 in mathematics, steinhausâmoser notation is a notation for expressing certain extremely large numbers. it is an extension of steinhaus's polygon notation. == definitions == :20px|n in a triangle a number  in a triangle means . :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  inside one triangle, which is equivalent to  raised to the power of . steinhaus only defined the triangle, the square, and a circle 20px|n in a circle, 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: â© 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: **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 =&nbsp;m(2,1,5) **megiston =&nbsp;m(10,1,5) **moser =&nbsp;m(2,1,m(2,1,5)) ==mega== a mega, â¡, is already a very large number, since â¡ = 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 &times; 10616)...))) [254 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}\times 256^{256^{257}}}=256^{256^{257+256^{257}}}â256^{\,\!256^{256^{257}}} similarly: *m(256,4,3) â {\,\!256^{
Large_numbers Mathematical_notation