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− | + | '''Hyper-Log notation''' or '''Hyper-L notation''' is an extension of the Log function that itself is a fast growing function, but not fast enough.<ref>[https://sites.google.com/site/jonathansgoogology/notations/hyper-l-notation Hyper-L Notation - Jonathan's Googology]</ref> Hyper-L notation is as follows. The standard notation is L(n) to indicate it is a hyper-L function. |
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+ | Representation: L(n) |
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+ | Rule: \(L(n)\) = \(Antilog(n)\), also known as \(10^n\) |
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+ | Examples: |
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− | <p style="font-weight:normal;margin-top:1em;margin-bottom:1em;font-size:16px;">Representation: L(n)</p> |
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+ | *\(L(5)\) = [[100,000]] |
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− | <p style="font-weight:normal;font-size:16px;">Rule: \(L(n)\)= reverse \(log(n)\) or \(revlog(n)\)</p> |
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+ | *\(L(33)\) = [[decillion]] |
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+ | *\(L(100)\) = [[googol]] |
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+ | *\(L(googol)\) = [[googolplex]] |
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− | <p style="font-weight:normal;font-size:16px;"> |
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+ | Representation: \(L((n))\) |
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− | </p> |
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− | <p style="font-weight:normal;font-size:16px;">Examples:</p> |
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+ | Examples: |
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− | <p style="font-weight:normal;font-size:16px;">\(L(5)\) = 100,000</p> |
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− | <p style="font-weight:normal;font-size:16px;">\(L(33)\)= decillion</p> |
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+ | *L((33)) = \(10^{\text{decillion}^{\text{decillion}^{...^{\text{decillion}<nowiki>}}}}</nowiki> (... = \text{decillion})\) |
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+ | *L((100)) = \(10^{\text{googol}^{\text{googol}^{\text{googol}^{...^{\text{googol}<nowiki>}}}}}</nowiki> (... = \text{googol})\) |
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+ | *L((googol)) = \(10^{\text{googolplex}^{\text{googolplex}^{...^{\text{googolplex}<nowiki>}}}}</nowiki> (... = \text{googolplex})\) |
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+ | == Third Order == |
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− | <p style="font-weight:normal;font-size:16px;">\(L(100)\)= googol</p> |
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+ | Representation: L(((n))) |
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+ | Rule: \(L(((n))) = 10\uparrow^{Antilog(n)}Antilog(n)\) |
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− | <p style="font-weight:normal;font-size:16px;">\(L(googol)\)= googolplex</p> |
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+ | Examples: |
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− | <p style="font-weight:normal;font-size:16px;"> |
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− | </p> |
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+ | *L(((5))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{100{,}000}100{,}000\) |
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− | <p style="font-weight:normal;font-size:16px;"> |
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+ | *L(((33))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{decillion}}\text{decillion}\) |
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− | </p> |
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+ | *L(((100))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{googol}}\text{googol}\) |
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+ | *L(((googol))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{googolplex}}\text{googolplex}\) |
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+ | == Approximations == |
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+ | For first order (\(L(n)\)): |
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+ | {| border="0" cellpadding="1" cellspacing="1" class="article-table" |
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+ | |- |
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+ | ! scope="col"|Notation |
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+ | ! scope="col"|Lower bound |
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+ | ! scope="col"|Upper bound |
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+ | |- |
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+ | |[[Hyper-E notation]] |
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+ | |colspan="2" align="center"|E(n) |
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+ | |- |
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+ | |[[Arrow notation]] |
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+ | |colspan="2" align="center"|\(10\uparrow n\) |
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+ | |} |
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− | + | == Sources == |
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+ | <references /> |
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− | <p style="font-weight:normal;margin-top:1em;margin-bottom:1em;font-size:16px;">Representation: \(L((n))\)</p> |
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− | |||
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− | |||
− | <p style="font-weight:normal;font-size:16px;"> |
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− | </p> |
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− | |||
− | <p style="font-weight:normal;font-size:16px;">Examples:</p> |
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− | |||
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− | |||
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− | |||
− | <p style="font-weight:normal;font-size:16px;">L((33))= 10^decillion^decillion^...^decillion (... = decillion)</p> |
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− | |||
− | <p style="font-weight:normal;font-size:16px;">L((100))= 10^googol^googol^googol^...^googol (... = googol)</p> |
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− | |||
− | <p style="font-weight:normal;font-size:16px;">L((googol))= 10^googolplex^googolplex^...^googolplex (... = googolplex)</p> |
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− | |||
− | <p style="font-weight:normal;font-size:16px;"> |
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− | </p> |
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− | |||
− | <p style="font-weight:normal;font-size:16px;"> |
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− | </p> |
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− | |||
− | <p style="font-weight:normal;font-size:16px;"> |
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− | </p> |
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− | |||
− | |||
− | |||
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− | <p style="font-weight:normal;margin-top:1em;margin-bottom:1em;font-size:16px;">Representation: L(((n)))</p> |
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− | |||
− | <p style="font-weight:normal;font-size:16px;">Rule: <span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span style="vertical-align:0px;display:inline-block;line-height:normal;width:7.815em;"><span style="vertical-align:0px;display:inline-block;line-height:normal;width:6.513em;height:0px;font-size:19.2px;"><span style="vertical-align:0px;line-height:normal;"><span style="vertical-align:0px;line-height:normal;"><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">10</span><span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑<span style="vertical-align:0px;line-height:normal;">↑</span></span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑<span style="vertical-align:0px;line-height:normal;">↑</span></span><span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">(</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">N</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">)</span></span></span></span></span></span>with revlog(n) up arrows</p> |
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− | |||
− | <p style="font-weight:normal;font-size:16px;"> |
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− | </p> |
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− | |||
− | <p style="font-weight:normal;font-size:16px;">Examples:</p> |
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− | |||
− | <p style="font-weight:normal;font-size:16px;">L(((5)))= <span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span style="vertical-align:0px;display:inline-block;line-height:normal;width:10.836em;"><span style="vertical-align:0px;display:inline-block;line-height:normal;width:9.013em;height:0px;font-size:19.2px;"><span style="vertical-align:0px;line-height:normal;"><span style="vertical-align:0px;line-height:normal;"><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">10</span><span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑<span style="vertical-align:0px;line-height:normal;">↑</span></span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑<span style="vertical-align:0px;line-height:normal;">↑</span></span><span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">(</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">100</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">,</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">000</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">)</span></span></span></span></span></span>with 100,000 up arrows</p> |
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− | <p style="font-weight:normal;font-size:16px;">L(((33)))= <span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span style="vertical-align:0px;display:inline-block;line-height:normal;width:11.253em;"><span style="vertical-align:0px;display:inline-block;line-height:normal;width:9.378em;height:0px;font-size:19.2px;"><span style="vertical-align:0px;line-height:normal;"><span style="vertical-align:0px;line-height:normal;"><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">10</span><span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑<span style="vertical-align:0px;line-height:normal;">↑</span></span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑<span style="vertical-align:0px;line-height:normal;">↑</span></span><span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">(</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">d</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">e</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">c</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">i</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">l</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">l</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">i</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">n</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">)</span></span></span></span></span></span>with decillion up arrows</p> |
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− | <p style="font-weight:normal;font-size:16px;">L(((100)))= <span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span style="vertical-align:0px;display:inline-block;line-height:normal;width:10.003em;"><span style="vertical-align:0px;display:inline-block;line-height:normal;width:8.336em;height:0px;font-size:19.2px;"><span style="vertical-align:0px;line-height:normal;"><span style="vertical-align:0px;line-height:normal;"><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">10</span><span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑<span style="vertical-align:0px;line-height:normal;">↑</span></span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑<span style="vertical-align:0px;line-height:normal;">↑</span></span><span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">(</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">g</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">g</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">l</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">)</span></span></span></span></span></span> with googol up arrows</p> |
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− | <p style="font-weight:normal;font-size:16px;">L(((googol)))= <span style="line-height:normal;word-spacing:normal;word-wrap:normal;white-space:nowrap;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;"><span style="vertical-align:0px;display:inline-block;line-height:normal;width:12.19em;"><span style="vertical-align:0px;display:inline-block;line-height:normal;width:10.159em;height:0px;font-size:19.2px;"><span style="vertical-align:0px;line-height:normal;"><span style="vertical-align:0px;line-height:normal;"><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">10</span><span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑<span style="vertical-align:0px;line-height:normal;">↑</span></span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">.</span><span style="padding-left:0.159em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">↑<span style="vertical-align:0px;line-height:normal;">↑</span></span><span style="padding-left:0.263em;vertical-align:0px;line-height:normal;font-family:MathJax_Main;">(</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">g</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">g</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">o</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">l</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">p</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">l</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">e</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Math-italic;">x</span><span style="vertical-align:0px;line-height:normal;font-family:MathJax_Main;">)</span></span></span></span></span></span>with googolplex up arrows</p> |
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− | <p style="font-weight:normal;font-size:16px;"> |
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− | </p> |
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− | <p style="font-weight:normal;font-size:16px;"> |
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− | </p> |
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− | <p style="font-weight:normal;font-size:16px;"> |
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− | </p> |
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− | <p style="font-weight:normal;font-size:16px;"> |
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− | </p> |
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− | |||
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− | === Sources === |
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− | [https://sites.google.com/site/jonathansgoogology/notations/hyper-l-notation JTOnstead's Hyper L Notation] |
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[[Category:Notations]] |
[[Category:Notations]] |
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[[Category:Notations by JTOnstead20]] |
[[Category:Notations by JTOnstead20]] |
Revision as of 21:42, 29 June 2021
Hyper-Log notation or Hyper-L notation is an extension of the Log function that itself is a fast growing function, but not fast enough.[1] Hyper-L notation is as follows. The standard notation is L(n) to indicate it is a hyper-L function.
First Order
Representation: L(n)
Rule: \(L(n)\) = \(Antilog(n)\), also known as \(10^n\)
Examples:
- \(L(5)\) = 100,000
- \(L(33)\) = decillion
- \(L(100)\) = googol
- \(L(googol)\) = googolplex
Second Order
Representation: \(L((n))\)
Rule: \(L((n))\) = \(10^{Antilog(n)^{Antilog(n)^{...^{Antilog(n)}}}}; where ...=Antilog(n)\)
Examples:
- L((3)) = \(10^{1000^{1000^{...^{1000}}}} (... = 1{,}000)\)
- L((5)) = \(10^{100,000^{100,000^{...^{100,000}}}} (... = 100{,}000)\)
- L((33)) = \(10^{\text{decillion}^{\text{decillion}^{...^{\text{decillion}}}}} (... = \text{decillion})\)
- L((100)) = \(10^{\text{googol}^{\text{googol}^{\text{googol}^{...^{\text{googol}}}}}} (... = \text{googol})\)
- L((googol)) = \(10^{\text{googolplex}^{\text{googolplex}^{...^{\text{googolplex}}}}} (... = \text{googolplex})\)
Third Order
Representation: L(((n)))
Rule: \(L(((n))) = 10\uparrow^{Antilog(n)}Antilog(n)\)
Examples:
- L(((5))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{100{,}000}100{,}000\)
- L(((33))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{decillion}}\text{decillion}\)
- L(((100))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{googol}}\text{googol}\)
- L(((googol))) = \(10\underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}_{\text{googolplex}}\text{googolplex}\)
Approximations
For first order (\(L(n)\)):
Notation | Lower bound | Upper bound |
---|---|---|
Hyper-E notation | E(n) | |
Arrow notation | \(10\uparrow n\) |