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- calculus - What is infinity divided by infinity? - Mathematics Stack . . .
One advantage of approach (2) is that it allows one to discuss indeterminate forms in concrete fashion and distinguish several cases depending on the nature of numerator and denominator: infinitesimal, infinite, or appreciable finite, before discussing the technical notion of limit which tends to be confusing to beginners
- Uncountable vs Countable Infinity - Mathematics Stack Exchange
As far as I understand, the list of all natural numbers is countably infinite and the list of reals between 0 and 1 is uncountably infinite Cantor's diagonal proof shows how even a theoretically complete list of reals between 0 and 1 would not contain some numbers My friend understood the concept, but disagreed with the conclusion
- intuition - One divided by Infinity? - Mathematics Stack Exchange
$\begingroup$ Arithmetic with $\infty$ is usually a convention rather than a piece of mathematics (For example, some mathematicians (in measure theory) take $\infty\cdot 0 = 0$ and reason that this should be the case since $\infty\cdot 0$ represents the "area" of an infinite line in the plane with $0$ width and hence should be $0$ since area = height$\times$ width)
- proof verification - Union of infinite sets and infinity - Mathematics . . .
Nothing new happens in the case of infinite unions An element belongs to the union of a family of sets if and only if it belongs to at least one of them Share
- calculus - Infinite Geometric Series Formula Derivation - Mathematics . . .
Infinite Geometric Series Formula Derivation Ask Question Asked 12 years, 3 months ago Modified 4 years
- probability - Questions about the Infinite Monkey Theorem - Mathematics . . .
By the equivalence to a countably infinite row of monkeys playing within a single second, all 88 keys must be played by a subset (in fact, a countably infinite number of subsets) of monkeys in a second This argument applies every second, so every second, all 88 keys are played by each of a countably infinite number of subsets of monkeys
- linear algebra - What, exactly, does it take to make an infinite . . .
In infinite dimensions, we can have cases where the identity operator's inverse isn't bounded A consequence of this is that the identity's inverse isn't continuous (it can be proven that an operator having bounded operator norm is equivalent to it being continuous)
- elementary set theory - What do finite, infinite, countable, not . . .
Clearly every finite set is countable, but also some infinite sets are countable Note that some places define countable as infinite and the above definition In such cases we say that finite sets are "at most countable"
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