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

<VAR >n</VAR > in a triangle means <VAR >nn</VAR > . <VAR >n</VAR > in a square is equivalent to "the number <VAR >n</VAR > inside <VAR >n</VAR > triangles, which are all nested." <VAR >n</VAR > in a pentagon is equivalent to "the number <VAR >n</VAR > inside <VAR >n</VAR > squares, which are all nested." etc.: <VAR >n</VAR > written in an ( <VAR >m</VAR > + 1 )-sided polygon is equivalent to "the number <VAR >n</VAR > inside <VAR >n</VAR > nested <VAR >m</VAR > -sided polygons". In a series of nested polygons, they are associated inward. The number <VAR >n</VAR > inside two triangles is equivalent to <VAR >nn</VAR > inside one triangle, which is equivalent to <VAR >nn</VAR > raised to the power of <VAR >nn</VAR > . Steinhaus defined only the triangle, the square, and the circle, which is equivalent to the pentagon defined above.

Special values

Steinhaus defined: 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: <VAR>M</VAR>(<VAR >n</VAR >, <VAR >m</VAR >, <VAR >p</VAR >) be the number represented by the number <VAR >n</VAR > in <VAR >m</VAR > nested <VAR >p</VAR > -sided polygons; then the rules are:

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(3.2317 × 10616)...))) [255 triangles] ... Using the other notation: mega = With the function f(x)=x^x 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: 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 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: ...

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: