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Jun
13
comment Why is it that this gives a good approximation of $\pi$?
What is a reason and what is a coincidence? This is realted to the mathematical phenomenon of almost integers, calculations involving seemingly random irrationals that turn out to be almost an integer (your choice could be rearranged to say $\frac{\ln^2 11}{\ln 100 \pi} \sim 1$). A reasonable 'why' answer might come from equations that involve floors (like powers of the Fibonacci constant).
Jun
12
comment What are “instantaneous” rates of change, really?
To reword what others have said, 'instantaneous' is just a metaphor to help with intuition. In the metaphor, there's a contradiction, movement over no time at all which really isn't movement. So that's a problem with the intuition, not the formal mathematics.
Jun
2
comment Why doesn't the Taylor series always converge?
Can you give some of these well-known examples?
May
26
comment General solution to a system of non linear equations with a specific pattern
Have you tried solving less complex systems, for example all these systems without the first equation ($a^2 + b^2 + ...$), or just the first two equations? or where $x^i$ all equal 0? Those kinds of simplifications may (or may not) be easier to solve and may give hints on how to solve the bigger problems.
May
8
comment How many ways can $133$ be written as sum of only $1s$ and $2s$
Why isn't the answer about integer partitions $\leq 2$ rather than tuples with entries $\leq 2$?
Apr
23
comment A fun problem by Arnold using the Poincaré recurrence theorem
@rhetoricalphysicist: OK, I see the point now. Can you comment a hint as to how to use Poincare or even give an answer then?
Apr
23
comment A fun problem by Arnold using the Poincaré recurrence theorem
Why Poincare's recurrence thorem? Use Benford's law instead (hattip Travis).
Mar
23
comment Where did mathematicians learn how to do truth tables?
Are you saying there is evidence that some Greek philosophers had the concept of truth functions (functions whose inputs and outputs are something like true and false)_and_ graphical representation of truth tables (a tabular representation)? I don't doubt the first, but I do the latter. Also, the question I have is about the intellectual provenance of the truth-table display in modern mathematics, not the multiple possibly non-influencing reinventions across the world.
Mar
16
comment Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers
@user86418 the derivation after $x+1=2y$ is elementary. Getting to that point is also elementary but missing; just say where it comes from (why +1? Why 2y?)
Mar
16
comment Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers
@user86418: nice... that's the super simple start. But you should add that to the answer to make it complete (i.e. where $x+1 = 2y$ comes from, both $x+1$ and $2y$ equal a side of the square).
Mar
15
comment Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers
@user86418 I'm trying to see how 3/4/5 comes in a flash rather than through hard work. One flash would be as you say, to consider the center of the circle, rather than the square, and the sides of that triangle. But then how does one figure the units. Even if 5 for the radius, I do not see any flash that leads to 3 and 4 for the legs except blind guessing. Where does $x+1 = 2y$ come from, your very first step, without already knowing it's a 3/4/5 triangle?
Mar
15
comment Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers
'Notice the... (3,4,5) triangle'? Is there a quick observation that leads to that? Like the others, that seems to be an observation after all the calculations for the solution have been done.
Mar
7
comment Odd number of students in odd number of classes
From the intro, there's no way to distinguish girls from boys so by symmetry, I don't see how pt a can possibly be true. Are you leaving out something like $b = \sum b_i > g = \sum g_i$?
Feb
13
comment Prove and element must be the identity element?
In other words, the theorem is that in a group if there is an element a that is the identity for any particular element, then it's the identity element for all elements. Why? There's the symbolic proof, which may suffice, but do you want to know the meaning?
Feb
9
comment Where did mathematicians learn how to do truth tables?
Lost: 1) you should edit your answer to paraphrase or quote the relevant text in those links so we have an idea of what you're talking about (answers on SO are expected to be self-contained). 2) Saying that Wittgenstein met Turing is little evidence of an intellectual connection, especially with respect to what essentially boils down to notation.
Feb
9
comment Where did mathematicians learn how to do truth tables?
Edit your answer instead of trying to comment.
Feb
9
comment Where did mathematicians learn how to do truth tables?
Can you give the details of your comment?
Jan
19
comment What do mathematicians mean by “equipped”
Which is to say that 'equipped' is metaphorical language, not a technical term.
Jan
6
comment Why is it important to have the closed form of a generating function?
A side consideration is simply that it is a general mathematical style that finite is desired more than the infinite. You have a better handle on things if they have been reduced to a finite thing. Ellipses are a sign of inscrutability. Of course, with notation, ellipses can be encapsulated into symbols with iteration, and then progress is measured in terms of reduction in bound variables (removing the iteration). Sure a gf can be more inscrutable, but part of math is having multiple formalisms to operate on. That's general strategy. Yuval's excellent answer gives specifics for ops on gfs.
Dec
12
comment What's the difference between a bijection and an isomorphism?
In short, a bijection (or -jection) is a kind of function on sets. An isomorphism (or -morphism) is a kind of function that is structure preserving. Or rather one term is in the language of set theory, the other is in category theory. Underlying all isomorphisms is some bijection, but a bijection doesn't necessarily preserve structure in the category you care about. If your category is Sets, then they're the same.