In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Steinhaus–Moser notation is a
notation
In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefore, ...
for expressing certain
large number
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, physical cosmology, cosmology, cryptograph ...
s. It is an extension (devised by
Leo Moser
Leo Moser (11 April 1921, Vienna – 9 February 1970, Edmonton) was an Austrian-Canadian mathematician, best known for his polygon notation.
A native of Vienna, Leo Moser immigrated with his parents to Canada at the age of three. He received his ...
) of
Hugo Steinhaus
Hugo Dyonizy Steinhaus ( ; ; January 14, 1887 – February 25, 1972) 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 Jan Kazimierz Unive ...
's polygon notation.
Definitions
:
20px, n in a triangle a number in a triangle means n
n.
:
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 Associated may refer to:
*Associated, former name of Avon, Contra Costa County, California
* Associated Hebrew Schools of Toronto, a school in Canada
*Associated Newspapers, former name of DMG Media, a British publishing company
See also
*Associati ...
inward. The number inside two triangles is equivalent to n
n inside one triangle, which is equivalent to n
n raised to the power of n
n.
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:
**
**
**
* and
**mega =
**megiston =
**moser =
Mega
A mega, ②, is already a very large number, since ② =
square(square(2)) = square(triangle(triangle(2))) =
square(triangle(2
2)) =
square(triangle(4)) =
square(4
4) =
square(256) =
triangle(triangle(triangle(...triangle(256)...)))
56 triangles=
triangle(triangle(triangle(...triangle(256
256)...)))
55 triangles~
triangle(triangle(triangle(...triangle(3.2 × 10
616)...)))
54 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
In mathematics, function composition is an operation that takes two function (mathematics), functions and , and produces a function such that . In this operation, the function is function application, applied to the result of applying the ...
, 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) ≈
*M(256,6,3) ≈
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 notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.
In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperati ...
.
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 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\to3\to4\to5\to6.
As wit ...
,
:
and, in
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.
In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperati ...
,
:
Therefore, Moser's number, although incomprehensibly large, is vanishingly small compared to
Graham's number
Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers such as Skewes's number and Moser's number, both of which ar ...
:
Proof that G >> M
/ref>
:
See also
* Ackermann function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total ...
References
External links
Robert Munafo's Large Numbers
(Steinhaus referred to this number as "megiston" with no "r".)
Steinhaus-Moser Notation - Pointless Large Number Stuff
{{DEFAULTSORT:Steinhaus-Moser notation
Mathematical notation
Large numbers