The next level of algebriac equations

[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

[KiNPiN]

[KiNPiN]

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.

your 4 formulas would only be equal to each other if A and B were (0,0) or (1,1) or (2,2).

also your "exutat" does not simplify the exponential process.

[AGThePoet]

"Simplify" may not have been an exact term; Exutat limits exponenting to make it more convinient, just exponents do to multiplication and multiplication does to addition.

Alex

[KiNPiN]

no, it doesnt. think about it. lets say a=3 and b=4; and you want to simplify the exponential process of 3^4(3^4)(3^4). according to your exutat formula, it would be 3^3 which completely makes no sense whatsoever.

[AGThePoet]

One thing i was thinkking about... what if exutat didn't do (a to the a power), what if was (a to the a power a times)? I know as the operations go up the answers go up exponentially, so i think this one could be a better fit (or maybe not, this is all rather confusing...)

Alex

[Lord Henry Wotton]

Your exutat should be a^(b*n)
Where n=the number of times a^b appears in our original exponential equation.