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 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 to "the number n inside n squares, which are all nested."

etc.: n written in an (m + 1)-sided polygon is equivalent to "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 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: ②

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 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

)

=

n

n

{\displaystyle M(n,1,3)=n^{n}}

M

(

n

,

1

,

p

+

1

)

=

M

(

n

,

n

,

p

)

{\displaystyle M(n,1,p+1)=M(n,n,p)}

M

(

n

,

m

+

1

,

p

)

=

M

(

M

(

n

,

1

,

p

)

,

m

,

p

)

{\displaystyle M(n,m+1,p)=M(M(n,1,p),m,p)}

and

mega =

M

(

2

,

1

,

5

)

{\displaystyle M(2,1,5)}

megiston =

M

(

10

,

1

,

5

)

{\displaystyle M(10,1,5)}

moser =

M

(

2

,

1

,

M

(

2

,

1

,

5

)

)

{\displaystyle M(2,1,M(2,1,5))}

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)...))) [254 triangles]

...