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.[1]
20px|n in a triangle a number in a triangle means nn.
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 with "the number inside squares, which are all nested."
etc.: written in an -sided polygon is equivalent with "the number inside nested -sided polygons". In a series of nested polygons, they are associated inward. The number inside two triangles is equivalent to nn inside one triangle, which is equivalent to nn raised to the power of nn.
Steinhaus defined only the triangle, the square, and the circle 20px|n in a circle, which is equivalent to the pentagon defined above.
Steinhaus defined:
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:
M(n,1,3)=nn
M(n,1,p+1)=M(n,n,p)
M(n,m+1,p)=M(M(n,1,p),m,p)
M(2,1,5)
M(10,1,5)
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)=xx
f256(256)=f258(2)
We have (note the convention that powers are evaluated from right to left):
(256256
256256 | |
) |
256257 | |
=256 |
256257 | |
(256 |
| |||||
) |
| |||||||||||
=256 |
| |||||
=256 |
| |||||
256 |
| |||||||||
{256 |
| |||||||||||||
{256 |
| |||||||||||||||||
{256 |
Thus:
M(256,256,3) ≈ (256\uparrow)256257
(256\uparrow)256
f(n)=256n
Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈
256\uparrow\uparrow257
After the first few steps the value of
nn
256n
10n
M(256,1,3) ≈ 3.23 x 10616
1.99 x 10619 | |
M(256,2,3) ≈ 10 |
log10616
| |||||
M(256,3,3) ≈ 10 |
619
1.99 x 10619
| |||||||||
M(256,4,3) ≈ 10 |
M(256,256,3) ≈ (10\uparrow)2551.99 x 10619
(10\uparrow)255
f(n)=10n
10\uparrow\uparrow257<mega<10\uparrow\uparrow258
It has been proven that in Conway chained arrow notation,
moser<3 → 3 → 4 → 2,
and, in Knuth's up-arrow notation,
moser<f3(4)=f(f(f(4))),wheref(n)=3\uparrown3.
Therefore, Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number:[2]
moser\ll3 → 3 → 64 → 2<f64(4)=Graham'snumber.