Steinhaus–Moser notation

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In mathematics, Steinhaus–Moser 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:
*[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)]

Contents

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

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 [f(x)=x^x] we have mega = [f^(256) = f^(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^}]
M(256,3,3) = [(256^})^}}=256^\times 256^}}=256^}}]≈[256^}}]

Similarly:

M(256,4,3) ≈ [}}}}]
M(256,5,3) ≈ [}}}}}]

etc.

Thus:

mega = [M(256,256,3)\approx(256\uparrow)^257], where [(256\uparrow)^] 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^]
[M(256,2,3)\approx10^}] ([\log _ 616] is added to the 616)
[M(256,3,3)\approx10^}}] ([619] is added to the [1.99\times 10^], which is negligible; therefore just a 10 is added at the bottom)

[M(256,4,3)\approx10^}}}]

...

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

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,

[\mbox < 3\rightarrow 3\rightarrow 65\rightarrow 2]

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]

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Steinhaus–Moser notation

Mega

Moser's number

See also

External links