Steinhaus–Moser notation


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

Definitions

etc.: written in an -sided polygon is equivalent to "the number inside nested -sided polygons". In a series of nested polygons, they are associated inward. The number inside two triangles is equivalent to inside one triangle, which is equivalent to raised to the power of.
Steinhaus defined only the triangle, the square, and the circle [image:Circle-n.svg|20px|n in a circle], which is 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". Megagon is here the name of a polygon with "mega" sides.
Alternative notations:
  • use the functions square and triangle
  • let be the number represented by the number in nested -sided polygons; then the rules are:
  • *
  • *
  • *
  • and
  • *mega =
  • *megiston =
  • *moser =

Mega

A mega, ②, is already a very large number, since ② =
square = square =
square =
square =
square =
square =
triangle) =
triangle) ~
triangle)
...
Using the other notation:
mega =
With the function we have mega = where the superscript denotes a functional power, not a numerical power.
We have :
Similarly:
etc.
Thus:
Rounding more crudely, 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 . Using base 10 powers we get:
...
  • 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: