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For example: $2+2=4$, $2*2=4$, $2^2=4$, even $2$ in tetration of $2$ equals $4$. (Tetration example: $3$ of $4$ equals to $(((3)^3)^3)^3$)

Mathematic have a way to prove those things like: $2$ [any operator here] $2 = 4$? But any form of this case. (Like replace those $2$ with $x$ and $y$ and $4$ with $z$)

I need an example of how to prove the example given here to understand the basic I think.

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In general, these operations are known as hyperoperations, we'll here denote them $x H_n y$. It's fairly tautological that $2H_n2=4$, but it can be seen as inductively following from the definitions:

Base case: $2 + 2 = 4$.

Inductive step: Suppose $2H_n2=4$. Then $2H_{n+1}2=2H_n2$ by definition, and $2H_{n+1}2=4$. $\square$

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  • $\begingroup$ How by definition $2H_{n+1}2=2H_n2$ is true? $\endgroup$
    – KugBuBu
    Commented Sep 29, 2014 at 18:53
  • $\begingroup$ @KugBuBu In the most typical case, $aH_nb = aH_{n-1}(aH_n(b-1))$. Plugging in, you get $$2H_{n+1}2=2H_n(2H_{n+1}(1)) = 2H_n2$$ with the last equality following again from the definition. $\endgroup$
    – theage
    Commented Sep 29, 2014 at 22:16
  • $\begingroup$ I got it, there's just no steps between, it breaks down to the same thing. $\endgroup$
    – KugBuBu
    Commented Sep 30, 2014 at 9:56
  • $\begingroup$ Just to note that your example is wrong - 3 of 4 is $3^\left(3^\left(3^3\right)\right)$ , not $((3^3)^3)^3)$. $\endgroup$ Commented Jan 13, 2019 at 10:59

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