Steinhaus–Moser notation explained

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]

Definitions

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.

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:

M(n, 1 , 3 ) = n ^{n} M(n, 1 ,p +1 ) = M(n,n,p) M(n,m +1 ,p) = M(M(n, 1 ,p),m,p)

• and
  • mega = 

M( 2 , 1 , 5 )

• • megiston = 

M( 10 , 1 , 5 )

• • moser = 

M( 2 , 1 ,M( 2 , 1 , 5 ))

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 × 10⁶¹⁶)...))) [255 triangles]...

Using the other notation:

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

With the function

f(x) = x ^{x} we have mega = f ^{256} ( 256 ) = f ^{258} ( 2 ) 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) =

( 256 ^{256} ) ^{256 ^{256}} =256 ^{256 ^{257}}

• M(256,3,3) =

( 256 ^{256 ^{257}} ) ^{256 ^{256 ^{257}}} =256 ^{256 ^{257} x 256 ^{256 ^{257}}} =256 ^{256 ^{257+256 ^{257}}} ≈256 ^{256 ^{256 ^{257}}} Similarly:

• M(256,4,3) ≈

{ 256 ^{256 ^{256 ^{256 ^{257}}}} }

• M(256,5,3) ≈

{ 256 ^{256 ^{256 ^{256 ^{256 ^{257}}}}} }

• M(256,6,3) ≈

{ 256 ^{256 ^{256 ^{256 ^{256 ^{256 ^{257}}}}}} }etc.

Thus:

• mega =

M( 256 , 256 , 3 ) ≈ ( 256 \uparrow) ^{256} 257 , where ( 256 \uparrow) ^{256} denotes a functional power of the function f(n) =256 ^{n} .

Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈

256 \uparrow\uparrow 257 , using Knuth's up-arrow notation.

After the first few steps the value of

n ^{n} is each time approximately equal to 256 ^{n} . In fact, it is even approximately equal to 10 ^{n} (see also approximate arithmetic for very large numbers). Using base 10 powers we get:M( 256 , 1 , 3 ) ≈ 3 . 23 x 10 ^{616} M( 256 , 2 , 3 ) ≈ 10 ^{1 . 99 x 10 ^{619}} (log _{10} 616 is added to the 616)M( 256 , 3 , 3 ) ≈ 10 ^{10 ^{1 . 99 x 10 ^{619}}} (619 is added to the 1 . 99 x 10 ^{619} , which is negligible; therefore just a 10 is added at the bottom)M( 256 , 4 , 3 ) ≈ 10 ^{10 ^{10 ^{1 . 99 x 10 ^{619}}}} ...

• mega =

M( 256 , 256 , 3 ) ≈ ( 10 \uparrow) ^{255} 1 . 99 x 10 ^{619} , where ( 10 \uparrow) ^{255} denotes a functional power of the function f(n) =10 ^{n} . Hence 10 \uparrow\uparrow 257 < mega < 10 \uparrow\uparrow 258

Moser's number

It has been proven that in Conway chained arrow notation,

moser < 3 → 3 → 4 → 2 ,

and, in Knuth's up-arrow notation,

moser < f ^{3} ( 4 ) = f(f(f( 4 ))), where f(n) = 3 \uparrow ^{n} 3 .

Therefore, Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number:^[2]

moser \ll 3 → 3 → 64 → 2 < f ^{64} ( 4 ) = Graham'snumber .

See also

• Ackermann function

References

1. Hugo Steinhaus, Mathematical Snapshots, Oxford University Press 1969³,, pp. 28-29
2. http://www-users.cs.york.ac.uk/~susan/cyc/b/gmproof.htm Proof that G >> M

External links

• Robert Munafo's Large Numbers
• Factoid on Big Numbers
• Megistron at mathworld.wolfram.com (Steinhaus referred to this number as "megiston" with no "r".)
• Circle notation at mathworld.wolfram.com
• Steinhaus-Moser Notation - Pointless Large Number Stuff