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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
Uncountable vs Countable Infinity - Mathematics Stack Exchange My friend and I were discussing infinity and stuff about it and ran into some disagreements regarding countable and uncountable infinity As far as I understand, the list of all natural numbers is
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
linear algebra - Invertibility of infinite-dimensional matrix . . . How do you extend your definitions to that case, and how infinite is n n? Assuming n =N n = N, the product of two such matrices is still not always well-defined, take for instance the matrix whose entries are all 1 1 and multiply it with itself With multiplication somewhat undefined, I just feel a little uneasy about the term "invertible"
calculus - Infinite limits - Mathematics Stack Exchange Obviously it depends on the definition of "exists" Some authors explicitly work over the extended real line with ±∞ ± ∞ adjoined, so that such infinite limits do explicitly "exist" as first-class values But there is no consensus One needs to pay attention to the author's definitions and conventions Perhaps it is worth mention - even though this case is rather trivial - that adjoining
calculus - Does $1. 0000000000\cdots 1$ with an infinite number of $0 . . . A decimal representation of a number has digits indexed by natural numbers Which exactly is the position of that last 1 1? Is it the first after the decimal point? The second? The third? Each digit must have its position, which must be a natural number In other words, when specifying a real number by a decimal representation, you can pick any infinite sequence of digits of your choosing, but
When does it make sense to say that something is almost infinite? 4 If "almost infinite" makes any sense in any context, it must mean "so large that the difference to infinity doesn't matter " One example where this could be meaningful is if you have parallel resistors and one is so large compared to the others that it doesn't measurably affect the total resistance
Is there an infinite dimensional inner product space without an . . . Couple this with the Baire category argument mentioned in Green Park's answer that in any infinite-dimensional Hilbert space, a Hamel basis is uncountable We conclude that in an infinite-dimensional separable Hilbert space, there cannot be an orthonormal Hamel basis