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 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 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 nⁿ inside one triangle, which is equivalent to nⁿ raised to the power of nⁿ.

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:
  • $M(n,1,3) = n^n$
  • $M(n,1,p+1) = M(n,n,p)$
  • $M(n,m+1,p) = M(M(n,1,p),m,p)$
• 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(3.2 × 10⁶¹⁶)...))) [254 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) ≈ ${\,\!256^{256^{256^{256^{257}}}}}$
• M(256,5,3) ≈ ${\,\!256^{256^{256^{256^{256^{257}}}}}}$

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:

• $M(256,1,3)\approx 3.23\times 10^{616}$
• $M(256,2,3)\approx10^{\,\!1.99\times 10^{619}}$ ($\log _{10} 616$ is added to the 616)
• $M(256,3,3)\approx10^{\,\!10^{1.99\times 10^{619}}}$ ($619$ is added to the $1.99\times 10^{619}$, which is negligible; therefore just a 10 is added at the bottom)
• $M(256,4,3)\approx10^{\,\!10^{10^{1.99\times 10^{619}}}}$

...

• 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,

$\mathrm{moser} < 3\rightarrow 3\rightarrow 4\rightarrow 2,$

and, in Knuth's up-arrow notation,

$\mathrm{moser} < f^{3}(4) = f(f(f(4))), \text{ where } f(n) = 3 \uparrow^n 3.$

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

$\mathrm{moser} \ll 3\rightarrow 3\rightarrow 64\rightarrow 2 < f^{64}(4) = \text{Graham's number}.$

See also

• Ackermann function

External links

• Robert Munafo's Large Numbers
• Factoid on Big Numbers
• Megistron at mathworld.wolfram.com
• Circle notation at mathworld.wolfram.com