Steinhaus–Moser notation
Encyclopedia
In mathematics
, Steinhaus
–Moser
notation
is a means of expressing certain extremely large numbers. It is an extension of Steinhaus’s polygon
notation.
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 inside one triangle, which is equivalent to raised to the power of .
Steinhaus only defined the triangle, the square, and a circle , equivalent to the pentagon defined above.
Moser’s number is the number represented by "2 in a megagon", where a megagon is a polygon with "mega" sides.
Alternative notations:
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)...))) [255 triangles] =
...
Using the other notation:
mega = M(2,1,5) = M(256,256,3)
With the function
we have mega = 
where the superscript denotes a functional power, not a numerical power.
We have (note the convention that powers are evaluated from right to left):
Similarly:
etc.
Thus:
Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈
, using Knuth's up-arrow notation
.
After the first few steps the value of
is each time approximately equal to 
. In fact, it is even approximately equal to 
(see also approximate arithmetic for very large numbers). Using base 10 powers we get:
...
,
and, in Knuth's up-arrow notation
,
Therefore Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number
:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Steinhaus
Hugo Steinhaus
Władysław Hugo Dionizy Steinhaus was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the University of Lwów, where he helped establish what later became known as the Lwów School of Mathematics...
–Moser
Leo Moser
Leo Moser was an Austrian-Canadian mathematician, best known for his polygon notation....
notation
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
is a means of expressing certain extremely large numbers. It is an extension of Steinhaus’s polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
notation.
Definitions
- a number in a triangle means .
- a number in a square is equivalent with "the number inside triangles, which are all nested."
- 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 inside one triangle, which is equivalent to raised to the power of .
Steinhaus only defined the triangle, the square, and a circle , equivalent to the pentagon defined above.
Special values
Steinhaus defined:- mega is the number equivalent to 2 in a circle: ②
- megiston is the number equivalent to 10 in a circle: ⑩
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 be the number represented by the number in nested -sided polygons; then the rules are:
- and
-
- mega =
- moser =
- mega =
-
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(3.2 × 10616)...))) [255 triangles] =
...
Using the other notation:
mega = M(2,1,5) = M(256,256,3)
With the function
we have mega = 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) =
- M(256,3,3) =
≈
Similarly:
- M(256,4,3) ≈
- M(256,5,3) ≈
etc.
Thus:
- mega =
, where 
denotes a functional power of the function 
.
Rounding more crudely (replacing the 257 at the end by 256), we get mega ≈
, using Knuth's up-arrow notationKnuth's up-arrow notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. It is closely related to the Ackermann function and especially to the hyperoperation sequence. The idea is based on the fact that multiplication can be viewed as iterated...
.
After the first few steps the value of
is each time approximately equal to . In fact, it is even approximately equal to (see also approximate arithmetic for very large numbers). Using base 10 powers we get:
(
is added to the 616)
(
is added to the 
, which is negligible; therefore just a 10 is added at the bottom)
...
- mega =
, where 
denotes a functional power of the function 
. Hence 
Moser's number
It has been proven that in Conway chained arrow notationConway chained arrow notation
Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g. 2 → 3 → 4 → 5 → 6.As with most...
,
and, in Knuth's up-arrow notation
Knuth's up-arrow notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. It is closely related to the Ackermann function and especially to the hyperoperation sequence. The idea is based on the fact that multiplication can be viewed as iterated...
,
Therefore Moser's number, although incomprehensibly large, is vanishingly small compared to Graham's number
Graham's number
Graham's number, named after Ronald Graham, is a large number that is an upper bound on the solution to a certain problem in Ramsey theory.The number gained a degree of popular attention when Martin Gardner described it in the "Mathematical Games" section of Scientific American in November 1977,...
: