Steinhaus–Moser notation

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

Definitions

a number n in a triangle means nn.
a number n in a square is equivalent to "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 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 , equivalent to the pentagon defined above.

Special values

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:
• and
• mega =
• megiston =
• moser =

Mega

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 we have mega = 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) =
• M(256,3,3) =

Similarly:

• M(256,4,3) ≈
• M(256,5,3) ≈

etc.

Thus:

• mega = , where denotes a functional power of the function .

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

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

• ( is added to the 616)
• ( is added to the , which is negligible; therefore just a 10 is added at the bottom)

...

• mega = , where denotes a functional power of the function . Hence

Moser's number

It has been proven that in Conway chained arrow notation,

and, in Knuth's up-arrow notation,

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