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why the mantissas used below are all roots of powers of 10

  • To: mathgroup at smc.vnet.net
  • Subject: [mg131865] why the mantissas used below are all roots of powers of 10
  • From: Marvin Burns <marvin at marvinrayburns.com>
  • Date: Sat, 19 Oct 2013 04:12:36 -0400 (EDT)
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  • Reply-to: Marvin Burns <marvin at marvinrayburns.com>

I sent this in yesterday but only part of the message was posted.

From Marvin Ray Burns to mathgroup

We will use the  Mathematica code 
Table[{x, (10 MantissaExponent[N[Sin[10^(-100 - 1/x)], 10]][[1]])^x}, {x, 1, 10}] // TableForm
I was playing around with small sines and posted this on another blog and didn't get much response. However, I would like to know why the mantissas used below are all roots of powers of 10.

There seems to be patterns for sin(10^-k) for rational k;
Here we have the "floats."
n sin(10^(-n-1/2))
1 0.03161750640
2 0.003162272390
3 0.0003162277607
4 0.00003162277660
5 0.000003162277660
6 0.0000003162277660
7 0.00000003162277660
Basically the mantissa  of 316227766 is being "floated out," because for small x sin(x)~x.
Here we use the mantissa:
Noticing that 3.162277660^2~=10 we have below a more subtle and beautiful pattern for sin(10^-k), using sufficiently large integral value for k. Here we use 100 but 9 is usually sufficient.

x        (10 Mantissa[sin(10^(-100 - 1/x))])^x
1       1.* 10^1
2       1.* 10^1
3       1.* 10^2
4       1.* 10^3
5       1.* 10^4
6       1.* 10^5
etc.

x        (10 Mantissa[sin(10^(-100 - 2/x))])^x
1        1.* 10^0
2        1.* 10^2
3        1.* 10^1
4        1.* 10^2
5        1.* 10^3
6        1.* 10^4
etc.
x       (10 Mantissa[sin(10^(-100 - 3/x))])^x
1       1.*  10^1
2       1.*  10^1
3       1.*  10^3
4       1.*  10^1
5       1.*  10^2
6       1.*  10^3
etc.

x        (10 Mantissa[sin(10^(-100 - 4/x))])^x
1       1.*  10^1
2       1.*  10^0
3       1.*  10^2
4       1.*  10^4
5       1.*  10^1
6       1.*  10^2
7       1.*  10^3
etc.

x (10 Mantissa[sin(10^(-100 - 5/x))])^x
1       1.*   10^1
2       1.*  10^1
3       1.*  10^1
4       1.*  10^3
5       1.*  10^4
6       1.*  10^1
7       1.*  10^2
8       1.*  10^3
etc.

The Mathematica code for this is 
Table[{x, (10 MantissaExponent[N[Sin[10^(-100 - 1/x)], 10]][[1]])^  x}, {x, 1, 10}] // TableForm
Change 1/x to 2/x,3/x, etc .
Can anyone figure out the pattern here?
Replacing 1/x with (3/2)/x  and ^x to ^(2x) we find
x       (10 Mantissa[sin(10^(-100 - (3/2)/x))])^(2x)
1       1.*  10^1
2       1.*  10^1
3       1.*  10^3
4       1.*  10^5
5       1.*  10^7
6       1.*  10^9
etc.

Replacing 1/x with (5/2)/x  and ^x to ^(2x) we find
x (10 Mantissa[sin(10^(-100 - (5/2)/x))])^(2x)
1      1.*  10^1
2      1.*  10^3
3      1.*  10^1
4      1.*  10^3
5      1.*  10^5
6      1.*  10^7
etc.

Replacing 1/x with (5/3)/x  and ^x to ^(3x) we find
x        (10 Mantissa[sin(10^(-100 - (5/3)/x))])^(3x)
1      1.*  10^1
2      1.*  10^1
3      1.*  10^4
4      1.*  10^7
5      1.*  10^10
6      1.*  10^12
etc.



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