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comment Prove there exists $m > 2010$ such that $f(m)$ is not prime
@AlexM. It's simply that modulo anything, congruence is compatible with addition and multiplication. So if $x\equiv y$, then $x^k\equiv y^k$, $a_kx^k\equiv a_ky^k$, and $f(x)\equiv f(y)$.
Apr
18
comment Standard deviation and mean question
If you really want to understand it, a more focused question will be of more use to you. Could you edit your question to describe, for each of the four questions, your thoughts on it, what elements from your course you think might be relevant, and where you're getting stuck? As it stands the question is hard to answer because it's impossible to know just what you need explained - so the only kind of answer possible would be "here is the solution", which likely wouldn't clarify anything for you.
Apr
14
comment Can abstract algebra be used to prove points are constructible?
@user54301 Oh, you're thinking of a slightly different necessary condition than I was. You need a stronger necessary condition than "A point is constructible only if it's in an extension of degree $2^n$". Namely, you need the condition "A point is constructible only if it's in a tower of extensions, each extension of which is of degree $2$". This doesn't require a more complicated proof than for the weaker condition. In fact, all the proofs I know prove the stronger condition first, but then reformulate it as the weaker condition because it's simpler.
Apr
14
comment Heuristic explanation for oscillatory behavior of first $n$ primes' multiples
$\mu a_k - k$ seems like a somewhat arbitrary function to want to plot. Is it supposed to represent something meaningful?
Apr
14
comment Are all continuous one one functions differentiable?
But it's not bijective.
Apr
13
comment Can abstract algebra be used to prove points are constructible?
@user54301 That's exactly what I said in my answer.
Apr
13
comment How do mathematicians find formulas?
I think this is probably the ideal way to answer the question. As I mentioned in another comment, the information that the asker needs here is the fact that mathematical formulae are deduced using logic. I was considering trying to explain that in an answer, but I think it's one of those things that's easier to show than to explain. It's hard for people to grasp what it means to prove something by logic until they see an example.
Apr
13
comment How do mathematicians find formulas?
Some of the answers below just go to show how as people who think about math all day, we forget that most laymen simply aren't aware of the most basic fact that math is based on logical reasoning, and not empiricism like physics. I can still remember "finding" theorems in middle school by drawing a mess of lines with a ruler and protractor and measuring until I found two things that matched up, genuinely believing that that's what mathematicians did.
Apr
13
comment How do mathematicians find formulas?
I think you've misunderstood the question. The OP appears to simply want to know how formulae are justified: is it deductive, empirical, or what? You seem to be assuming it's understood that the justification is logical/deductive and moving on to explaining methodology.
Apr
13
comment Intersection of two circular arcs with same center
@tjvg1991 Do you know the start points and end points? If so, what format are those points in? That is, do you know their coordinates, or their angular positions, or what? Obviously if you don't know the start and end points, the problem is impossible.
Apr
12
comment How to find a definite integral over a symmetric interval without finding the antiderivative?
Why would the integral not exist? It's a continuous function on a closed, bounded interval.
Apr
8
comment Generator of group, find the inverse, solve equation
Of course we can find the exact number - $p$ is explicitly given, so in principle you could just try every number between $1$ and $p$ and check if it's a generator, thus counting the number of generators.
Apr
8
comment A pedagogical proof that 9's can be ignored when calculating digital roots
One approach would simply be to namedrop "modular arithmetic" and say that it involves advanced math. This would hopefully have the effect of giving the students a sense of wonder and building their interest in mathematics.
Apr
5
comment Has lack of mathematical rigour killed anybody before?
In fact, statistics in general is probably a good place to look for examples of this sort of thing.
Apr
5
comment is “$a^0 = 1$” a definition or there exists a proof?
In a way, it's obvious that it's a definition. If $a^n=aaa...a$, $n$ times, then $a^0$ just flat out doesn't make sense, so it clearly needs to be given a special meaning before it can be said to have any value at all.
Apr
5
comment is “$a^0 = 1$” a definition or there exists a proof?
@EnjoysMath Are you being serious?
Apr
5
comment What does the notation $\mathbb R[x]$ mean?
$\mathbb R[x]$ couldn't possible mean $\mathbb R^n$. The notation $\mathbb R^n$ includes a variable $n$ which appears nowhere in the notation $\mathbb R[x]$.
Apr
3
comment Why only the numerator is derived?
To whom it may concern: please don't downvote questions because you think they're elementary.
Apr
3
comment Finding the curvature of a line.
Not every curve can be aligned with a circle. Unless what you're really asking is "given the width/height of a circular arc, how can I determine how big the circle it's from is?".
Apr
2
comment Why is the construction of the real numbers important?
@industry7 That would certainly be another approach to motivating the concept of a real number. I chose my "if you needed a square root of two, you'd just pick an approximation" story because it's purely arithmetical (even if I chose to illustrate it with geometry). We just seem to be using the word "obvious" in different ways.