Steinhaus–Moser notation

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In mathematics, SteinhausMoser notation is a means of expressing certain extremely large numbers. It is an extension of Steinhaus' polygon notation.

n in a triangle (a number n in a triangle) means nn.

n in a square (a number n in a square) is equivalent with "the number n inside n triangles, which are all nested."

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. 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 n in 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:
    • M(n,1,3) = nn
    • M(n,1,p + 1) = M(n,n,p)
    • M(n,m+1,p) = M\big(M(n,1,p),m,p\big)
and
    • mega = M(2,1,5)
    • moser = M\big(2,1,M(2,1,5)\big)

[edit] 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 f(x) = xx we have mega = f256(256) = f258(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) = (256^{\,\!256})^{256^{256}}=256^{256^{257}}
  • M(256,3,3) = (256^{\,\!256^{257}})^{256^{256^{257}}}=256^{256^{257}\times 256^{256^{257}}}=256^{256^{257+256^{257}}}256^{\,\!256^{256^{257}}}

Similarly:

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

etc.

Thus:

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

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

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

  • M(256,1,3)\approx 3.23\times 10^{616}
  • M(256,2,3)\approx10^{\,\!1.99\times 10^{619}} (log10616 is added to the 616)
  • M(256,3,3)\approx10^{\,\!10^{1.99\times 10^{619}}} (619 is added to the 1.99\times 10^{619}, which is negligible; therefore just a 10 is added at the bottom)
  • M(256,4,3)\approx10^{\,\!10^{10^{1.99\times 10^{619}}}}

...

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

[edit] Moser's number

It has been proven, using the Conway chained arrow notation, that

\mbox{moser} < 3\rightarrow 3\rightarrow 64\rightarrow 2

Therefore Moser's number, although extremely large, is smaller than Graham's number.

[edit] External links