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 hyper 0 MphLee Fellow Posts: 184 Threads: 19 Joined: May 2013 03/09/2021, 10:28 PM Ok, at the beginning I was convinced that the grouping operation on fingers was just the arithmetic mean. Quote:To start you would put a fingers in one hand and b fingers in the other. Next you would figure out how many groups of a fingers you had total. Then you would perform the sum (# of groups) + (# in each group ) which is the same as (# of groups) + a. In that interpretation we have that given a sequence $a_i\in\mathbb C$ for $i\in I$, we can see this sequence as $|I|$ group of fingers where every group $i\in I$ has $a_i$ fingers in it. So we define an "hyper 0" operator $\underset{i\in I}{\rm O}$ $\underset{i\in I}{\rm O}a_i=\frac{\sum_{i\in I}a_i}{|I|}+|I|$ if $\forall i,j\in I: a_i=a_j$ then  $\underset{i=1}{\overset{n}{\rm O}}a=a+n$ for $|I|=2$ (the 2-ary version) we get $a_1{\rm O}a_2=\frac{a_1+a_2}{2}+2$   and   $a{\rm O}a=a+2$ But then an example of computation proposed is Quote:3[0]2 You have 5 fingers total.That is 1 and 2/3s groups of 3. So the answer is 5/3+3 = 14/3 So the groups are meant to be weighted and the operation is clearly not commutative anymore. In fact the operation proposed is the following. Let $a_i\in\mathbb C$ for $I=\{1,2,3,..., n\}$. Define $\lambda_i:=a_i/a_1$ $\underset{i=1}{\overset{n}{\rm O}}a_i:=a_1+\sum_{i=1}^n\lambda_i$ if $1\leq\forall i,j\leq n: a_i=a_j$ then  $\underset{i=1}{\overset{n}{\rm O}}a=a+\sum_{i=1}^n1=a+n$ for $n=2$,  $a_1=a\neq 0$ and $a_2=b$ $a{\rm O}b=a+(1+\frac{b}{a})$   and   $a{\rm O}a=a+2$ It is clear that the solutions work in some way for preaddition. It is not clear to me how these two solutions can meet the requirment of fundamentality Quote:3. This operation should be something that is fundamental since they require summation and ratios to be defined. MathStackExchange account:MphLee Fundamental Law $(\sigma+1)0=\sigma (\sigma+1)$ « Next Oldest | Next Newest »

 Messages In This Thread hyper 0 - by dantheman163 - 10/25/2009, 11:29 PM RE: hyper 0 - by andydude - 10/26/2009, 01:43 AM RE: hyper 0 - by MphLee - 03/09/2021, 10:28 PM

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