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Jul
22
comment Rational number to the power of irrational number = irrational number. True?
@makerofthings7 It's probably a combination of many things: (a) you speak the way people around you speak, in mathematics or anywhere else - this is just a kind of jargon, (b) there are lots of non-native English speakers reading and writing English papers, so clarity and consistency is very important, (c) "prove that x exists such that..." sounds very sloppy to me, because "such" modifies "x" ("prove that such an x exists that..."??), and in any case it might mislead you into thinking you were given a formula or algorithm for x earlier on, and are being asked to check it makes sense.
Jul
21
comment Finding multiplicative inverse modulo n using matrix method
This method really just solves the two simultaneous equations $97x = 1$ (true for some specific $x = 97^{-1}$) and $224x = 0$ (true for any $x$, so in particular for $x = 97^{-1}$) without using division. It ends up with $1x = 97$ (and $2x = -30$).
Jul
20
comment Combinatorial - Ways to create subcommittees of a certain size out of a committee?
That's fine. You should find that it simplifies to $\frac{10!}{6!3!1!}$. (You can interpret this as "arrange the 10 people in any order; the first 6 (in any order) will form one committee, the next three (in any order) will form one committee, and the next one (in any order) will form one committee". This is called a multinomial coefficient.)
Jul
20
comment Second Linear Algebra Text for Budding Mathematician
That said, I don't know how far Gilbert Strang's course goes. I might be imagining you've gone much further than you have.
Jul
20
comment Second Linear Algebra Text for Budding Mathematician
@Mohino In my experience, linear algebra is very useful as a tool, and it's not really something you can "learn more of". The basic classification of vector spaces and the maps between them is standard; anything else seems to me to be esoteric tools for very specific applications. Still, I'm happy to be proved wrong. You might like to consider studying rings and modules (generalisations of fields and vector spaces respectively; somewhat harder, and a large current area of mathematical research). It may help to learn some group theory first.
Jul
20
comment Transcendence basis and spanning.
Asaf's comment still holds: $\sqrt{2}$ is algebraically dependent on $5\sqrt{2}$, which is not included in $\mathbb{Q}[\sqrt{3}]$.
Jul
19
comment What is basis of $\mathbb{R}$
A single open interval won't be enough, no. (If you take the open interval (0,1) as a base for your topology, (2,5) isn't open, but it should be.) You can take the set of all open intervals, though. Or you could take the set of all unions of two open intervals. Or you could take all open sets. There are many bases.
Jul
18
comment Extensions of number fields
Finite integral domains are fields.
Jul
18
comment How to describe the family $\tau$ of all open sets of $(\mathbb R^2,\delta)$
$B(p, \varepsilon)$ should only really be defined for $\varepsilon > 0$, so you can throw that problem away immediately. Otherwise, yes, you're right. All $\delta$-open balls are a singleton set (but not $\{0\}$), or a Euclidean open ball around zero, or a union of the two. So any set that doesn't contain zero is open; and any set that contains zero and a small Euclidean open ball around it is open. A non-open set would be, for example, $\{(x,y) : y \leq x^2\}$.
Jul
18
comment Is a graph with only one node a connected graph?
Yes. (It's either connected or disconnected...)
Jul
17
comment Prove that at a party with at least two people, there are two people who know the same number of people…
@CptSupermrkt It looks like a proof to me. Proofs don't need to be formal, they simply need to be watertight. ;)
Jul
17
comment Help in a proof in Hungerford's book
Suppose u is a unit. Then there is some v such that uv = 1. Apply deg to both sides of that equality, and work out deg u.
Jul
16
comment Proof that for any interval (a,b) with a<b in the real numbers contains both rational and irrational numbers?
Okay, will do...
Jul
16
comment Can we conclude any information about the isomorphism classes of groups of order $n!$?
I mean I computed the first few, inputted it into the OEIS, and it told me that the 7th term "seems to be unknown". How out-of-date that is I don't know, and it doesn't provide its sources, so nor do I. :) oeis.org/A133777
Jul
16
comment Impossible identity? $ \tan{\frac{x}{2}}$
Paul: would you have written $\frac{x}{2} = \tan^{-1}(t)$? If so, you're almost there. Look at the graph of $\tan$, and notice that adding $\pi$ doesn't change anything (i.e. $\tan(x/2) = \tan(x/2 + \pi)$ - so we can't quite work out which $x/2$ this $t$ came from, but we can work it out up to adding or subtracting a few $\pi$s.
Jul
16
comment Impossible identity? $ \tan{\frac{x}{2}}$
(@PaulthePirate: you might find it helpful to latch onto a particular example. One I found helpful: suppose you're driving in your car, and you are at position $x$ at time $t$, then your velocity is $dx/dt$ ("the change in $x$ with respect to $t$"). Of course, $x$ varies with $t$, not vice-versa (you may revisit the same position at many different times, but not vice-versa). (If you drew a graph, you'd probably draw $t$ on the horizontal axis and $x$ on the vertical axis, showing that $t$ is somehow varying of its own accord and $x$ depends on $t$, right?))
Jul
16
comment Can we conclude any information about the isomorphism classes of groups of order $n!$?
For n = 1, 2, ..., 6, the number of groups of order n! is 1, 1, 2, 15, 47, 840. The 7th term seems to be unknown. You may also like this: icm.tu-bs.de/ag_algebra/software/small/number.html . I couldn't find anything more general.
Jul
16
comment What's the difference between $\mathbb{R}^2$ and the complex plane?
@fhyve Right - I guess I was just trying to hammer home the point that any structure that can be imposed on one can be imposed on the other (for obvious but somehow stupid reasons), but that that new structure isn't necessarily interesting. I can give [0,1] a few topologies and algebraic operations at random, but (it seems to me that) this isn't interesting because the structures don't interact. Imposing the structure of $\mathbb{C}$ on $\mathbb{R}^2$ somehow isn't interesting in the same sort of way. You get two multiplications that never talk to each other, for example.
Jul
16
comment What's the difference between $\mathbb{R}^2$ and the complex plane?
@Cancan No problem! Sorry if this was too theoretical - I got a little carried away. ;)
Jul
15
comment Determine whether the function is a linear transformation:
Read "T(u+v) must equal T(u) + T(v) for all u and v in your space", etc.