Steinhaus–Moser notation

In mathematics, Steinhaus–Moser notation is a means of 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 with "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) = nⁿ
  • 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)
  • 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)...))) [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 = f²⁵⁶(256) = f²⁵⁸(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) = 256ⁿ.

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 nⁿ is each time approximately equal to 256ⁿ. In fact, it is even approximately equal to 10ⁿ (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 ₁₀616 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) = 10ⁿ. 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,

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

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

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

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