Steinhaus–Moser notation

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In mathematics, Steinhaus–Moser notation is a notation for expressing certain extremely large numbers. It is an extension of Steinhaus's polygon notation.

Definitions[edit]

n in a triangle a number n in a triangle means nn.
n in a square a number n in a square is equivalent to "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 nested m-sided polygons". 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.

Special values[edit]

Steinhaus defined:

  • mega is the number equivalent to 2 in a circle:
  • 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) = n^n
    • M(n,1,p+1) = M(n,n,p)
    • M(n,m+1,p) = M(M(n,1,p),m,p)
  • and
    • mega = M(2,1,5)
    • megiston = M(10,1,5)
    • moser = M(2,1,M(2,1,5))

Mega[edit]

A mega, 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)...))) [254 triangles] = ...

Using the other notation:

mega = M(2,1,5) = M(256,256,3)

With the function f(x)=x^x we have mega = 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) = (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)=256^n.

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

After the first few steps the value of n^n is each time approximately equal to 256^n. In fact, it is even approximately equal to 10^n (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}} (\log _{10} 616 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)=10^n. Hence 10\uparrow\uparrow 257 < \text{mega} < 10\uparrow\uparrow 258

Moser's number[edit]

It has been proven that in Conway chained arrow notation,

\mathrm{moser} < 3\rightarrow 3\rightarrow 4\rightarrow 2,

and, in Knuth's up-arrow notation,

\mathrm{moser} < f^{3}(4) = f(f(f(4))), \text{ where } f(n) = 3 \uparrow^n 3.

Therefore Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number:

\mathrm{moser} \ll  3\rightarrow 3\rightarrow 64\rightarrow 2 < f^{64}(4) = \text{Graham's number}.

See also[edit]

External links[edit]