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Steinhaus–Moser notation

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In mathematics, Steinhaus–Moser 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. (a number n in a square) is equivalent with "the number n inside n triangles, which are all nested." (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." In a series of nested polygons, they are associated inward. The number n inside two triangles is equivalent to nⁿ inside one triangle, which is equivalent to nⁿ raised to the power of nⁿ. Steinhaus only defined the triangle, the square, and a circle , equivalent to the pentagon defined above. Steinhaus defined:

"mega" is the number equivalent to 2 in a circle: ②
"megiston" is the number equivalent to 10 in a circle: ⑩

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>

1 Mega
2 Moser's number
3 See also
4 External links

Mega

Note that ② is already a very large number, since ② = 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)...))) [255 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 proven 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>