Im studying calculus this year and one of the lectures included undefined values (*forbidden and unwanted* not my words btw). These included:
1. 0/0
2. oo – oo
3. oo/oo
4. 0*oo
5. 0^0
6. oo^0
7. 1^oo
All of these are extremely weird to me and I don’t really understand them, but the one that strikes me the most is the last one. As a former math competitor and regarded as “gifted” in math, I feel stupid not being able to comprehend this, but most importantly it shatters my belief that math can explain everything and that is has all the answers.
I don’t see infinity as a really big number, I understand it as a concept, and what confuses me the most is seeing infinity treated as number. 1^oo for me doesn’t make any sense. It seems mathematically absurd. Infinity isn’t number, it’s a quantity that can’t be measured. And if it is treated as a number shouldn’t 1^oo = 1?
How come these are “undefined”? Someone please answer, Im losing my mind over this. All explanations are welcome, shallow or deep.
edit: to clarify oo = infinity
In: Mathematics
Algebraically, these all contain an instance of 0/0. Which is bad.
We can say that 1/0, 5/0, -8/0 etc are all equal to infinity and everything is totally fine and consistent and works out (though, you do need +infinity = -infinity for this to be nice). Dividing by zero, in most cases, actually causes no issue and using limits is a way to make sense of this in a precise way without having to actually consider infinity as a point.
The issues pop up when you divide zero by zero. All of the “proofs” of inconsistent things involving division by zero end up happening because they try to divide zero by zero somewhere, not simply because they are dividing b zero.
All of these expressions use 0/0 in them, so the are indeterminate, since there isn’t really any nice, consistent way to assign a value to 0/0 – either trivial, hyper complicated and unhelpful, or leads to contradiction.
1. 0/0
2. oo – oo = (1/0) – (1/0) = (0-0)/0 = 0/0
3. oo/oo = (1/0) / (1/0) = 0/0
4. 0*oo = 0*(1/0) = 0/0
5. 0^0 = A^(oo*0) = A^(0/0) – for any 0<A<1
6. oo^0 = B^(oo*0) = B^(0/0) – for any B>1
7. 1^oo = A^(0*oo) = A^(0/0) – for any A!=0
For any of these to have a value, even if it is infinity, you need to make sense of 0/0, which cannot be done.
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