# Steinhaus-Moser notation

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

Missing image
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"

Missing image
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:
• [itex]M(n,1,3) = n^n[itex]
• [itex]M(n,1,p+1) = M(n,n,p)[itex]
• [itex]M(n,m+1,p) = M\big(M(n,1,p),m,p\big)[itex]
and
• mega = [itex]M(2,1,5)[itex]
• moser = [itex]M\big(2,1,M(2,1,5)\big)[itex]
 Contents

## 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 [itex]f(x)=x^x[itex] we have mega = [itex]f^{256}(256) = f^{258}(2)[itex] 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) = [itex](256^{\,\!256})^{256^{256}}=256^{256^{257}}[itex]
• M(256,3,3) = [itex](256^{\,\!256^{257}})^{256^{256^{257}}}=256^{256^{257}\times 256^{256^{257}}}=256^{256^{257+256^{257}}}[itex]≈[itex]256^{\,\!256^{256^{257}}}[itex]

Similarly:

• M(256,4,3) ≈ [itex]{\,\!256^{256^{256^{256^{257}}}}}[itex]
• M(256,5,3) ≈ [itex]{\,\!256^{256^{256^{256^{256^{257}}}}}}[itex]

etc.

Thus:

• mega = [itex]M(256,256,3)\approx(256\uparrow)^{256}257[itex], where [itex](256\uparrow)^{256}[itex] denotes a functional power of the function [itex]f(n)=256^n[itex].

Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈ [itex]256\uparrow\uparrow 257[itex], using Knuth's up-arrow notation.

Note that after the first few steps the value of [itex]n^n[itex] is each time approximately equal to [itex]256^n[itex]. In fact, it is even approximately equal to [itex]10^n[itex] (see also approximate arithmetic for very large numbers). Using base 10 powers we get:

• [itex]M(256,1,3)\approx 3.23\times 10^{616}[itex]
• [itex]M(256,2,3)\approx10^{\,\!1.99\times 10^{619}}[itex] ([itex]\log _{10} 616[itex] is added to the 616)
• [itex]M(256,3,3)\approx10^{\,\!10^{1.99\times 10^{619}}}[itex] ([itex]619[itex] is added to the [itex]1.99\times 10^{619}[itex], which is negligible; therefore just a 10 is added at the bottom)
• [itex]M(256,4,3)\approx10^{\,\!10^{10^{1.99\times 10^{619}}}}[itex]

...

• mega = [itex]M(256,256,3)\approx(10\uparrow)^{255}1.99\times 10^{619}[itex], where [itex](10\uparrow)^{255}[itex] denotes a functional power of the function [itex]f(n)=10^n[itex]. Hence [itex]10\uparrow\uparrow 257 < \mbox{mega} < 10\uparrow\uparrow 258[itex]

## 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,

[itex]\mbox{moser} < 3\rightarrow 3\rightarrow 65\rightarrow 2[itex]

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