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

:20px|n in a triangle a number in a triangle means .

:20px|n in a square a number in a square is equivalent to "the number inside triangles, which are all nested."

:20px|n in a pentagon a number in a pentagon is equivalent to "the number inside squares, which are all nested."

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

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:
  • 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 be the number represented by the number in nested -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(2<sup>2</sup>)) =

square(triangle(4)) =

square(4<sup>4</sup>) =

square(256) =

triangle(triangle(triangle(...triangle(256)...))) [256 triangles] =

triangle(triangle(triangle(...triangle(256<sup>256</sup>)...))) [255 triangles] ~

triangle(triangle(triangle(...triangle(3.2317 &times; 10<sup>616</sup>)...))) [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:

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

See also

  • Ackermann function

References