Eillion Notation

There are so many illion-related notations in googology, but it's time for me to make one. This is based on recursion and hyperrecursion and works a bit like the fast-growing hierarchy.

Base definition

E(n) = 10^(3*10^(3*10^(...(3*10^n)...))+3) with n 3*10's

E0(n) = E(n)

Em(n) = Em-1(Em-1(...(Em-1(n))...)) with n sets of brackets

En(n) = E[1](n) = order type w

Array Notation

2-entry array notation

We can now set forth three rules, where @ is a string of numbers and separators. All numbers in all arrays using this notation must be nonzero positive integers.

• Rule 1. Tailing rule: E[@,1](n) = E[@](n)

• Rule 2. Recursion rule: E[m,k](n) = E[m-1,k](E[m-1,k](...(E[m-1,k](n))...)) with n nests

• Rule 3. Hyperoperation rule: E[n,k](n) = E[1,k+1](n)

The limit of this notation is order type w^2.

linear array notation e.g. E[a,b,c,...,n](x)

Now we'll have to edit rule 2. 

• E[m@](n) = E[m-1@](E[m-1@](...(E[m-1@](n))...)) with n nests

• If rule 1, 2, 3 or the rule above does not apply, we start a thing called process which starts from the first number after the opening bracket.

  • Case A. If the number is 1, jump to the next entry. 

  • Case B. If the number is greater than 1, decrease the entry by 1 and change the previous entry to the number in the () bracket. Check if rules apply. 

  • E.g. E[1,2,3](3) = E[3,1,3](3)

This brings the limit of the notation up to w^w.

My planned extensions to this notation has the array behave like the one in Username5243's Array Notation, except at epsilon-zero, the first-order comma becomes a double comma, a second-order comma becomes a triple comma, and the limit would be akin to order type psi_0(Omega_w).

Googolisms

The prefixes all follow Denis Maksudov's naming system.

1. E(0) = 1, because 10^0 = 1/zeralillion

2. E(1) = 10^33 (decillion)/unalillion

3. E(2) = hectillion/balillion

4. E(3) = kalillion/tralillion

5. E(4) = quadralillion

6. E(5) = quintalillion

7. E(9) = hairillion (nonalillion)

8. E(1000) = killalillion

9. E(10^6) = megalillion

10. E(10^21) = zettalillion

11. E(10^24) = yottalillion

12. E1(2) = E(E(2)) = hendunillion

13. E1(3) = E(E(E(3))) = hentrunillion

14. Rpillager's seven-e = E(E(E(10^6))) (the seven e stands for the original E(E(E(10^6))) by Harry Foundalis) (note that this is smaller than the original)

15. E1(10) = heckillion

16. E2(10) = deckillion

17. E3(10) = treckillion

18. E4(10) = quadrackillion

19. E5(10) = quintackillion

20. E10(10) = E[1](10) = hyprunillion = dekackillion

21. E1000(10) = killackillion

22. E10^6(10) = megackillion

23. E[1](meameamealokkapoowa) = meameamealokkapockillion

and so on.

MORE COMING SOON

Copyright 2023 Redstonepillager. Work is licensed under CC-BY-SA.

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