In mathematics, Moser's polygon 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:
• "megistron" 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) = nⁿ
  • M(n,1,p + 1) = M(n,n,p)
  • 

and

• • mega = M(2,1,5)
  • moser = 

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 = 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) =
• M(256,3,3) = ≈

Similarly:

• M(256,4,3) ≈
• M(256,5,3) ≈

etc.

Thus:

• mega = , where denotes a functional power of the function f(n) = 256ⁿ.

Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈ , 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:

• 
• (log₁₀616 is added to the 616)
• (619 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 f(n) = 10ⁿ. Hence

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,

See also

• Conway chained arrow
• Knuth's up-arrow notation

External

• 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