{"history":[],"article":{"id":"62b0071f-185a-4fc6-8f27-8ace87bd9cb4","title":"Steinhaus–Moser notation","slug":"steinhaus-moser-notation","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **Steinhaus–Moser [notation](https://bliptext.com/articles/mathematical-notation)** is a [notation](https://bliptext.com/articles/mathematical-notation) for expressing certain [large numbers](https://bliptext.com/articles/large-number).\nIt is an extension (devised by [Leo Moser](https://bliptext.com/articles/leo-moser)) of [Hugo Steinhaus](https://bliptext.com/articles/hugo-steinhaus)'s polygon notation.\n\n# Definitions\nn\nin a **triangle** means\nnn\n.\nn\nin a **square** is equivalent to \"the number\nn\ninside\nn\ntriangles, which are all nested.\"\nn\nin a **pentagon** is equivalent to \"the number\nn\ninside\nn\nsquares, which are all nested.\"\netc.:\nn\nwritten in an (\nm + 1\n)-sided polygon is equivalent to \"the number\nn\ninside\nn\nnested\nm\n-sided polygons\". In a series of nested polygons, they are [associated](https://bliptext.com/articles/association-mathematics) inward. The number\nn\ninside two triangles is equivalent to\nnn\ninside one triangle, which is equivalent to\nnn\nraised to the power of\nnn\n.\nSteinhaus defined only the triangle, the square, and the **circle**, which is equivalent to the pentagon defined above.\n# [n in a circle | upload.wikimedia.org/wikipedia/commons/f/f0/Circle-n.svg]\n\n# Special values\nSteinhaus defined:\n**Moser's number** is the number represented by \"2 in a megagon\".\n**Megagon** is here the name of a polygon with \"mega\" sides (not to be confused with the [polygon with one million sides](https://bliptext.com/articles/megagon)).\nAlternative notations:\nM(n, m, p)\nbe the number represented by the number\nn\nin\nm\nnested\np\n-sided polygons; then the rules are:\n\n# Mega\nA mega, ②, is already a very large number, since ② =\nsquare(square(2)) = square(triangle(triangle(2))) =\nsquare(triangle(22)) =\nsquare(triangle(4)) =\nsquare(44) =\nsquare(256) =\ntriangle(triangle(triangle(...triangle(256)...))) [256 triangles] =\ntriangle(triangle(triangle(...triangle(256256)...))) [255 triangles] ~\ntriangle(triangle(triangle(...triangle(3.2317 × 10616)...))) [255 triangles]\n...\nUsing the other notation:\nmega =\nWith the function f(x)=x^x we have mega = where the superscript denotes a [functional power](https://bliptext.com/articles/iterated-function), not a numerical power.\nWe have (note the convention that powers are evaluated from right to left):\nSimilarly:\netc.\nThus:\nRounding more crudely (replacing the 257 at the end by 256), we get mega ≈, using [Knuth's up-arrow notation](https://bliptext.com/articles/knuth-s-up-arrow-notation).\nAfter the first few steps the value of n^n is each time approximately equal to 256^n.\nIn fact, it is even approximately equal to 10^n (see also [approximate arithmetic for very large numbers](https://bliptext.com/articles/large-numbers)).\nUsing base 10 powers we get:\n...\n\n# Moser's number\nIt has been proven that in [Conway chained arrow notation](https://bliptext.com/articles/conway-chained-arrow-notation),\nand, in [Knuth's up-arrow notation](https://bliptext.com/articles/knuth-s-up-arrow-notation),\nTherefore, Moser's number, although incomprehensibly large, is vanishingly small compared to [Graham's number](https://bliptext.com/articles/graham-s-number):","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:06:22.196Z","updated_at":"2025-01-25T19:06:22.196Z","search_vector":"'moser':2 'notat':3 'steinhaus':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **Steinhaus–Moser [notation](https://bliptext.com/articles/mathematical-notation)** is a [notation](https://bliptext.com/articles/mathematical-notation) for expressing certain [large numbers](https://bliptext.com/articles/large-number).\nIt is an extension (devised by [Leo Moser](https://bliptext.com/articles/leo-moser)) of [Hugo Steinhaus](https://bliptext.com/articles/hugo-steinhaus)'s polygon notation.\n\n# Definitions\n n \nin a **triangle** means\n nn \n.\n n \nin a **square** is equivalent to \"the number\n n \ninside\n n \ntriangles, which are all nested.\"\n n \nin a **pentagon** is equivalent to \"the number\n n \ninside\n n \nsquares, which are all nested.\"\netc.:\n n \nwritten in an (\n m + 1\n)-sided polygon is equivalent to \"the number\n n \ninside\n n \nnested\n m \n-sided polygons\". In a series of nested polygons, they are [associated](https://bliptext.com/articles/association-mathematics) inward. The number\n n \ninside two triangles is equivalent to\n nn \ninside one triangle, which is equivalent to\n nn \nraised to the power of\n nn \n.\nSteinhaus defined only the triangle, the square, and the **circle**, which is equivalent to the pentagon defined above.\n# [n in a circle | upload.wikimedia.org/wikipedia/commons/f/f0/Circle-n.svg]\n\n# Special values\nSteinhaus defined:\n**Moser's number** is the number represented by \"2 in a megagon\".\n**Megagon** is here the name of a polygon with \"mega\" sides (not to be confused with the [polygon with one million sides](https://bliptext.com/articles/megagon)).\nAlternative notations:\n M ( n , m , p )\nbe the number represented by the number\n n \nin\n m \nnested\n p \n-sided polygons; then the rules are:\n\n# Mega\nA mega, ②, is already a very large number, since ② =\nsquare(square(2)) = square(triangle(triangle(2))) =\nsquare(triangle(22)) =\nsquare(triangle(4)) =\nsquare(44) =\nsquare(256) =\ntriangle(triangle(triangle(...triangle(256)...))) [256 triangles] =\ntriangle(triangle(triangle(...triangle(256256)...))) [255 triangles] ~\ntriangle(triangle(triangle(...triangle(3.2317 × 10616)...))) [255 triangles]\n...\nUsing the other notation:\nmega =\nWith the function f(x)=x^x we have mega = where the superscript denotes a [functional power](https://bliptext.com/articles/iterated-function), not a numerical power.\nWe have (note the convention that powers are evaluated from right to left):\nSimilarly:\netc.\nThus:\nRounding more crudely (replacing the 257 at the end by 256), we get mega ≈, using [Knuth's up-arrow notation](https://bliptext.com/articles/knuth-s-up-arrow-notation).\nAfter the first few steps the value of n^n is each time approximately equal to 256^n.\nIn fact, it is even approximately equal to 10^n (see also [approximate arithmetic for very large numbers](https://bliptext.com/articles/large-numbers)).\nUsing base 10 powers we get:\n...\n\n# Moser's number\nIt has been proven that in [Conway chained arrow notation](https://bliptext.com/articles/conway-chained-arrow-notation),\nand, in [Knuth's up-arrow notation](https://bliptext.com/articles/knuth-s-up-arrow-notation),\nTherefore, Moser's number, although incomprehensibly large, is vanishingly small compared to [Graham's number](https://bliptext.com/articles/graham-s-number):","revisionCount":0},"hasMore":false,"nextCursor":null}

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