n_kelthuzad said:
I am working on infinity recently. Trying to define the 'indirect' result of infinity as 'range of numbers'. So its like: if there is a set A of infinite elements, f(x)=a\wedgeb\wedgec\wedged... (a,b,c,d...\inA);
However, one cannot say a=f(x) or b=f(x) and so on.
e.g. 1 = e^2i∏ (I know the argument so don't need to remind me e^0i∏)
and 1 = e^0
both equations are true but only remains in the equation. if you pull them out:
2i∏ \neq 0 (well that should be true if i^1 have 'impact' on the real plane)
And what should be the correct way to express this?
TY,
Victor Lu, 16, BHS, CHCH, NZ
Hello n_kelthuzad
In terms of a function, a function always has the probably that it returns one value for every possible input combination.
In terms of different values producing the same function multiple times or even infinitely many times (like a sine or cosine wave) then if you want to find when this happens, if you have a function that is analytic and continuous (think f(x) can be written as some kind of series expansion that is valid across all of the domain which is usually the whole real line or x axis).
What you do is you have to break the function up into 'invertible' parts. This requires calculus and solving for when the derivative is zero. Once you do this then what you get is a function that is 'split' up into parts where you can find the inverse.
In other words if you have f(x) and one part is where a < x < b (or when x is between a and b) then what you do is you are given f(x) and you want to find x and you use more advanced mathematics to do this (usually taught in high school or university): if you want to know this I'll do my best to explain it as simply as I can.
So let's say you have say a sine-wave (which is what the e^ix basically looks like for both the real and the complex parts): you use the derivatives to get the invertible parts (for a cosine wave its between 0 and pi, for a sine wave its between -pi and pi for the first part and then it just keeps repeating in both directions forever) and then what you do is you use what is called a root finder which is programmed into a computer to get a solution for that 'part'
Now this idea will allow you to figure out exactly how many solutions your f(x) gives for the whole domain of the function (all your x's that you can use) and you can find out what x values give them. But usually what we do is for certain functions is we know (usually because of the nature of the derivatives) where we have to check and using other mathematics and computers we get the answers quickly.
The same is also true for complex functions, but you need to use complex calculus and not real calculus (calculus that works no complex numbers), but the idea is the same.
So to express this (if the above is the right idea) we do this:
Let f(x) = a for some known f(x) where a is a constant and we will assume that it is valud (in other words there is at least one value of x where f(x) = a).
Then what we do is find x0, x1, x2, ..., xn, ... where we have
f(xi) = a for all xi where x i=0 is x0, x i= 1 = x1 and so on.
The above is how we say that f(x) = a has the solutions given by the values x0,x1,x2, all the way up to the number of solutions (which might be infinite!). Just to clarify, this means:
f(x0) = a
f(x1) = a
f(x2) = a
f(x3) = a
and so on and we always assume that every x0, x1, and so on is always different from the rest.