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__Sign Up__- Suppose we are dividing some number, lets say 5, by 0.
- What kind of useful way could we represent this and how would we go about getting the inverse?
- Lets see
- 5/0 = inf*0 rem 5
- Assuming 0 goes into 5 infinite times.
- while
- 0/0 = inf*0 rem 0
- and
- 0/0 = k
- where 'inf' represents a non-terminating process, or the set of all numbers in some set.
- therefore
- 5/0 = 0*0 rem 5*1
- but its a special kind of zero, because it too
- represents the output of a process.
- Consider it a production process.
- That remainder gets carried around as a mathematical object
- 0/0 = inf*0 rem 5
- 5 * inf0 = 5/0
- 5/0 = inf*0 rem 5
- Why?
- Because 0 goes into 5 either 0 times, or an infinite number of times, with 5 as the remainder.
- Actually its
- 5/0 = inf*0 rem 5*1
- That one at the end is just the multiplicative identity, which I put in there
- because I am obsessive.
- Now think of 0/0 not as some heresy (you godless heathens), but like a point on the graph of a function with inputs x, and y.
- Its a special dimension, sort of like the imaginary numbers, and i.
- Or the matrix (lol).
- We will represent it with the special symbol 'k'.
- Now how do we take this, (inf*0 rem 5*1) and get 5/0 back out? And why are we using this structure? We'll see shortly.
- We get back to 5/0 like so:
- First we swap the 'inf' and the 5.
- inf*0 rem 5*1 --> inf/0 = 5*0 rem inf*1
- And then we make the 5 the numerator, and the infinity the denominator.
- --> 5/inf = 0 --> 5/(5/inf) == 5/0
- And this works because 5*0 is just 0, and 5/inf *approaches* zero, which only
- works if 5/(inf) actually equals 5/(inf*1)
- where the 1 is the multiplicative identity. Which seems like a tautology, but has a purpose.
- Infinite division must approach 0. And we can represent n/0 as the *output* of some other division that resulted in n on the left side, and 0 on the right side, even if we can't compute it further than that because its indeterminant.
- or, if we want more abstract representation:
- inf*1 rem n*0 --> inf/0 = n*0 rem inf*1 --> n/inf = 0 --> n/(n/inf) == n/0
- where inf/0 equals n*0 rem inf*1 for *all* n, though its not reversible if we
- don't have the value of n in n/0 or inf*1 rem n*0
- For example, take finding the limit of x and sin(0)
- An indeterminate expression seems to have more than one answer, depending on how you evaluate it.
- lim(x->0) (sin(x)/x)
- Seems to be zero if you use direct substitution: sin(0)/0. However, sin x and x are almost equal to each other as you get close to zero, which actually makes the limit 1.
- Lets work the problem a little bit.
- where n = sin(0)
- sin(0)/0 == inf*0 rem sin(0)*1 --> sin(0)*0 rem inf*1
- sin(0)*0 rem inf*1 --> sin(0)/inf = 0 --> sin(0)/(sin(0/inf) == n/0 == n
- NOW ASSUME something strange.
- Assume our n is actually a number thats infinite on a given number line--only *relative* to
- *some other* number, m.
- It has a fixed value, and yet its infinite. How?
- That number, m, is none other than our indeterminant point k=0/0
- When we substitute 0 for our new dimension, k, then n/0 = n makes more sense (and I'll show you how in a moment).
- And this works because
- inf*1 rem n*0 --> inf/0 = n*0 rem inf*1 --> n/inf = 0 --> n/(n/inf) == n/0
- Thats n/inf = 0 --> n/(n/inf) == n/0
- Think about it, k represents a special point on a function, where inf gets replaced with k
- such that n/(n/(n/inf)) --> 0/(0/(0/0))
- which after rearrangement...
- 0/0 = inf*0 rem 0*1 -- > (0*0 rem inf*1)
- which becomes
- (0*0 rem inf*1)
- and when we plug that into our solution 0/(0/(0/0))
- we get this:
- (0*0 rem inf*1)/((0*0 rem inf*1)/((0*0 rem inf*1)/(0*0 rem inf*1)))
- That 'remainder' inf*1 exists on the same imaginary number line as k.
- Its not determinant on the *ordinary* number line, because it breaks some rules otherwise.
- But it *is* determinant on this new number line
- It translates to
- inf/(inf/(inf/inf))
- And what does that come out to?
- It comes out to 1.
- Eat it haters.
- I'll take my noble prize at the special olympics now.
- I want it delivered in the form of twenty two thousand ziplining tickets.
- So all the kids in wheelchairs will get to experience not what its like to *walk*
- for once in their life.
- But what its like to FLY.

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