# Does this pattern of arithmetic exist?

I have not studied mathematics very deeply, and I'm not familiar with the terminology. However, I have thought about a kind of pattern and wonder if anything like that exists or is being used today.

The original thought of mine is that multiplication is a type of addition, and exponentiation is a type of multiplication. In this sense addition is addition on the first level, multiplication is addition on the second level, and exponentiation is addition on the third level. Logically then there should be a fourth, fifth, sixth, etc level of addition too.

What do you think of this?

Example:

1st level: 3 + 3 = 6
2nd level: 3 * 3 = 9 derived from 3 + 3 + 3
3rd level: 3 ^ 3 = 27 derived from (3 * 3) * 3
[using arbitrary sign ¤ for 4th level]
4th level: 3 ¤ 3 = 19683 derived from (3 ^ 3) ^ 3
[using arbitrary sign \ for 5th level]
5th level: 3 \ 3 = (3 ¤ 3) ¤ 3 = 19683 ¤ 3 = (19683 ^ 19683) ^ 19683

1st level: 4 + 3 = 7
2nd level: 4 * 3 = 12 derived from 4 + 4 + 4
3rd level: 4 ^ 3 = 64 derived from (4 * 4) * 4
4th level: 4 ¤ 3 = 4294967296 derived from (4 ^ 4) ^ 4
5th level: 4 \ 3 = (4 ¤ 4) ¤ 4 = 4294967296 ¤ 4 =
= ((4294967296^4294967296) ^ 4294967296) ^ 4294967296

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@ribot Oh, I didn't see that. I'd say the former is the more interesting notion, though, as $(3^3)^3$ simplifies to $3^{(3 \cdot 3)}$.. –  Cocopuffs Dec 16 '12 at 19:02