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.

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" Steinhaus only defined the triangle, the square, and a circle n in a cicle, equivalent to the pentagon defined above. Steinhaus defined:

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

$M(n,1,3) = n^n$
$M(n,1,p+1) = M(n,n,p)$
$M(n,m+1,p) = M\big(M(n,1,p),m,p\big)$

and

mega = $M(2,1,5)$
moser = $M\big(2,1,M(2,1,5)\big)$

Mega

Note that 2 in a circle is already a very large number, since 2 in a circle = square(square(2)) = square(triangle(triangle(2))) = square(triangle(2²)) = square(triangle(4)) = square(4⁴) = square(256) = triangle(triangle(triangle(...triangle(256)...))) triangles = triangle(triangle(triangle(...triangle(256²⁵⁶)...))) triangles = triangle(triangle(triangle(...triangle(3.2 × 10⁶¹⁶)...))) 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. Note that 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 $10\uparrow\uparrow 257 < \mbox{mega} < 10\uparrow\uparrow 258$

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,

\mbox{moser} < 3\rightarrow 3\rightarrow 65\rightarrow 2

External

Factoid on Big Numbers
Robert Munafo's Big Numbers, which hints Steinhaus and Moser came up with this notation jointly in the '70s.

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