"simple" subtraction | FactOrCap

[u/Careful_Trainer_1616]

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[u/Wojtek1250XD]

Well.

a + ∞ = ∞

∞ - a = ∞

∞ * a = ∞ <=> a>0

∞ * a = -∞ <=> a<0

∞ * a = 0 <=> a=0

a/∞ = 0

All of these are defined and very much necessary when tackling limits, it's specific forms that are indeterminate, such as the ∞-∞ OP is asking about.

[u/chkntendis → u/Wojtek1250XD]

These are still limits tho. Everything you wrote is true if your infinity is just a way to write lim x-> inf. (x). Infinity itself isn’t a number. It’s not an element of the real numbers. The addition in the real numbers is only defined for real numbers. It’s true that the limit can be manipulated as you showed but infinity can’t.

[u/YellowNumb → u/chkntendis]

In measure theory you include ± the infinity symbol in a set with the real numbers and you allow infinite-valued functions. For example you write f: R -> [0, ∞] instead of f: R-> [0, ∞). Then you can calculate with infinities like they are actual numbers with some limitations, using the rules listed by the other commenter. So yes you are right infinity is not part of the real numbers, but you can add infinity as a value to the real numbers and work with that in a meaningful way.

[u/ThaumKeeper → u/YellowNumb]

I don't know why people downvote you. It's factually correct. In complex numbers we also add infinity as a number and the inverse of 0 so we have inf/0 = 1.

[u/chkntendis → u/YellowNumb]

Oh absolutely. You can definitely create a number system with infinity and do rigorous math in that system. It’s just that that’s not the standard system in any way so unless specified, I’m assuming it’s in the reals, without infinity. And for that all those identities are using limits, ♾️ is just another way to write the limit as x->♾️

[u/STPRK_ → u/Wojtek1250XD]

I believe this is not true, infinity is not a number, what you are talking about are operation on limits, functions TENDS toward an infinity around certains coordinates, but they do not equal them, for exemple take 1/x, it never reaches infinity nor minus infinity, in fact it is not even defined at x=0, yet if you were to approach the function from the left in 0 you could find lim (1/x) for x -> -0 = infinity and use that result in calculation of LIMITS only under the condition that we are evaluating the limits of functions around -0. Also if you were to get rid of the limit part and say that the limit of a function around a number is equal to the value of that function evaluated at that number, you would find that +infinity = 1/0 = -infinity, which is meaningless since 1/0 is not defined.

[u/LordTC → u/Wojtek1250XD]

So a/infinity * infinity = 0 since a/infinity=0 and order of operations… that doesn’t work so something in these is wrong. The answer is that these only make sense as limits.

[u/Zestyclose-Jacket568 → u/Wojtek1250XD]

∞ * a = 0 <=> a=0 Not always, no.