# Steinhaus–Moser notation

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

## Definitions

- a number
`n`in a**triangle**means`n`^{n}.

- 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 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` nested `m`-sided polygons". In a series of nested polygons, they are associated inward. The number `n` inside two triangles is equivalent to `n`^{n} inside one triangle, which is equivalent to `n`^{n} raised to the power of `n`^{n}.

Steinhaus only defined the triangle, the square, and a **circle** , 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", 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: - 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^{2})) =
square(triangle(4)) =
square(4^{4}) =
square(256) =
triangle(triangle(triangle(...triangle(256)...))) [256 triangles] =
triangle(triangle(triangle(...triangle(256^{256})...))) [255 triangles] ~
triangle(triangle(triangle(...triangle(3.2 × 10^{616})...))) [254 triangles] =
...

Using the other notation:

mega = M(2,1,5) = M(256,256,3)

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):

- 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 .

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

## External links

- Robert Munafo's Large Numbers
- Factoid on Big Numbers
- Megistron at mathworld.wolfram.com (Note that Steinhaus referred to this number as "megiston" with no "r".)
- Circle notation at mathworld.wolfram.com