ハイパーE表記 (ハイパーEひょうき、英語: Hyper-E Notation) は、Sbiis Saibian が考案した巨大数を表記する方法である[1]。1つ以上の正の整数の数列 an の引数をハイペリオン記号#で区切ったものであり、E(b)a1#a2#...#anと表記し、bを基数と呼ぶ。基数が省略されたときは10がデフォルトであり、よく省略される。
拡張記法として、拡張ハイパーE表記、連鎖E表記、及び拡張連鎖E表記がある。
E ( b ) a = b a {\displaystyle E(b)a=b^{a}}
E ( b ) a 1 # a 2 # ⋯ # a n # 1 = E ( b ) a 1 # a 2 # ⋯ # a n {\displaystyle E(b)a_{1}\#a_{2}\#\cdots \#a_{n}\#1=E(b)a_{1}\#a_{2}\#\cdots \#a_{n}}
E ( b ) a 1 # ⋯ # a n − 2 # a n − 1 # a n = E ( b ) a 1 # ⋯ # a n − 2 # ( E ( b ) a 1 # ⋯ # a n − 2 # a n − 1 # ( a n − 1 ) ) {\displaystyle E(b)a_{1}\#\cdots \#a_{n-2}\#a_{n-1}\#a_{n}=E(b)a_{1}\#\cdots \#a_{n-2}\#(E(b)a_{1}\#\cdots \#a_{n-2}\#a_{n-1}\#(a_{n}-1))}
E 2 = 10 2 = 100 {\displaystyle E2=10^{2}=100}
E E 5 = E 10 5 = E 10 5 = 10 10 5 {\displaystyle EE5=E10^{5}=E10^{5}=10^{10^{5}}}
E 3 # 2 = E E 3 # 1 = E E 3 = E 10 3 = E 1000 = 10 1000 {\displaystyle E3\#2=EE3\#1=EE3=E10^{3}=E1000=10^{1000}}
E ( b ) a 1 # h ( 1 ) ⋯ # h ( n − 1 ) a n # h ( n ) 1 = E ( b ) a 1 # h ( 1 ) ⋯ # h ( n − 1 ) a n {\displaystyle E(b)a_{1}\#^{h(1)}\cdots \#^{h(n-1)}a_{n}\#^{h(n)}1=E(b)a_{1}\#^{h(1)}\cdots \#^{h(n-1)}a_{n}}
E ( b ) a 1 # h ( 1 ) ⋯ # h ( n − 3 ) a n − 2 # h ( n − 2 ) a n − 1 # a n = E ( b ) a 1 # h ( 1 ) ⋯ # h ( n − 3 ) a n − 2 # h ( n − 2 ) ( E ( b ) a 1 # h ( 1 ) ⋯ # h ( n − 3 ) a n − 2 # h ( n − 2 ) a n − 1 # ( a n − 1 ) ) {\displaystyle E(b)a_{1}\#^{h(1)}\cdots \#^{h(n-3)}a_{n-2}\#^{h(n-2)}a_{n-1}\#a_{n}=E(b)a_{1}\#^{h(1)}\cdots \#^{h(n-3)}a_{n-2}\#^{h(n-2)}(E(b)a_{1}\#^{h(1)}\cdots \#^{h(n-3)}a_{n-2}\#^{h(n-2)}a_{n-1}\#(a_{n}-1))}
E ( b ) a 1 # h ( 1 ) ⋯ # h ( n − 2 ) a n − 1 # h ( n − 1 ) a n = E ( b ) a 1 # h ( 1 ) ⋯ # h ( n − 2 ) a n − 1 # h ( n − 1 ) − 1 a n − 1 # h ( n − 1 ) ( a n − 1 ) {\displaystyle E(b)a_{1}\#^{h(1)}\cdots \#^{h(n-2)}a_{n-1}\#^{h(n-1)}a_{n}=E(b)a_{1}\#^{h(1)}\cdots \#^{h(n-2)}a_{n-1}\#^{h(n-1)-1}a_{n-1}\#^{h(n-1)}(a_{n}-1)}
E 3 # 2 3 {\displaystyle \quad \,E3\#^{2}3}
= E 3 # 3 # 2 1 {\displaystyle =E3\#3\#^{2}1}
= E 3 # 3 {\displaystyle =E3\#3}
= E ( E 3 # 2 ) {\displaystyle =E(E3\#2)}
= E ( E ( E 3 # 1 ) ) {\displaystyle =E(E(E3\#1))}
= E ( E 1000 ) {\displaystyle =E(E1000)}
= E 10 1000 {\displaystyle =E10^{1000}}
= 10 10 1000 {\displaystyle =10^{10^{1000}}}
E 2 # 3 3 {\displaystyle \quad \,E2\#^{3}3}
= E 2 # 2 2 # 3 2 {\displaystyle =E2\#^{2}2\#^{3}2}
= E 2 # 2 2 # 2 2 # 3 1 {\displaystyle =E2\#^{2}2\#^{2}2\#^{3}1}
= E 2 # 2 2 # 2 2 {\displaystyle =E2\#^{2}2\#^{2}2}
= E 2 # 2 2 # 2 # 2 1 {\displaystyle =E2\#^{2}2\#2\#^{2}1}
= E 2 # 2 2 # 2 {\displaystyle =E2\#^{2}2\#2}
= E 2 # 2 ( E 2 # 2 2 # 1 ) {\displaystyle =E2\#^{2}(E2\#^{2}2\#1)}
= E 2 # 2 ( E 2 # 2 2 ) {\displaystyle =E2\#^{2}(E2\#^{2}2)}
= E 2 # 2 ( E 2 # 2 # 2 1 ) {\displaystyle =E2\#^{2}(E2\#2\#^{2}1)}
= E 2 # 2 ( E 2 # 2 ) {\displaystyle =E2\#^{2}(E2\#2)}
= E 2 # 2 ( E ( E 2 # 1 ) ) {\displaystyle =E2\#^{2}(E(E2\#1))}
= E 2 # 2 ( E ( E 2 ) ) {\displaystyle =E2\#^{2}(E(E2))}
= E 2 # 2 ( E 100 ) {\displaystyle =E2\#^{2}(E100)}
= E 2 # 2 10 100 {\displaystyle =E2\#^{2}10^{100}}
= E 2 # 2 # ⋯ 2 # 2 ⏟ 10 100 こ {\displaystyle =E\underbrace {2\#2\#\cdots 2\#2} _{10^{100}{\text{こ}}}}
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