Steinhaus-Moser notation
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In mathematics, Moser's polygon notation is a means of expressing certain extremely large numbers. It is an extension of Steinhaus's polygon notation.
(a number n in a triangle) means nn
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Square-n.png
n in a square
(a number n in a square) is equivalent with "the number n inside n triangles, which are all nested"
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Pentagon-n.png
n in a pentagon
(a number n in a pentagon) is equivalent with "the number n inside n squares, which are all nested"
etc.: n written in an (m+1)-sided polygon is equivalent with "the number n inside n m-sided polygons, which are all nested"
Steinhaus only defined the triangle, the square, and a circle Missing image
Circle-n.png
n in a cicle
, equivalent to the pentagon defined above.
Steinhaus defined:
- "mega" is the number equivalent to 2 in a circle:
- "megistron" is the number equivalent to 10 in a circle:
Moser's number is the number represented by "2 in a megagon", where a "megagon" is a polygon with "mega" sides.
Alternative notations:
- use the functions square(x) and triangle(x)
- let M(n,m,p) be the number represented by the number n in m nested p-sided polygons; then the rules are:
- <math>M(n,1,3) = n^n<math>
- <math>M(n,1,p+1) = M(n,n,p)<math>
- <math>M(n,m+1,p) = M\big(M(n,1,p),m,p\big)<math>
- and
- mega = <math>M(2,1,5)<math>
- moser = <math>M\big(2,1,M(2,1,5)\big)<math>
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Mega
Note that is already a very large number, since = square(square(2)) = square(triangle(triangle(2))) = square(triangle(22)) = square(triangle(4)) = square(44) = square(256) = triangle(triangle(triangle(...triangle(256)...))) [256 triangles] = triangle(triangle(triangle(...triangle(256256)...))) [255 triangles] = triangle(triangle(triangle(...triangle(3.2 × 10616)...))) [254 triangles] = ...
Using the other notation:
mega = M(2,1,5) = M(256,256,3)
With the function <math>f(x)=x^x<math> we have mega = <math>f^{256}(256) = f^{258}(2)<math> where the superscript denotes a functional power, not a numerical power.
We have (note the convention that powers are evaluated from right to left):
- M(256,2,3) = <math>(256^{\,\!256})^{256^{256}}=256^{256^{257}}<math>
- M(256,3,3) = <math>(256^{\,\!256^{257}})^{256^{256^{257}}}=256^{256^{257}\times 256^{256^{257}}}=256^{256^{257+256^{257}}}<math>≈<math>256^{\,\!256^{256^{257}}}<math>
Similarly:
- M(256,4,3) ≈ <math>{\,\!256^{256^{256^{256^{257}}}}}<math>
- M(256,5,3) ≈ <math>{\,\!256^{256^{256^{256^{256^{257}}}}}}<math>
etc.
Thus:
- mega = <math>M(256,256,3)\approx(256\uparrow)^{256}257<math>, where <math>(256\uparrow)^{256}<math> denotes a functional power of the function <math>f(n)=256^n<math>.
Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈ <math>256\uparrow\uparrow 257<math>, using Knuth's up-arrow notation.
Note that after the first few steps the value of <math>n^n<math> is each time approximately equal to <math>256^n<math>. In fact, it is even approximately equal to <math>10^n<math> (see also approximate arithmetic for very large numbers). Using base 10 powers we get:
- <math>M(256,1,3)\approx 3.23\times 10^{616}<math>
- <math>M(256,2,3)\approx10^{\,\!1.99\times 10^{619}}<math> (<math>\log _{10} 616<math> is added to the 616)
- <math>M(256,3,3)\approx10^{\,\!10^{1.99\times 10^{619}}}<math> (<math>619<math> is added to the <math>1.99\times 10^{619}<math>, which is negligible; therefore just a 10 is added at the bottom)
- <math>M(256,4,3)\approx10^{\,\!10^{10^{1.99\times 10^{619}}}}<math>
...
- mega = <math>M(256,256,3)\approx(10\uparrow)^{255}1.99\times 10^{619}<math>, where <math>(10\uparrow)^{255}<math> denotes a functional power of the function <math>f(n)=10^n<math>. Hence <math>10\uparrow\uparrow 257 < \mbox{mega} < 10\uparrow\uparrow 258<math>
Moser's number
It has been proved that Moser's number, although extremely large, is smaller than Graham's number.
Therefore, using the Conway chained arrow notation,
- <math>\mbox{moser} < 3\rightarrow 3\rightarrow 65\rightarrow 2<math>
See also
External
- Factoid on Big Numbers (http://www-users.cs.york.ac.uk/~susan/cyc/b/big.htm)
- Robert Munafo's Big Numbers (http://home.earthlink.net/~mrob/pub/math/largenum-3.html), which hints Steinhaus and Moser came up with this notation jointly in the '70s.
- Megistron at mathworld.wolfram.com (http://mathworld.wolfram.com/Megistron.html)
- Circle notation at mathworld.wolfram.com (http://mathworld.wolfram.com/CircleNotation.html)sl:Steinhausov mnogokotniški zapis