Steinhaus–Moser notation

Template:Short description 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 Template:Math in a triangle means Template:Math.

a number Template:Math in a square is equivalent to "the number Template:Math inside Template:Math triangles, which are all nested."

a number Template:Math in a pentagon is equivalent to "the number Template:Math inside Template:Math squares, which are all nested."

etc.: Template:Math written in an (Template:Math)-sided polygon is equivalent to "the number Template:Math inside Template:Math nested Template:Math-sided polygons". In a series of nested polygons, they are associated inward. The number Template:Math inside two triangles is equivalent to Template:Math inside one triangle, which is equivalent to Template:Math raised to the power of Template:Math.

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: Template:Tooltip
• 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 Template:Math be the number represented by the number Template:Math in Template:Math nested Template:Math-sided polygons; then the rules are:
  • $M(n,1,3)=n^n$
  • $M(n,1,p+1)=M(n,n,p)$
  • $M(n,m+1,p)=M(M(n,1,p),m,p)$
• and
  • mega = $M(2,1,5)$
  • megiston = $M(10,1,5)$
  • moser = $M(2,1,M(2,1,5))$

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)...))) [255 triangles] ...

Using the other notation:

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

With the function $f(x)=x^x$ we have mega = $f^{256}(256)=f^{258}(2)$ where the superscript denotes a functional power, not a numerical power.

We have (note the convention that powers are evaluated from right to left):

• $M(256,2,3)=$ $(256^{256}{)}^{256^{256}}=256^{256^{257}}$
• $M(256,3,3)=$ $(256^{256^{257}}{)}^{256^{256^{257}}}=256^{256^{257}\times 256^{256^{257}}}=256^{256^{257+256^{257}}}$≈$256^{256^{256^{257}}}$

Similarly:

• $M(256,4,3)\approx$ $256^{256^{256^{256^{257}}}}$
• $M(256,5,3)\approx$ $256^{256^{256^{256^{256^{257}}}}}$
• $M(256,6,3)\approx$ $256^{256^{256^{256^{256^{256^{257}}}}}}$

etc.

Thus:

• mega = $M(256,256,3)\approx (256\uparrow {)}^{256}257$, where $(256\uparrow {)}^{256}$ denotes a functional power of the function $f(n)=256^n$.

Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈ $256\uparrow \uparrow 257$, 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:

• $M(256,1,3)\approx 3.23\times 10^{616}$
• $M(256,2,3)\approx 10^{1.99\times 10^{619}}$ (${\log }_{10}616$ is added to the 616)
• $M(256,3,3)\approx 10^{10^{1.99\times 10^{619}}}$ ($619$ is added to the $1.99\times 10^{619}$, which is negligible; therefore just a 10 is added at the bottom)
• $M(256,4,3)\approx 10^{10^{10^{1.99\times 10^{619}}}}$

...

• mega = $M(256,256,3)\approx (10\uparrow {)}^{255}1.99\times 10^{619}$, where $(10\uparrow {)}^{255}$ denotes a functional power of the function $f(n)=10^n$. Hence $10\uparrow \uparrow 257<\text{mega}<10\uparrow \uparrow 258$

Moser's number

It has been proven that in Conway chained arrow notation,

$\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{r}<3\to 3\to 4\to 2,$

and, in Knuth's up-arrow notation,

$\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{r}<f^3(4)=f(f(f(4))),\text{ where }f(n)=3{\uparrow }^{n}3.$

Therefore, Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number:

$\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{r}\ll 3\to 3\to 64\to 2<f^{64}(4)=\text{Graham's number}.$

See also

• Ackermann function

External links

• Robert Munafo's Large Numbers
• Factoid on Big Numbers
• Megistron at mathworld.wolfram.com (Steinhaus referred to this number as "megiston" with no "r".)
• Circle notation at mathworld.wolfram.com
• Steinhaus-Moser Notation - Pointless Large Number Stuff