Is there a theory about repeating the same arithmetic operation? E.G. multiplication is a repetition of addition, and exponentiation is the repetition of multiplication.
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2Continuous has another meaning in math. Maybe “repeated” is a better term? – Thomas Andrews Apr 08 '20 at 19:28
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Yes thanks, mind spasm! :) – Ilya Grushevskiy Apr 08 '20 at 19:30
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1Have a look at the Wikipedia pages on the Ackermann function and Knuth's up-arrow notation. – Rob Arthan Apr 08 '20 at 21:39
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I've no idea why this question is the subject of down-votes. It's a perfectly good question that leads to some interesting mathematics. – Rob Arthan Apr 08 '20 at 21:41
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As Rob Arthan pointed out in the comments, Knuth's up arrow notation is the way to go, and the sequence is called the hyperoperation sequence.
Ilya Grushevskiy
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My answer is maybe less about giving an answer, and maybe more about giving some perspective. Maths is just convention; numbers don't have a meaning thought you can think about it the way you like.
Actually, are you looking for algebra theory ? It all comes down to "how" you define operators. What you say that all operators can be reduced and it works well, it's perhaps because you are talking about complex numbers, but it might be true that:
$$ a^5 \neq a \times a \times a \times a \times a $$
Take a look at rings if you haven't yet, and fields. Those are sets upon which you define an operator. There is no reason why the addition (or first law) would have anything in common with multiplication (second law). Actually, in group theory, one writes
$$ g^3 = g + g + g $$
if $g$ is an element of the group, and $+$ is the law inside the group. LHS is just a common, useful (short cut) notation.
Marine Galantin
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-1. The tag is arithmetic, meaning OP is most likely not looking for answers related to abstract algebra. Unhelpful. – Micah Windsor Apr 09 '20 at 17:03
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Also, don't submit an edit 'correcting' OP's grammar if you think "repeating" is spelled "repetiting". – Micah Windsor Apr 09 '20 at 17:06
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@MicahWindsor Hi, I understand what you ve written, my proposition was to explore other realms, since the question could be understand either as the question of someone that already knows a lot about the field, or a totally newcommer. I took the bet about the second hypothesis. If someone don't know what algebra is, he will think arithmetic is all about operators, when actually it is the concern of algebra. What for grammar, sorry for that, actually I was more focused on the meaning of the question, not on grammar / spelling where I don't think I am good enough to give anyone a lesson. – Marine Galantin Apr 09 '20 at 17:09
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@MicahWindsor I approved your edit btw :) thank you for the precision! – Marine Galantin Apr 09 '20 at 17:11
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If the intention isn't to answer, your post belongs in the comment section. And thank you for approving the edit. – Micah Windsor Apr 09 '20 at 17:14