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From Anarchopedia

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.

(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."

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:
• "megiston" 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:
  • Failed to parse (Cannot write to or create math temp directory): M(n,1,3) = n^n

• • Failed to parse (Cannot write to or create math temp directory): M(n,1,p+1) = M(n,n,p)

• • Failed to parse (Cannot write to or create math temp directory): M(n,m+1,p) = M\big(M(n,1,p),m,p\big)

and

• • mega = Failed to parse (Cannot write to or create math temp directory): M(2,1,5)

• • moser = Failed to parse (Cannot write to or create math temp directory): M\big(2,1,M(2,1,5)\big)

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

[edit] 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 Failed to parse (Cannot write to or create math temp directory): f(x)=x^x

we have mega = Failed to parse (Cannot write to or create math temp directory): f^{256}(256)  = f^{258}(2)
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) = Failed to parse (Cannot write to or create math temp directory): (256^{\,\!256})^{256^{256}}=256^{256^{257}}

• M(256,3,3) = Failed to parse (Cannot write to or create math temp directory): (256^{\,\!256^{257}})^{256^{256^{257}}}=256^{256^{257}\times 256^{256^{257}}}=256^{256^{257+256^{257}}}

≈Failed to parse (Cannot write to or create math temp directory): 256^{\,\!256^{256^{257}}}

Similarly:

• M(256,4,3) ≈ Failed to parse (Cannot write to or create math temp directory): {\,\!256^{256^{256^{256^{257}}}}}

• M(256,5,3) ≈ Failed to parse (Cannot write to or create math temp directory): {\,\!256^{256^{256^{256^{256^{257}}}}}}

etc.

Thus:

• mega = Failed to parse (Cannot write to or create math temp directory): M(256,256,3)\approx(256\uparrow)^{256}257

, where Failed to parse (Cannot write to or create math temp directory): (256\uparrow)^{256}

denotes a functional power of the function Failed to parse (Cannot write to or create math temp directory): f(n)=256^n

.

Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈ Failed to parse (Cannot write to or create math temp directory): 256\uparrow\uparrow 257 , using Knuth's up-arrow notation.

Note that after the first few steps the value of Failed to parse (Cannot write to or create math temp directory): n^n

is each time approximately equal to Failed to parse (Cannot write to or create math temp directory): 256^n

. In fact, it is even approximately equal to Failed to parse (Cannot write to or create math temp directory): 10^n

(see also approximate arithmetic for very large numbers). Using base 10 powers we get:

• Failed to parse (Cannot write to or create math temp directory): M(256,1,3)\approx 3.23\times 10^{616}

• Failed to parse (Cannot write to or create math temp directory): M(256,2,3)\approx10^{\,\!1.99\times 10^{619}}

(Failed to parse (Cannot write to or create math temp directory): \log _{10} 616
is added to the 616)

• Failed to parse (Cannot write to or create math temp directory): M(256,3,3)\approx10^{\,\!10^{1.99\times 10^{619}}}

(Failed to parse (Cannot write to or create math temp directory): 619
is added to the Failed to parse (Cannot write to or create math temp directory): 1.99\times 10^{619}

, which is negligible; therefore just a 10 is added at the bottom)

• Failed to parse (Cannot write to or create math temp directory): M(256,4,3)\approx10^{\,\!10^{10^{1.99\times 10^{619}}}}

...

• mega = Failed to parse (Cannot write to or create math temp directory): M(256,256,3)\approx(10\uparrow)^{255}1.99\times 10^{619}

, where Failed to parse (Cannot write to or create math temp directory): (10\uparrow)^{255}

denotes a functional power of the function Failed to parse (Cannot write to or create math temp directory): f(n)=10^n

. Hence Failed to parse (Cannot write to or create math temp directory): 10\uparrow\uparrow 257 < \mbox{mega} < 10\uparrow\uparrow 258

[edit] 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,

Failed to parse (Cannot write to or create math temp directory): \mbox{moser} < 3\rightarrow 3\rightarrow 65\rightarrow 2

[edit] See also

• Conway chained arrow
• Knuth's up-arrow notation

[edit] External links

• Factoid on Big Numbers
• Robert Munafo's Big Numbers, which hints Steinhaus and Moser came up with this notation jointly in the '70s.
• Megistron at mathworld.wolfram.com
• Circle notation at mathworld.wolfram.com