Steinhaus-Moser notation

From Academic Kids

In mathematics, Moser's polygon notation is a means of expressing certain extremely large numbers. It is an extension of Steinhaus's polygon notation.

n in a triangle (a number n in a triangle) means nn

Missing image
Square-n.png
n in a square

(a number n in a square) is equivalent with "the number n inside n triangles, which are all nested"

Missing image
Pentagon-n.png
n in a pentagon

(a number n in a pentagon) is equivalent with "the number n inside n squares, which are all nested"

etc.: n written in an (m+1)-sided polygon is equivalent with "the number n inside n m-sided polygons, which are all nested"

Steinhaus only defined the triangle, the square, and a circle Missing image
Circle-n.png
n in a cicle

, equivalent to the pentagon defined above.

Steinhaus defined:

  • "mega" is the number equivalent to 2 in a circle: 2 in a circle
  • "megistron" is the number equivalent to 10 in a circle: 10 in a circle

Moser's number is the number represented by "2 in a megagon", where a "megagon" is a polygon with "mega" 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:
    • <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\big(M(n,1,p),m,p\big)<math>
and
    • mega = <math>M(2,1,5)<math>
    • moser = <math>M\big(2,1,M(2,1,5)\big)<math>
Contents

Mega

Note that 2 in a circle is already a very large number, since 2 in a circle = 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(3.2 × 10616)...))) [254 triangles] = ...

Using the other notation:

mega = M(2,1,5) = M(256,256,3)

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):

  • M(256,2,3) = <math>(256^{\,\!256})^{256^{256}}=256^{256^{257}}<math>
  • 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>

Similarly:

  • M(256,4,3) ≈ <math>{\,\!256^{256^{256^{256^{257}}}}}<math>
  • M(256,5,3) ≈ <math>{\,\!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.

Note that 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 < \mbox{mega} < 10\uparrow\uparrow 258<math>

Moser's number

It has been proved that Moser's number, although extremely large, is smaller than Graham's number.

Therefore, using the Conway chained arrow notation,

<math>\mbox{moser} < 3\rightarrow 3\rightarrow 65\rightarrow 2<math>

See also

External

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