What Comes Before and After Addition, Multiplication, and Exponentiation?

[Loren Booda]

What operation precedes, and what operation follows the sequence below?

...addition, multiplication, exponentiation...​

 

[Tide]

Is this a trick question?

I could say that a left brace precedes and a right brace follows.

 

[Loren Booda]

...addition, multiplication, exponentiation... are all operations. Are there operations (like factorialization, say) that extrapolate logically from this given sequence of operations?

 

[dimachka]

parentheses?

 

[-Job-]

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.

 

[KingNothing]

There is no operation that precedes addition. Addition is the most basic operation.

As far as after exponentiation, there is an operation that I believe is called "towers". It doesn't really interact with anything very nicely, so we don't use it much. There is also something called the ackermann function which denotes 'higher operations', one might say.

 

[-Job-]

I still like the increment.
Ok, so an interesting fact is:
k*n = k + k + k ... + k (n-1 additions)
k^n = k * k * k ... * k (n-1 multiplications)
k$n = k^k^k^k ... ^ k (n-1 $)

k*(2-n) = k - k - k ... - k (n-1 subtractions)
k^(2-n) = k / k / k ... / k (n-1 divisions)

We actually need name for the inverse operation to exponentication. Like:
addition : subbtraction
multiplication : division
exponentiation : inponentiation?

 

[JasonRox]

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.

Nice one.

I like that answer.

 

[xcoder66@yahoo.com]

The common link between these three operators is that they all increment from zero point.

Addition increments in a variable manner, but only allows one incrementation. 0 + 6. The right operand causes displacement of six. Multiplication increments in a variable manner, but allows more than one incremention per operation. 3 * 6. Three incrementations of the value six. Exponentiation increments in a restricted manner, but also allows multiple incrementation. 2^3. This increments to the value of 8.

As for what comes before addition I would say it is: Nothing Operator. Addition causes the tiniest displacement on an infinite number line. What could be less than fundamental displacement? There is no answer and if there was, you would begin to say one, and then you'd be adding.

And after exponentiation? I'm not sure that there is a clear pattern with this operator series. I don't think the operators logically follow each other, except for fundamental incrementation. But how they increment must show the logical pattern, so I don't really believe they are a logical series in that sense. I think they are a consequential convenience for us.

But I'd love for someone to point out a significant operator patern in this series, because I don't see it.

 

[tmc]

The one following the series would be a "Power tower", as someone mentionned.

From mathworld:
http://mathworld.wolfram.com/PowerTower.html

 

[xcoder66@yahoo.com]

The one following the series would be a "Power tower", as someone mentioned.
From mathworld:
http://mathworld.wolfram.com/PowerTower.html

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.

 

[tmc]

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.

It logically follows from the following:

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.

 

[xcoder66@yahoo.com]

It logically follows from the following:

It logically follows, only if you isolate the fact that each operator evolves by creating a multiple of the previous operator.

A power operand doesn't allow the versatility of two changing operands, it only allows variable times of multiplication.

Since the two first operators allow for variation of both operands and the power operator doesn't, it cannot be logically a series on that point.

But if you ignore that and only focus on that fact that there is a multiple fundamental consistent between them, yes, I would agree half way.

 

[-Job-]

Are you referring to x||y ? (||is my powertower notation) I'm not sure if i understand what you're saying.

 

[xcoder66@yahoo.com]

Are you referring to x||y ? (||is my powertower notation) I'm not sure if i understand what you're saying.

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.

1. The most fundamental quality between all operators, which is identical in each operator.

2. The most fundamental quality that evolves, which can be recognized as a predeccessor of the previous operator in the series.

3. A quality that is unique to an operator, in that it doesn't exist in another operator.

I think the most fundamental quality that is present in them all is they all increment, which is basic displacement in a positive or negative direction.

Concerning category 2, multiplication evolves from addition, because of the "multiple concept". Multiplication is a shorthand for multiple additions. I think powers use the same "multiple concept", because it becomes multiple multiplications.

Concerning 3. I think that both operands being able to be different values is a unique quality that only exists in the addition and multiplication operators. It disappears abruptly in the power operator.

Of couse, there are other implications this categorical way of analyzing the operators produces, but I just wanted to limit it to the brief point I made earlier, because I'm not sure I made a very good explanation.

As for the tower operator, I checked the link, but I don't quite understand the explanation of it's effect upon values, so I can't express an opinion.