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
You can take one to an infinite power, you just can’t call it infinity.
The problem with “infinity” is that it doesn’t fall on the number line. It’s defined as the extreme limit that the number line approaches, but never actually reaches, and because of that, it’s not a number, in and of itself. That’s why you can’t use it in operations like exponentiation. To do that, you’d need a different way of thinking about numbers: something that can handle actual numbers which are infinitely large.
*This system exists*. They’re called surreal numbers, and they include values which are infinitely large or infinitessimally small. The most famous is probably ℵ₀ (pronounced “aleph null”) which could be called the answer to the question “How many natural numbers are there?”. Note the phrasing: this is about how many there are, not what the biggest one is. The reasons that this is important kind of fall out of ELI5 territory, but the end result is that this is a number, and you can plug it into exponentiation if you have something that understands it. According to Wolfram Alpha, 1 to the ℵ₀ power is 1. There are other surreal numbers too, some of them even larger than ℵ₀, and as long as everyone understands that you’re working in surreal numbers, the math works out.
One fun fact: surreal numbers are large enough that you can’t really change their values by adding or subtracting any finite value from them. This has led to the joke “ℵ₀ bottles of beer on the wall / ℵ₀ bottles of beer / Take one down, pass it around / ℵ₀ bottles of beer on the wall”.
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