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What is infinity divided by infinity? - Mathematics Stack Exchange I know that $\\infty \\infty$ is not generally defined However, if we have 2 equal infinities divided by each other, would it be 1? if we have an infinity divided by another half-as-big infinity, for
I have learned that 1 0 is infinity, why isnt it minus infinity? An infinite number? Kind of, because I can keep going around infinitely However, I never actually give away that sweet This is why people say that 1 0 "tends to" infinity - we can't really use infinity as a number, we can only imagine what we are getting closer to as we move in the direction of infinity
elementary set theory - What do finite, infinite, countable, not . . . A set A A is infinite, if it is not finite The term countable is somewhat ambiguous (1) I would say that countable and countably infinite are the same That is, a set A A is countable (countably infinite) if there exists a bijection between A A and N N (2) Other people would define countable to be finite or in bijection with N N
linear algebra - What, exactly, does it take to make an infinite . . . If your infinite dimensional space has an inner product and is complete with respect to the induced norm then it is an infinite dimensional Hilbert space That's all it takes to make an infinite dimensional Hilbert space
Is there a case infinite p-group is meaningful? An infinite dimensional vector space over a field with p p elements is probably the simplest example But there are some really wacky groups called "Tarski Monsters" that provide examples of infinite simple groups with every proper, non-trivial subgroup of order p p
What is the difference between infinite and transfinite? The reason being, especially in the non-standard analysis case, that "infinite number" is sort of awkward and can make people think about ∞ ∞ or infinite cardinals somehow, which may be giving the wrong impression But "transfinite number" sends, to me, a somewhat clearer message that there is a particular context in which the term takes place
Subspaces of an infinite dimensional vector space If V V is an infinite dimensional vector spaces, then it has an infinite basis Any proper subset of that basis spans a proper subspace whose dimension is the cardinality of the subset So, since an infinite set has both finite and infinite subsets, every infinite dimensional vector space has both finite and infinite proper subspaces