Steinhaus–Moser notation#Mega

:image:Triangle-n.svg a number {{math|n}} in a triangle means {{math|nⁿ}}.

:image:Square-n.svg a number {{math|n}} in a square is equivalent to "the number {{math|n}} inside {{math|n}} triangles, which are all nested."

:image:Pentagon-n.svg a number {{math|n}} in a pentagon is equivalent to "the number {{math|n}} inside {{math|n}} squares, which are all nested."

etc.: {{math|n}} written in an ({{math|m + 1}})-sided polygon is equivalent to "the number {{math|n}} inside {{math|n}} nested {{math|m}}-sided polygons". In a series of nested polygons, they are associated inward. The number {{math|n}} inside two triangles is equivalent to {{math|nⁿ}} inside one triangle, which is equivalent to {{math|nⁿ}} raised to the power of {{math|nⁿ}}.

Steinhaus defined only the triangle, the square, and the circle image:Circle-n.svg, which is equivalent to the pentagon defined above.

Steinhaus defined:

• mega is the number equivalent to 2 in a circle: {{tooltip|2=C(2) = S(S(2))|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 {{math|M(n, m, p)}} be the number represented by the number {{math|n}} in {{math|m}} nested {{math|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 = $M(2,1,M(2,1,5))$

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):

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

Similarly:

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

etc.

Thus:

• mega = $M(256,256,3)\backslash approx(256\backslash uparrow)^\{256\}257$, where $(256\backslash uparrow)^\{256\}$ 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 ≈ $256\backslash uparrow\backslash uparrow\; 257$, 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:

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

...

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

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:[http://www-users.cs.york.ac.uk/~susan/cyc/b/gmproof.htm Proof that G >> M]

:

• Ackermann function

• [http://www.mrob.com/pub/math/largenum.html Robert Munafo's Large Numbers]
• [http://www-users.cs.york.ac.uk/~susan/cyc/b/big.htm Factoid on Big Numbers]
• [http://mathworld.wolfram.com/Megistron.html Megistron at mathworld.wolfram.com] (Steinhaus referred to this number as "megiston" with no "r".)
• [http://mathworld.wolfram.com/CircleNotation.html Circle notation at mathworld.wolfram.com]
• [https://sites.google.com/site/pointlesslargenumberstuff/home/2/steinhausmoser Steinhaus-Moser Notation - Pointless Large Number Stuff]

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