The next level of the algebriac ladder (need some help)

[AGThePoet]

Hi I'm new here, I'm in 10th grade and using this topic for my science fair project in my junior year.

 

 

I was thinking about what would come next in the line up of:

 

Addition

Multiplication

Exponential

?

 

And tried to use wordplay such as:

 

Multiplication is adding a number to itself x times.

Expoential is multiplying a number to itself x times.

 

and continueing with the pattern:

 

? is exponenting a number to itself x times.

 

I wouldn't believe this for the longest time because I didn't think that the final change would simply be a simplified version of exponenting. So I tried to look deeper into the relationships between the terms. This is what I found:

 

 

 

 

In here we are all probably familiar with the terms addition, multiplication, and exponential. I do a lot in number theory, a lot of which I'll post in here at sometime or another. Anyways, I have always tried to find a relationship between addition stright to exponential and failed miserably. A while ago I came upon a conclusion: Addition, Multiplication, and Exponential (and their respective opposites) are all the same thing. Take multiplications relationship to addition:

 

 

2+2+2+2

is the same as

2*4

 

(this may seem excessively simple, hang with me it get's better)

 

If you really think about what multiplication is, it's simply taking addition and simplifying it. With multiplication you can add a number to itself eighty six times a lot easier than adding it that many. However, to use it like that it has to be a single number. You cannot convert

 

1+2+5+6

 

into multiplication (well, you can, but its not worth the time or effort). So multiplication is easier to do mass addition with than plain addition, but there are severe limitations on its usage.

 

 

Same thing with the multiplication/exponential relationship. You can change

 

3*3*3*3

into

3 to the fourth power

 

and make it easier; however, you cannot change

 

2*8*5*9

into an exponent.

 

This keeps the rule that advancing a level creates an easier formula, but limits the factors to just one number each time (in this case the threes)

 

 

So it only makes sense that (to finish the pattern) the ?/exponential relationship would be the same.

 

Follow this through and you will find that the last term which I took the libery of naming Exutat, hybrid of the latin words "exterus" (highest) and "commutatus" (change).

 

So the table is now complete:

addition is (a+B)

multiplication is (a*B)

exponential is (a to the b power)

and finally

Exutat is (a to the a power)

 

Follow the patterns and you will get the same answer.

 

If you have questions on any if this please don't be afraid to ask

If you can help me with this please do

If this isn't new at all and I just have never heard of it, please tell me and i will withdraw my claim upon it.

 

 

Thanks for hanging with me this far,

agthepoet

[GAHD]

Let me see if I have this straight:

ex. 1

3 Exutat 3 = (3^3)^3 = 72

can also be written as (3^3) (3^3) (3^3)

further elaborated to (3*3*3) (3*3*3) (3*3*3)

 

am I correct? or would the exponetials stack exponentially in the case of this equation?

ex.2 (3^3)^(3^3)^(3^3) = ?

[AGThePoet]

honestly im still trying to figure this thing out so if one seems more logical than the other let me know please

[GAHD]

lol, looks like I'm gonna be doing some number-crunching.

[AGThePoet]

I was also thinking that instead of exutat being (a to the a power), it might be (a to the a power a times), so instead of (if a=4)

 

NOTE: the lines look starigth up and down i geuss dont know why it posts like that but it should be going bottm left to top right (exponential).

 

exutat=

 

 

 

 

4)

4)

4)

(((4

 

 

so it would be (4*4*4*4)*(4*4*4*4)*(4*4*4*4)*(4*4*4*)

instead of (4*4*4*4)

 

 

not sure if that is close to being right, but thought I would throw it in.

 

AGThePoet

[Phire]

If I understand correctly you saying you could simplify 3 to the power 3 to the power of 10? Well, if I understand correctly, what you just described to me is one of the exponential rules. If lets say for example I have (3^6)*(3^10), then you can simply simplify (pun not intented) by adding the exponential together, so the answer would be (3^16). Now, if I had (3^6)^10, another exponential law would apply. You would get (3^60). So I don't see a need for any exutat. But still, pretty clever thought processing none the less. Keep it up. If your really interested in math, then I bet you can't wait for calculus and linear algerbra.

[halogenesis]

Yeah I agree Phire, it also looks to me just the long way around the exponetial law. (4^4)^4 in Ag's previous post would be just 4^16... because the exponents would just multiply.

 

But don't they learn that in Grade 10 math already? Because Agthepoet says he's in Grade 10... So maybe he's trying to say something else...? I'm not entirely clear.