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Steinhaus–Moser notation

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

Definitions

a number in a triangle means .
a number in a square is equivalent to "the number inside triangles, which are all nested."
a number in a pentagon is equivalent to "the number inside squares, which are all nested."

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 , 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 (not to be confused with the polygon with one million sides).

Alternative notations:

  • use the functions square(x) and triangle(x)
  • 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(2)) = square(triangle(triangle(2))) = square(triangle(2<sup>2</sup>)) = square(triangle(4)) = square(4<sup>4</sup>) = square(256) = triangle(triangle(triangle(...triangle(256)...))) [256 triangles] = triangle(triangle(triangle(...triangle(256<sup>256</sup>)...))) [255 triangles] ~ triangle(triangle(triangle(...triangle(3.2317 &times; 10<sup>616</sup>)...))) [255 triangles] ...

Using the other notation:

mega =

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):

  • ≈

Similarly:

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:

See also

References

External links