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What operation precedes, and what operation follows the sequence below?
...addition, multiplication, exponentiation...
Nice one.-Job- said:Multiplication is short for n-1 additions. Exponentiation is short for n-1 multiplications. Whatever comes next ought to be short for n-1 exponentiations. For example:
k*n = k + k + k ... + k (n-1 additions)
k^n = k * k * k ... * k (n-1 multiplications)
So the next one would be:
k$n = k^k^k^k ... ^ k (n-1 exponentiations)
I used $ for the operation symbol. Someone please come up with a name for it .
The one that precedes addition is trickier. Addition would have to be short for n such operations, so:
k+n = k % k % k ... % k (n-1 operations)
Maybe % is actually an increment operation like:
k+n = k++ k++ k++ ... k++ (n increments)
So an increment would be our simplest operation which doesn't seem unreasonable since it is unary.
I don't recognize it as a series. Incrementation inconsistently evolves between the multiplication and power operators, when compared to how it evolves between addition and multiplication operators. Addition simply displaces on the number line. Multiplication displaces, but is a notation for multiple additions. The way things increment breaks here, because raising things to a power causes a restriction: the number being multiplied by in a power operation has to be both the left and right operand in the power operation, whereas this wasn't a restriction in addition and multiplication. This quality "that left and right operand may vary" suddelnly drops out the of picture, which breaks a significant quality of the operators.tmc said:The one following the series would be a "Power tower", as someone mentioned.
From mathworld:
http://mathworld.wolfram.com/PowerTower.html
It logically follows from the following:xcoder66@yahoo.com said:I don't recognize it as a series. Incrementation inconsistently evolves between the multiplication and power operators, when compared to how it evolves between addition and multiplication operators. Addition simply displaces on the number line. Multiplication displaces, but is a notation for multiple additions. The way things increment breaks here, because raising things to a power causes a restriction: the number being multiplied by in a power operation has to be both the left and right operand in the power operation, whereas this wasn't a restriction in addition and multiplication. This quality "that left and right operand may vary" suddelnly drops out the of picture, which breaks a significant quality of the operators.
So, the tower has to tie everything together with some fundamental property that shows the logical evolution through all operators or it simply evolves from expotentiation. I'm still in the dark on how these really are a logical series, rather than just being considered a consequential convenience series.
Job said:Multiplication is short for n-1 additions. Exponentiation is short for n-1 multiplications. Whatever comes next ought to be short for n-1 exponentiations. For example:
k*n = k + k + k ... + k (n-1 additions)
k^n = k * k * k ... * k (n-1 multiplications)
So the next one would be:
k$n = k^k^k^k ... ^ k (n-1 exponentiations)
I used $ for the operation symbol. Someone please come up with a name for it .
The one that precedes addition is trickier. Addition would have to be short for n such operations, so:
k+n = k % k % k ... % k (n-1 operations)
Maybe % is actually an increment operation like:
k+n = k++ k++ k++ ... k++ (n increments)
So an increment would be our simplest operation which doesn't seem unreasonable since it is unary.
It logically follows, only if you isolate the fact that each operator evolves by creating a multiple of the previous operator.tmc said:It logically follows from the following:
I find operators fascinating. I think it might be helpful to enumerate what we are distinguishing here. Here's what I can think of, but I need everyone's help, because I couldn't expect to name every useful category of qualities for analysis of a candidate operator series.-Job- said:Are you referring to x||y ? (||is my powertower notation) I'm not sure if i understand what you're saying.