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a measure of irrationality

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  • Roger Lee Bagula
    Here is a new chaotic sequence. Herb Conn sent me a biography of Erdos and I learned about Ramsey theory. As a result I got this idea for rating the chaos in
    Message 1 of 3 , Feb 1, 2002
      Here is a new chaotic sequence.
      Herb Conn sent me a biography of Erdos
      and I learned about Ramsey theory.
      As a result I got this idea for rating the chaos in an irrational square root
      by how often it's irrational rotation goes in a positive direction in a row.
      So I found a way to measure "irrationality"
      in an irrational number!
      Since it worked on regular integers, I tried it on primes only.
      It works there as well.
      If the measure is one, then it is the most variable.
      If it is zero all the numbers are going in the negative direction!
      Respectfully, Roger L. Bagula
      tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
      619-5610814 :
      TFTN web sites maintained as an Emag:
      URL : http://home.earthlink.net/~tftn
      URL : http://www.geocities.com/ResearchTriangle/Thinktank/7279/
      URL : http://members.nbci.com/RogerLBagula/index.html
      URL : http://sites.netscape.net/rlbtftn/index.html
      URL : http://victorian.fortunecity.com/carmelita/435/
      URL : http://www.crosswinds.net/~translight/index.html
      True basic program:
      00 PRINT " in Mac use Bounder to set RTP(run time package)"
      110 PRINT " to 2,000,000 minimum heap for True Basic"
      120 SET MODE "color"
      130 SET WINDOW 0,1922,0,1062
      140 SET COLOR MIX(0) 1,1,1
      150 SET COLOR MIX(1) 0,0,0
      160 REM sieve
      170 PRINT "running sieve takes time"
      180 LET E=640*4
      190 DIM P(2500)
      200 LET Q=0
      210 FOR N=1 To 8*E
      220 IF N<4 Then
      230 LET Q=Q+1
      240 LET P(Q)=N
      250 GOTO 420
      260 END IF
      270 LET I=0
      280 LET T=2
      290 LET J=Int(N/T)
      300 LET K=J*T
      310 IF N=K Then GOTO 420
      320 LET I=I+1
      330 LET L=T*T
      340 IF L>N Then
      350 LET Q=Q+1
      360 IF q>e then GOTO 440
      370 LET P(Q)=N
      380 GOTO 420
      390 END IF
      400 LET T=I*2+1
      410 GOTO 290
      420 IF Q>E Then GOTO 440
      430 NEXT N
      440 DIM aa(100)
      450 PRINT" irrational rotation direction in prime square roots"
      460 PRINT " a Ramsey approach to a measure of irrationality"
      470 PRINT" by R. L. Bagula 1 feb 2002©"
      480 LET s1=750
      490 FOR a=2 to 78
      500 REM loop of square root integers"
      510 LET t0=0
      520 LET count =0
      530 LET maxcount=0
      540 FOR pp= 1 to 100
      550 REM irrational rotation as modulo one ( for 100 integers)
      560 LET t=mod(pp*sqr(p(a)),1)
      570 REM direction on the unit interval taken and counted
      580 IF t>t0 then
      590 SET COLOR "blue"
      600 LET count=count +1
      610 ELSE
      620 SET COLOR "red"
      630 LET count =0
      640 END IF
      650 REM last position taken
      660 LET t0=t
      670 REM check for number of counts in a row in the same positive direction
      680 IF count>maxcount then LET maxcount =count
      690 REM plot of points on a circle with positive blue and
      negative red
      700 LET x=cos(2*Pi*t)
      710 LET y=sin(2*Pi*t)
      720 PLOT 1922/2+s1*x,1062/2+s1*(1062/1922)*y;
      730 NEXT pp
      740 REM array of maximum counts formed
      750 PLOT
      760 LET aa(a)=maxcount
      770 CLEAR
      780 NEXT a
      790 PRINT" irrational rotation direction in prime square roots"
      800 PRINT" a Ramsey approach to a measure of irrationality"
      810 PRINT" by R. L. Bagula 1 feb 2002©"
      820 FOR a=2 to 78
      830 SET COLOR "black"
      840 PRINT "prime "; p(a);"maxcount ";aa(a);"ratio ";aa(a)/100
      850 NEXT a
      860 END
    • teoriadetodo
      Hi, Roger How does it rate compared to this simple idea? http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat2.html Can a unique irrationality order
      Message 2 of 3 , Feb 11, 2002
        Hi, Roger

        How does it rate compared to this simple idea?

        http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat2.html

        Can a unique irrationality order be established?
        What properties would such an order have?
        Akin to the real line? Other?

        Fernando


        --- In caostheory@y..., Roger Lee Bagula <tftn@e...> wrote:
        > Here is a new chaotic sequence.
        > Herb Conn sent me a biography of Erdos
        > and I learned about Ramsey theory.
        > As a result I got this idea for rating the chaos in an irrational
        square root
        > by how often it's irrational rotation goes in a positive direction
        in a row.
        > So I found a way to measure "irrationality"
        > in an irrational number!
        > Since it worked on regular integers, I tried it on primes only.
        > It works there as well.
        > If the measure is one, then it is the most variable.
        > If it is zero all the numbers are going in the negative direction!
        > Respectfully, Roger L. Bagula
        > tftn@e..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
        > 619-5610814 :
        > TFTN web sites maintained as an Emag:
        > URL : http://home.earthlink.net/~tftn
        > URL : http://www.geocities.com/ResearchTriangle/Thinktank/7279/
        > URL : http://members.nbci.com/RogerLBagula/index.html
        > URL : http://sites.netscape.net/rlbtftn/index.html
        > URL : http://victorian.fortunecity.com/carmelita/435/
        > URL : http://www.crosswinds.net/~translight/index.html
        > True basic program:
        > 00 PRINT " in Mac use Bounder to set RTP(run time package)"
        > 110 PRINT " to 2,000,000 minimum heap for True Basic"
        > 120 SET MODE "color"
        > 130 SET WINDOW 0,1922,0,1062
        > 140 SET COLOR MIX(0) 1,1,1
        > 150 SET COLOR MIX(1) 0,0,0
        > 160 REM sieve
        > 170 PRINT "running sieve takes time"
        > 180 LET E=640*4
        > 190 DIM P(2500)
        > 200 LET Q=0
        > 210 FOR N=1 To 8*E
        > 220 IF N<4 Then
        > 230 LET Q=Q+1
        > 240 LET P(Q)=N
        > 250 GOTO 420
        > 260 END IF
        > 270 LET I=0
        > 280 LET T=2
        > 290 LET J=Int(N/T)
        > 300 LET K=J*T
        > 310 IF N=K Then GOTO 420
        > 320 LET I=I+1
        > 330 LET L=T*T
        > 340 IF L>N Then
        > 350 LET Q=Q+1
        > 360 IF q>e then GOTO 440
        > 370 LET P(Q)=N
        > 380 GOTO 420
        > 390 END IF
        > 400 LET T=I*2+1
        > 410 GOTO 290
        > 420 IF Q>E Then GOTO 440
        > 430 NEXT N
        > 440 DIM aa(100)
        > 450 PRINT" irrational rotation direction in prime square roots"
        > 460 PRINT " a Ramsey approach to a measure of irrationality"
        > 470 PRINT" by R. L. Bagula 1 feb 2002©"
        > 480 LET s1=750
        > 490 FOR a=2 to 78
        > 500 REM loop of square root integers"
        > 510 LET t0=0
        > 520 LET count =0
        > 530 LET maxcount=0
        > 540 FOR pp= 1 to 100
        > 550 REM irrational rotation as modulo one ( for 100
        integers)
        > 560 LET t=mod(pp*sqr(p(a)),1)
        > 570 REM direction on the unit interval taken and counted
        > 580 IF t>t0 then
        > 590 SET COLOR "blue"
        > 600 LET count=count +1
        > 610 ELSE
        > 620 SET COLOR "red"
        > 630 LET count =0
        > 640 END IF
        > 650 REM last position taken
        > 660 LET t0=t
        > 670 REM check for number of counts in a row in the same
        positive direction
        > 680 IF count>maxcount then LET maxcount =count
        > 690 REM plot of points on a circle with positive blue and
        > negative red
        > 700 LET x=cos(2*Pi*t)
        > 710 LET y=sin(2*Pi*t)
        > 720 PLOT 1922/2+s1*x,1062/2+s1*(1062/1922)*y;
        > 730 NEXT pp
        > 740 REM array of maximum counts formed
        > 750 PLOT
        > 760 LET aa(a)=maxcount
        > 770 CLEAR
        > 780 NEXT a
        > 790 PRINT" irrational rotation direction in prime square roots"
        > 800 PRINT" a Ramsey approach to a measure of irrationality"
        > 810 PRINT" by R. L. Bagula 1 feb 2002©"
        > 820 FOR a=2 to 78
        > 830 SET COLOR "black"
        > 840 PRINT "prime "; p(a);"maxcount ";aa(a);"ratio ";aa(a)/100
        > 850 NEXT a
        > 860 END
      • Roger Lee Bagula
        Dear Fernando, I ll get back to you about the irrational left or right handiness of the Fibonacci sequence. I usually use f(0)=1,f(1)=1 instead of
        Message 3 of 3 , Feb 11, 2002
          Dear Fernando,
          I'll get back to you about the irrational left or right handiness of the
          Fibonacci sequence. I usually use f(0)=1,f(1)=1 instead of f(0)=0,f(1)=1
          as he does at this site.
          sqr(0) and sqr(1) aren't going to get you anything ... ha, ha...
          teoriadetodo wrote:
          >
          > Hi, Roger
          >
          > How does it rate compared to this simple idea?
          >
          > http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat2.html
          >
          > Can a unique irrationality order be established?
          > What properties would such an order have?
          > Akin to the real line? Other?
          >
          > Fernando
          >
          > --- In caostheory@y..., Roger Lee Bagula <tftn@e...> wrote:
          > > Here is a new chaotic sequence.
          > > Herb Conn sent me a biography of Erdos
          > > and I learned about Ramsey theory.
          > > As a result I got this idea for rating the chaos in an irrational
          > square root
          > > by how often it's irrational rotation goes in a positive direction
          > in a row.
          > > So I found a way to measure "irrationality"
          > > in an irrational number!
          > > Since it worked on regular integers, I tried it on primes only.
          > > It works there as well.
          > > If the measure is one, then it is the most variable.
          > > If it is zero all the numbers are going in the negative direction!
          > > Respectfully, Roger L. Bagula
          > > tftn@e..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
          > > 619-5610814 :
          > > TFTN web sites maintained as an Emag:
          > > URL : http://home.earthlink.net/~tftn
          > > URL : http://www.geocities.com/ResearchTriangle/Thinktank/7279/
          > > URL : http://members.nbci.com/RogerLBagula/index.html
          > > URL : http://sites.netscape.net/rlbtftn/index.html
          > > URL : http://victorian.fortunecity.com/carmelita/435/
          > > URL : http://www.crosswinds.net/~translight/index.html
          > > True basic program:
          > > 00 PRINT " in Mac use Bounder to set RTP(run time package)"
          > > 110 PRINT " to 2,000,000 minimum heap for True Basic"
          > > 120 SET MODE "color"
          > > 130 SET WINDOW 0,1922,0,1062
          > > 140 SET COLOR MIX(0) 1,1,1
          > > 150 SET COLOR MIX(1) 0,0,0
          > > 160 REM sieve
          > > 170 PRINT "running sieve takes time"
          > > 180 LET E=640*4
          > > 190 DIM P(2500)
          > > 200 LET Q=0
          > > 210 FOR N=1 To 8*E
          > > 220 IF N<4 Then
          > > 230 LET Q=Q+1
          > > 240 LET P(Q)=N
          > > 250 GOTO 420
          > > 260 END IF
          > > 270 LET I=0
          > > 280 LET T=2
          > > 290 LET J=Int(N/T)
          > > 300 LET K=J*T
          > > 310 IF N=K Then GOTO 420
          > > 320 LET I=I+1
          > > 330 LET L=T*T
          > > 340 IF L>N Then
          > > 350 LET Q=Q+1
          > > 360 IF q>e then GOTO 440
          > > 370 LET P(Q)=N
          > > 380 GOTO 420
          > > 390 END IF
          > > 400 LET T=I*2+1
          > > 410 GOTO 290
          > > 420 IF Q>E Then GOTO 440
          > > 430 NEXT N
          > > 440 DIM aa(100)
          > > 450 PRINT" irrational rotation direction in prime square roots"
          > > 460 PRINT " a Ramsey approach to a measure of irrationality"
          > > 470 PRINT" by R. L. Bagula 1 feb 2002©"
          > > 480 LET s1=750
          > > 490 FOR a=2 to 78
          > > 500 REM loop of square root integers"
          > > 510 LET t0=0
          > > 520 LET count =0
          > > 530 LET maxcount=0
          > > 540 FOR pp= 1 to 100
          > > 550 REM irrational rotation as modulo one ( for 100
          > integers)
          > > 560 LET t=mod(pp*sqr(p(a)),1)
          > > 570 REM direction on the unit interval taken and counted
          > > 580 IF t>t0 then
          > > 590 SET COLOR "blue"
          > > 600 LET count=count +1
          > > 610 ELSE
          > > 620 SET COLOR "red"
          > > 630 LET count =0
          > > 640 END IF
          > > 650 REM last position taken
          > > 660 LET t0=t
          > > 670 REM check for number of counts in a row in the same
          > positive direction
          > > 680 IF count>maxcount then LET maxcount =count
          > > 690 REM plot of points on a circle with positive blue and
          > > negative red
          > > 700 LET x=cos(2*Pi*t)
          > > 710 LET y=sin(2*Pi*t)
          > > 720 PLOT 1922/2+s1*x,1062/2+s1*(1062/1922)*y;
          > > 730 NEXT pp
          > > 740 REM array of maximum counts formed
          > > 750 PLOT
          > > 760 LET aa(a)=maxcount
          > > 770 CLEAR
          > > 780 NEXT a
          > > 790 PRINT" irrational rotation direction in prime square roots"
          > > 800 PRINT" a Ramsey approach to a measure of irrationality"
          > > 810 PRINT" by R. L. Bagula 1 feb 2002©"
          > > 820 FOR a=2 to 78
          > > 830 SET COLOR "black"
          > > 840 PRINT "prime "; p(a);"maxcount ";aa(a);"ratio ";aa(a)/100
          > > 850 NEXT a
          > > 860 END
          >
          >
          >
          > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

          --
          Respectfully, Roger L. Bagula
          tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
          619-5610814 :
          TFTN web sites maintained as an Emag:
          URL : http://home.earthlink.net/~tftn
          URL : http://www.geocities.com/ResearchTriangle/Thinktank/7279/
          URL : http://members.nbci.com/RogerLBagula/index.html
          URL : http://sites.netscape.net/rlbtftn/index.html
          URL : http://victorian.fortunecity.com/carmelita/435/
          URL : http://www.crosswinds.net/~translight/index.html
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