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| In [[mathematics]], '''Steinhaus–[[Leo Moser|Moser]] notation''' is a [[mathematical notation|notation]] for expressing certain extremely [[large number]]s. It is an extension of [[Hugo Steinhaus|Steinhaus]]'s polygon notation.
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| == Definitions ==
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| :[[image:Triangle-n.svg|20px|n in a triangle]] a number {{math|<VAR >n</VAR >}} in a '''triangle''' means {{math|<VAR >n</VAR ><sup><VAR >n</VAR ></sup>}}.
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| :[[image:Square-n.svg|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."
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| :[[image:Pentagon-n.svg|20px|n in a pentagon]] a number {{math|<VAR >n</VAR >}} in a '''pentagon''' is equivalent with "the number {{math|<VAR >n</VAR >}} inside {{math|<VAR >n</VAR >}} squares, which are all nested."
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| etc.: {{math|<VAR >n</VAR >}} written in an ({{math|<VAR >m</VAR > + 1}})-sided polygon is equivalent with "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 [[Association (mathematics)|associated]] inward. The number {{math|<VAR >n</VAR >}} inside two triangles is equivalent to {{math|<VAR >n</VAR ><sup><VAR >n</VAR ></sup >}} inside one triangle, which is equivalent to {{math|<VAR >n</VAR ><sup><VAR >n</VAR ></sup>}} raised to the power of {{math|<VAR >n</VAR ><sup><VAR >n</VAR ></sup>}}.
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| Steinhaus only defined the triangle, the square, and a '''circle''' [[image:Circle-n.svg|20px|n in a circle]], equivalent to the pentagon defined above.
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| == Special values ==
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| Steinhaus defined:
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| *'''mega''' is the number equivalent to 2 in a circle: {{h:title|<nowiki>C(2) = S(S(2))</nowiki>|②}}
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| *'''megiston''' is the number equivalent to 10 in a circle: ⑩
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| '''Moser's number''' is the number represented by "2 in a megagon", where a '''megagon''' is a polygon with "mega" sides.
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| Alternative notations:
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| *use the functions square(x) and triangle(x)
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| *let {{math|M(<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:
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| **<math>M(n,1,3) = n^n</math>
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| **<math>M(n,1,p+1) = M(n,n,p)</math>
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| **<math>M(n,m+1,p) = M(M(n,1,p),m,p)</math>
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| * and
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| **mega = <math>M(2,1,5)</math>
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| **megiston = <math>M(10,1,5)</math>
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| **moser = <math>M(2,1,M(2,1,5))</math>
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| ==Mega==
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| A mega, ②, is already a very large number, since ② =
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| square(square(2)) = square(triangle(triangle(2))) =
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| square(triangle(2<sup>2</sup>)) =
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| square(triangle(4)) =
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| square(4<sup>4</sup>) =
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| square(256) =
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| triangle(triangle(triangle(...triangle(256)...))) [256 triangles] =
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| triangle(triangle(triangle(...triangle(256<sup>256</sup>)...))) [255 triangles] ~
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| triangle(triangle(triangle(...triangle(3.2 × 10<sup>616</sup>)...))) [254 triangles] =
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| ...
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| Using the other notation:
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| mega = M(2,1,5) = M(256,256,3)
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| 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 [[composite function|functional power]], not a numerical power.
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| We have (note the convention that powers are evaluated from right to left):
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| *M(256,2,3) = <math>(256^{\,\!256})^{256^{256}}=256^{256^{257}}</math>
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| *M(256,3,3) = <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>
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| Similarly:
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| *M(256,4,3) ≈ <math>{\,\!256^{256^{256^{256^{257}}}}}</math>
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| *M(256,5,3) ≈ <math>{\,\!256^{256^{256^{256^{256^{257}}}}}}</math>
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| etc.
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| Thus:
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| *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>.
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| 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]].
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| 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 [[Large numbers#Approximate arithmetic for very large numbers|approximate arithmetic for very large numbers]]). Using base 10 powers we get:
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| *<math>M(256,1,3)\approx 3.23\times 10^{616}</math>
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| *<math>M(256,2,3)\approx10^{\,\!1.99\times 10^{619}}</math> (<math>\log _{10} 616</math> is added to the 616)
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| *<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)
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| *<math>M(256,4,3)\approx10^{\,\!10^{10^{1.99\times 10^{619}}}}</math>
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| ...
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| *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>
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| ==Moser's number<!--This section is linked from [[Moser's number]]-->==
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| It has been proven that in [[Conway chained arrow notation]],
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| :<math>\mathrm{moser} < 3\rightarrow 3\rightarrow 4\rightarrow 2,</math>
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| and, in [[Knuth's up-arrow notation]],
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| :<math>\mathrm{moser} < f^{3}(4) = f(f(f(4))), \text{ where } f(n) = 3 \uparrow^n 3.</math>
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| Therefore Moser's number, although incomprehensibly large, is vanishingly small compared to [[Graham's number]]:
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| :<math>\mathrm{moser} \ll 3\rightarrow 3\rightarrow 64\rightarrow 2 < f^{64}(4) = \text{Graham's number}.</math>
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| == See also ==
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| * [[Ackermann function]]
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| ==External links==
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| * [http://www.mrob.com/pub/math/largenum.html Robert Munafo's Large Numbers]
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| * [http://www-users.cs.york.ac.uk/~susan/cyc/b/big.htm Factoid on Big Numbers]
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| *[http://mathworld.wolfram.com/Megistron.html Megistron at mathworld.wolfram.com]
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| *[http://mathworld.wolfram.com/CircleNotation.html Circle notation at mathworld.wolfram.com]
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| {{Large numbers}}
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| {{DEFAULTSORT:Steinhaus-Moser notation}}
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| [[Category:Mathematical notation]]
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| [[Category:Large numbers]]
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Fascinated to meet you! The little name is Eusebio Ledbetter. To drive is a complete thing that I'm ultimately addicted to. South Carolina is where all of my home is and Since i don't plan on shifting it. I used as a way to be unemployed but actually I am a cashier but the promotion for no reason comes. I'm not good at webdesign but you might want to check my website: http://prometeu.net
Stop by my site; clash of clans hack tool no survey or password