Steinhaus-Moser notation

The Steinhaus-Moser notation is a way of representing very large numbers. It was proposed as circular notation by the Polish mathematician Hugo Steinhaus in 1950 and later expanded to include polygon notation by the Austrian Leo Moser . Both are based on the notation of high powers using geometric symbols.

Circle notation

The symbol denotes the number . Then stands for the number “ n in n nested triangles” as well as for “n in n nested squares”. ${\ displaystyle n ^ {n}}$

A 2 in a square would correspond to a 2 in two nested triangles, i.e. the number

{\ displaystyle \ left (2 ^ {2} \ right) ^ {\ left (2 ^ {2} \ right)} = 4 ^ {4} = 256}

.

But even the number can hardly be represented with the usual number system, since the exponents of the number themselves constantly grow exponentially (each newly formed number is raised to the power of itself, the number generated by it again with itself and so on). See also the section below .

Polygon notation

The basic structure of polygon notation or polygon notation is the same as that of circle notation, only the square is not followed by the circle as the largest element, but pentagons, hexagons, heptagons or even higher ones are added. This means that significantly larger numbers can be displayed. thus corresponds to n in n nested squares and is equivalent to in circle notation.

In general, " in a -sided polygon" stands for "the number n in n m -sided nested polygons". ${\ displaystyle n}$ ${\ displaystyle (m + 1)}$

Numbers named by Steinhaus and Moser

a mega is the number that corresponds to a 2 in a circle (or pentagon) (

).

a megiston is the number that corresponds to a 10 in a circle (or pentagon) (

).

Moser's number is the number that corresponds to a 2 in a megagon , i.e. a polygon with sides.

Alternative notation

${\ displaystyle M (n, m, p)}$ Let be the number represented by the number n in m nested p- sided polygons. The following applies:

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

${\ displaystyle \ mathrm {Mega} = M (2,1,5) = M (256,256,3)}$
${\ displaystyle \ mathrm {Moser} = M (2,1, M (2,1,5))}$

Mega

corresponds to a two in two squares, i.e. a two in two triangles that are all together in one square. That in turn corresponds to 256 in a square, i.e. 256 in 256 nested triangles, i.e.

{\ displaystyle \ left (\ left (\ left (256 ^ {256} \ right) ^ {\ left (256 ^ {256} \ right)} \ right) ^ {\ left (\ left (256 ^ {256} \ right) ^ {\ left (256 ^ {256} \ right)} \ right)} \ right) ^ {\ left (\ left (\ left (256 ^ {256} \ right) ^ {\ left (256 ^ {256} \ right)} \ right) ^ {\ left (\ left (256 ^ {256} \ right) ^ {\ left (256 ^ {256} \ right)} \ right)} \ right)} \ ldots }

(This is only the display after the fourth of the 256 triangles has been resolved!)

After solving the first triangle, continue with the number (in words: thirty-two quadrillion centillions). ${\ displaystyle 3 {,} 2 \ cdot 10 ^ {616}}$

In function notation, Mega could be represented as follows:

{\ displaystyle f (x): = x ^ {x}}

{\ displaystyle \ mathrm {Mega} = f ^ {256} (256) = f ^ {258} (2)}

(The superscript numbers stand for the composition of figures; f is linked to itself 256 times.)

The following is an attempt to approximate the number mega:

It should be noted that after the first exponentiation steps the value of is approximately the same . In fact, the value is roughly the same . It follows: ${\ displaystyle n ^ {n}}$ ${\ displaystyle 256 ^ {n}}$ ${\ displaystyle 10 ^ {n}}$