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What is infinity divided by infinity? - Mathematics Stack Exchange 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
limits - Can I subtract infinity from infinity? - Mathematics Stack . . . $\begingroup$ Can this interpretation ("subtract one infinity from another infinite quantity, that is twice large as the previous infinity") help us with things like $\lim_{n\to\infty}(1+x n)^n,$ or is it just a parlor trick for a much easier kind of limit? $\endgroup$ –
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"
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
What exactly is infinity? - Mathematics Stack Exchange Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries In the theory he developed, there are infinite sets of different sizes (called cardinalities) For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite "-Wikipedia: Infinity
Does infinite equal infinite? - Mathematics Stack Exchange (the principal exception I know of is the extended hyperreal line, which has many infinite numbers obeying the 'usual' laws of arithmetic, and a pair of additional numbers we call $+\infty$ and $-\infty$ that have the largest magnitude of all infinite numbers, and do not obey the 'usual' laws of arithmetic)
I have learned that 1 0 is infinity, why isnt it minus infinity? Infinite numbers do exist in the hyperreal number system which properly extends the real number system, but then their reciprocals are infinitesimals rather than zero Thus the idea of $\frac{1}{0}$ can be interpreted as saying that if $\epsilon$ is infinitesimal then $\frac{1}{\epsilon}$ is infinite
Example of infinite field of characteristic $p\\neq 0$ On the other hand, if we had $\overline{\mathbb{F}_p}\subseteq\mathbb{F}_p(T)$, then we would have that there were some $\frac{f}{g}\in \mathbb{F}_p(T)$ such that $\frac{f}{g}\notin\mathbb{F}_p$ and $\frac{f}{g}\in\overline{\mathbb{F}_p}$ (because $\overline{\mathbb{F}_p}$ is infinite and $\mathbb{F}_p$ is finite), and they would have to be