)
(P,M)
(L,I)
(L,S)
(D,Y)
(Y,V)
(S,M)
(P,J)
(R,A)
(R,H)
(Q,U)
(D,Q)
(D,Y)
(Q,P)
(O,G)
(G,R)
(G,Q)
(G,Q)
(C,D)
(S,Z)
(P,S)
(E,N)
(U,X)
(H,W)
(N,Z)
(V,A)
(C,F)
(X,A)
(N,I)
(R,M)
(Q,V)
(Y,X)
(A,A)
(T,T)
(X,F)
(G,P)
(K,O)
(Y,C)
(E,V)
(Q,U)
(U,E)
(N,F)
(T,C)
__________________________________
- MathsIsFun
- 2005-08-30 09:21:57
Hi John, you have made some very helpful posts and still a guest!
Base conversion of whole numbers is fairly easy if you think about remainders.
Let's a whole number that looks like PI but without "decimals"!
Watch this series of divisions (R means remainder, which is ignored in the next division):
31416 / 26 = 1208 R 8 1208 / 26 = 46 R 12 46 / 26 = 1 R 20
Now, think about the last answer (1 R 20), it means that 31416/26/26/26 = 1 (plus bits), in other words it tells us that we should put a "1" in the "power of 3" column!!!
Likewise we should put a 20 in the hundreds, 12 in the tens and 8 in the units.
Why?
Becasue our division work just before has really said that: 31416 = 1208 × 26 + 8 (so 8 belongs in the units, and from here on we are dealing with the next column of digits) 1208 = 46 × 26 + 12 (so 12 belongs in the second column, and from here we go to the third column) 46 = 1 × 26 + 20 (so 20 belongs in the thirtd column, and we put one in the fourth column)
Let's see if it has worked: 1 × 26³ = 17576 20 × 26² = 13520 12 × 26 = 312 8 x 1 = 8 Total: 31416
So, to do whole numbers you do repeated divisions and put the numbers in from right to left
Now, if you have followed THAT, then we get to the tricky part: "decimals" (hmmm... not an accurate word becuase it implies base-10 but you know what I mean)
To do "decimals", you use repeated multiplies and build from left to right
.1416 × 26 = 3.6816 .6816 × 26 = 17.7216 .7216 × 26 = 18.7616 etc...
The first answer says to put a 3 in the first "decimals" column, the second answer says to put a 17 in the second column etc ..
So, I guess the answer is something like C.CQR
As a check I calculated 3 + 3/26 + 17/26² + 18/26³ = 3.141556...
- John E. Franklin
- 2005-08-30 08:28:32
Imagine base-26 where Z=0, A=1, B=2, C=3, D=4, ..., T=20, U=21, V=22, W=23, X=24, Y=25. What approach should I take to convert the first 20 decimals of pi into this base-26? The answer would begin: C.C I guess. I am interested in pure methods and methods that only are useful to certain number of digits.
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