Solving for x: Unraveling the Rules of O(x)

[Alkatran]

We all know 2+2=4,2*2=4,2^2=4, etc. So, if we say O(1) means +, O(2) means *, etc. Then we know that 2 O(x) 2 = 4.

So, how do we solve for x? If we have 3 O(x) 5 = 24, what is x?

Let's say that - is O(-1), / is O(-2), etc (can you define something that uses what it describes to define itself? (the -)?)

That would mean
x O(y) z = t is the same as x = t O(-y) z
right? What other rules are there that are independent of the level of operation?

 

[TenaliRaman]

if i define aXb as a^2-b+(a-b)
and O(3) = X
then 3 O(3) 5 = 24

can you define something that uses what it describes to define itself? (the -)?)

in that case * and / are not the ones u are looking for ... u are preferably looking for something like + or - ??

That would mean
x O(y) z = t is the same as x = t O(-y) z
right? What other rules are there that are independent of the level of operation?

Hmm what exactly is level of operation in this example?

-- AI

 

[Alkatran]

+ is O(1)
- is O(-1)
* is O(2)
/ is O(-2)
^ is O(3)
sqr() is O(-3)
etc

if x O(y) z = t then t O(-y) z = x

x O(y) 2 = x O(y - 1) x
(2*2 = 2+2, 3*2 = 3+3, 5+2 = 5 ? 5 = 7)

x O(y) 3 = x O(y -1) x O(y - 1) x
(2*3 = 2+2+2, 3*3 = 3+3+3, 5+3 = 5 ? 5 ? 5 = 8)

I think in this system ?, or O(0), would be some form of increment?

 

[TenaliRaman]

5*2 = 5+5
(am i missing something?)

similarly, 5*3 = 5+5+5

Also Note that,
if x O(y) z = t then t O(-y) z = x
will fail for y=2
if i place z as 0 ...

It seems u are trying to develop some sort of system where one can represent one operation in terms of another or something like that ... am i right??

why don't u take the group theoretic approach?
tho i am unsure if something like this has been tried before ...

-- AI

 

[Alkatran]

5*2 = 5+5
(am i missing something?)

similarly, 5*3 = 5+5+5

Also Note that,
if x O(y) z = t then t O(-y) z = x
will fail for y=2
if i place z as 0 ...

It seems u are trying to develop some sort of system where one can represent one operation in terms of another or something like that ... am i right??

why don't u take the group theoretic approach?
tho i am unsure if something like this has been tried before ...

-- AI

Of course it will fail. You divided by 0! Just the same as you can't find the 0th root of a number. (what multiplied 0 times will equal...)

You could say
x O(y) z = undefined if y < -1 and z = 0

 

[TenaliRaman]

umm we are not really generalising here are we?
so what are we looking for?

-- AI

 

[Alkatran]

When you display it like this, it brings up the question what happens because of O() when you pass real numbers, imaginary numbers, 0, etc...