Steinhaus–Moser notation

{{short description|Notation for extremely large numbers}} 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.<ref>Hugo Steinhaus, ''Mathematical Snapshots'', Oxford University Press 1969³, {{ISBN|0195032675}}, pp. 28-29</ref>

== Definitions == :20px|n in a triangle a number {{math|<VAR >n</VAR >}} in a '''triangle''' means {{math|<VAR >nⁿ</VAR >}}.

:20px|n in a square a number {{math|<VAR >n</VAR >}} in a '''square''' is equivalent to "the number {{math|<VAR >n</VAR >}} inside {{math|<VAR >n</VAR >}} triangles, which are all nested."

:20px|n in a pentagon a number {{math|<VAR >n</VAR >}} in a '''pentagon''' is equivalent to "the number {{math|<VAR >n</VAR >}} inside {{math|<VAR >n</VAR >}} squares, which are all nested."

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

Steinhaus defined only the triangle, the square, and the '''circle''' 20px|n in a circle, which is equivalent to the pentagon defined above.

== Special values == 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|<VAR>M</VAR>(<VAR >n</VAR >, <VAR >m</VAR >, <VAR >p</VAR >)}} be the number represented by the number {{math|<VAR >n</VAR >}} in {{math|<VAR >m</VAR >}} nested {{math|<VAR >p</VAR >}}-sided polygons; then the rules are: **<math>M(n,1,3) = n^n</math> **<math>M(n,1,p+1) = M(n,n,p)</math> **<math>M(n,m+1,p) = M(M(n,1,p),m,p)</math> * and **mega =&nbsp;<math>M(2,1,5)</math> **megiston =&nbsp;<math>M(10,1,5)</math> **moser =&nbsp;<math>M(2,1,M(2,1,5))</math>

==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 &times; 10⁶¹⁶)...))) [255 triangles] ...

Using the other notation:

mega = <math>M(2,1,5) = M(256,256,3)</math>

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

We have (note the convention that powers are evaluated from right to left): *<math>M(256,2,3) =</math> <math>(256^{\,\!256})^{256^{256}}=256^{256^{257}}</math> *<math>M(256,3,3) =</math> <math>(256^{\,\!256^{257}})^{256^{256^{257}}}=256^{256^{257}\times 256^{256^{257}}}=256^{256^{257+256^{257}}}</math>≈<math>256^{\,\!256^{256^{257}}}</math> Similarly: *<math>M(256,4,3) \approx</math> <math>{\,\!256^{256^{256^{256^{257}}}}}</math> *<math>M(256,5,3) \approx</math> <math>{\,\!256^{256^{256^{256^{256^{257}}}}}}</math> *<math>M(256,6,3) \approx</math> <math>{\,\!256^{256^{256^{256^{256^{256^{257}}}}}}}</math> etc.

Thus: *mega = <math>M(256,256,3)\approx(256\uparrow)^{256}257</math>, where <math>(256\uparrow)^{256}</math> denotes a functional power of the function <math>f(n)=256^n</math>.

Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈ <math>256\uparrow\uparrow 257</math>, using Knuth's up-arrow notation.

After the first few steps the value of <math>n^n</math> is each time approximately equal to <math>256^n</math>. In fact, it is even approximately equal to <math>10^n</math> (see also approximate arithmetic for very large numbers). Using base 10 powers we get: *<math>M(256,1,3)\approx 3.23\times 10^{616}</math> *<math>M(256,2,3)\approx10^{\,\!1.99\times 10^{619}}</math> (<math>\log _{10} 616</math> is added to the 616) *<math>M(256,3,3)\approx10^{\,\!10^{1.99\times 10^{619}}}</math> (<math>619</math> is added to the <math>1.99\times 10^{619}</math>, which is negligible; therefore just a 10 is added at the bottom) *<math>M(256,4,3)\approx10^{\,\!10^{10^{1.99\times 10^{619}}}}</math> ... *mega = <math>M(256,256,3)\approx(10\uparrow)^{255}1.99\times 10^{619}</math>, where <math>(10\uparrow)^{255}</math> denotes a functional power of the function <math>f(n)=10^n</math>. Hence <math>10\uparrow\uparrow 257 < \text{mega} < 10\uparrow\uparrow 258</math>

==Moser's number<!--This section is linked from Moser's number-->==

It has been proven that in Conway chained arrow notation,

:<math>\mathrm{moser} < 3\rightarrow 3\rightarrow 4\rightarrow 2,</math>

and, in Knuth's up-arrow notation,

:<math>\mathrm{moser} < f^{3}(4) = f(f(f(4))), \text{ where } f(n) = 3 \uparrow^n 3.</math>

Therefore, Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number:<ref>[http://www-users.cs.york.ac.uk/~susan/cyc/b/gmproof.htm Proof that G >> M]</ref>

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

== See also == * Ackermann function

== References == <references />

==External links== * [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]

{{Hyperoperations}} {{Large numbers}}

{{DEFAULTSORT:Steinhaus-Moser notation}} Category:Mathematical notation Category:Large numbers