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Difference between revisions of "Googology"
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Here are some examples of such numbers using the Ackermann’s Generalized Exponential notation, g(a, b, c) = g(a - 1, g(a, b - 1, c), c), g(0, a, b) = a + b, g(1, a, b) = ab: | Here are some examples of such numbers using the Ackermann’s Generalized Exponential notation, g(a, b, c) = g(a - 1, g(a, b - 1, c), c), g(0, a, b) = a + b, g(1, a, b) = ab: | ||
+ | :g(2, 100, 10) = ten-to-the-hundredth = | ||
+ | :g(2, 2, g(3, 2, 10)) = to-the-second-ten squared = | ||
+ | :g(2, 50, g(1, 2, 50)) = twice-fifty-to-the-fiftieth. | ||
− | g(2, 100, 10) = | + | :great googol = g(2, 3/2, g(2, 100, 10)) = g(2, 1000, 10) |
+ | :googolteen = g(2, 100, 10) + 10 | ||
+ | :googolty = g(2, 101, 10) | ||
+ | :googolplex = g(2, g(2, 100, 10), 10) = g(2, g(2, g(2, 2, 10), 10), 10) = g(3, 2, 2, 10) | ||
+ | :eleventyplex = g(2, 110, 10) | ||
+ | :googolex = g(2, 60, 120) > 5g(2, 124, 10) | ||
+ | :googoc = g(2, 100, 200) > g(2, 230, 10) | ||
+ | :googoci = g(2, 101, 202) > 6g(2, 232, 10) | ||
+ | :googocci = g(2, 201, 402) > 2g(2, 523, 10) | ||
+ | :googoccic = g(2, 299, 598) > g(2, 830, 10) | ||
+ | :googoxem = g(2, 990, 1980) >5g(2, 3263, 10) | ||
+ | :googolmox = g(2, 1003, 2006) > g(2, 3312, 10) | ||
+ | :googomump = g(2, 2000, 4000) > g(2, 7204, 10) | ||
+ | :googolplox = g(2, g(2, 50, 100), 3) > g(2, 2.002, g(3, 3, 10)) | ||
+ | :googolbar = g(2, 50000, 100000) = g(2, 250000, 10) | ||
+ | :googocbar = g(2, 100000, 200000) > 5g(2, 530103, 10) | ||
+ | :googodbar = g(2, 500000, 1000000) = g(2, 3000000, 10) | ||
+ | :googombar = g(2, 1000000, 2000000) > 9g(2, 6301029, 10) | ||
+ | :googomembar = g(2, 2000000, 4000000) > 9g(2, 13204119, 10) | ||
+ | :umpbarbar = g(1, g(2, 1000, 1000), 1000) = g(3, 2, 1000) + 3, 10) = g(2, 3003, 10) | ||
+ | :googolplexian = g(2, g(2, g(2, g(2, 2, 10), 10), 10), 10) = g(4, 1, 2, 2, 10) | ||
+ | :googolplux = g(g(2, 100, 10), g(2, 100, 10), 1, 1, 2, 100, 10) = g(g(3, 2, g(2, 100, 10)), 1, 1, 2, 100, 10) | ||
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Which extrapolates, since c = g(2, 2, x) and m = g(2, 3, x) to: | Which extrapolates, since c = g(2, 2, x) and m = g(2, 3, x) to: | ||
− | googolpluc = g(g(g(2, 2, 3), 2, g(2, 100, 10)), 1, 1, 2, 100, 10) | + | :googolpluc = g(g(g(2, 2, 3), 2, g(2, 100, 10)), 1, 1, 2, 100, 10) |
− | + | :googolplum = g(g(g(3, 3, 3), 2, g(2, 100, 10)), 1, 1, 2, 100, 10) | |
− | googolplum = g(g(g(3, 3, 3), 2, g(2, 100, 10)), 1, 1, 2, 100, 10) | + | :baggoogol = g(2, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) |
− | + | :traggoogol = g(3, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | |
− | baggoogol = g(2, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | + | :quadraggoogol = g(4, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) |
− | + | :centaggoogol = g(100, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | |
− | traggoogol = g(3, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | + | :dentaggoogol = g(500, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) |
− | + | :mentaggoogol = g(1000, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | |
− | quadraggoogol = g(4, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | + | :bentaggoogol = g(g(1, 2, 1000), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) |
− | + | :trentaggoogol = g(g(1, 3, 1000), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | |
− | centaggoogol = g(100, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | + | :yottaggoogol = g(g(2, 80, 10), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) |
− | + | ||
− | dentaggoogol = g(500, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | + | |
− | + | ||
− | mentaggoogol = g(1000, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | + | |
− | + | ||
− | bentaggoogol = g(g(1, 2, 1000), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | + | |
− | + | ||
− | trentaggoogol = g(g(1, 3, 1000), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) | + | |
− | + | [[Category:Large numbers]][[Category:Mathematics]] |
Latest revision as of 03:02, 5 September 2010
Googology is the study of numbers between a googol and a googolplex and higher.
Here are some examples of such numbers using the Ackermann’s Generalized Exponential notation, g(a, b, c) = g(a - 1, g(a, b - 1, c), c), g(0, a, b) = a + b, g(1, a, b) = ab:
- g(2, 100, 10) = ten-to-the-hundredth =
- g(2, 2, g(3, 2, 10)) = to-the-second-ten squared =
- g(2, 50, g(1, 2, 50)) = twice-fifty-to-the-fiftieth.
- great googol = g(2, 3/2, g(2, 100, 10)) = g(2, 1000, 10)
- googolteen = g(2, 100, 10) + 10
- googolty = g(2, 101, 10)
- googolplex = g(2, g(2, 100, 10), 10) = g(2, g(2, g(2, 2, 10), 10), 10) = g(3, 2, 2, 10)
- eleventyplex = g(2, 110, 10)
- googolex = g(2, 60, 120) > 5g(2, 124, 10)
- googoc = g(2, 100, 200) > g(2, 230, 10)
- googoci = g(2, 101, 202) > 6g(2, 232, 10)
- googocci = g(2, 201, 402) > 2g(2, 523, 10)
- googoccic = g(2, 299, 598) > g(2, 830, 10)
- googoxem = g(2, 990, 1980) >5g(2, 3263, 10)
- googolmox = g(2, 1003, 2006) > g(2, 3312, 10)
- googomump = g(2, 2000, 4000) > g(2, 7204, 10)
- googolplox = g(2, g(2, 50, 100), 3) > g(2, 2.002, g(3, 3, 10))
- googolbar = g(2, 50000, 100000) = g(2, 250000, 10)
- googocbar = g(2, 100000, 200000) > 5g(2, 530103, 10)
- googodbar = g(2, 500000, 1000000) = g(2, 3000000, 10)
- googombar = g(2, 1000000, 2000000) > 9g(2, 6301029, 10)
- googomembar = g(2, 2000000, 4000000) > 9g(2, 13204119, 10)
- umpbarbar = g(1, g(2, 1000, 1000), 1000) = g(3, 2, 1000) + 3, 10) = g(2, 3003, 10)
- googolplexian = g(2, g(2, g(2, g(2, 2, 10), 10), 10), 10) = g(4, 1, 2, 2, 10)
- googolplux = g(g(2, 100, 10), g(2, 100, 10), 1, 1, 2, 100, 10) = g(g(3, 2, g(2, 100, 10)), 1, 1, 2, 100, 10)
Which extrapolates, since c = g(2, 2, x) and m = g(2, 3, x) to:
- googolpluc = g(g(g(2, 2, 3), 2, g(2, 100, 10)), 1, 1, 2, 100, 10)
- googolplum = g(g(g(3, 3, 3), 2, g(2, 100, 10)), 1, 1, 2, 100, 10)
- baggoogol = g(2, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
- traggoogol = g(3, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
- quadraggoogol = g(4, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
- centaggoogol = g(100, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
- dentaggoogol = g(500, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
- mentaggoogol = g(1000, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
- bentaggoogol = g(g(1, 2, 1000), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
- trentaggoogol = g(g(1, 3, 1000), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
- yottaggoogol = g(g(2, 80, 10), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))