Misplaced Pages

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

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:
  • ${\displaystyle M(n,1,3)=n^{n}}$
  • ${\displaystyle M(n,1,p+1)=M(n,n,p)}$
  • ${\displaystyle M(n,m+1,p)=M(M(n,1,p),m,p)}$
• and
  • mega = ${\displaystyle M(2,1,5)}$
  • megiston = ${\displaystyle M(10,1,5)}$
  • moser = ${\displaystyle M(2,1,M(2,1,5))}$

Mega

A mega, ②, is already a very large number, since ② = square(square(2)) = square(triangle(triangle(2))) = square(triangle(2)) = square(triangle(4)) = square(4) = square(256) = triangle(triangle(triangle(...triangle(256)...))) = triangle(triangle(triangle(...triangle(256)...))) ~ triangle(triangle(triangle(...triangle(3.2317 × 10)...))) ...

Using the other notation:

mega = ${\displaystyle M(2,1,5)=M(256,256,3)}$

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

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

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

Similarly:

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

etc.

Thus:

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

Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈ ${\displaystyle 256\uparrow \uparrow 257}$, using Knuth's up-arrow notation.

After the first few steps the value of ${\displaystyle n^{n}}$ is each time approximately equal to ${\displaystyle 256^{n}}$. In fact, it is even approximately equal to ${\displaystyle 10^{n}}$ (see also approximate arithmetic for very large numbers). Using base 10 powers we get:

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

...

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

Moser's number

It has been proven that in Conway chained arrow notation,

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

and, in Knuth's up-arrow notation,

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

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

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

See also

• Ackermann function

References

1. Hugo Steinhaus, Mathematical Snapshots, Oxford University Press 1969, ISBN 0195032675, pp. 28-29
2. Proof that G >> M

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

• Mathematical notation
• Large numbers