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Mar
29
revised Probability binomial word problem
edited body
Mar
29
comment Probability binomial word problem
You can view your sample space as the one containing all possible values for all $X_i$, but the simplest way would be to see that $$P(S_n>0)=\left(\frac{49}{50}\right)^n$$ and then find the condition on $n$ that this imposes if you want $P(S_n>0)\ge\frac{1}{2}$
Mar
29
answered Probability binomial word problem
Mar
25
comment Find all solutions for $7x^2 \equiv 3 \mod5$, if any.
Note that there are only $5$ numbers modulo $5$, so finding the solutions directly is always easy.
Mar
21
comment Visualizing mathematics and geometry
I've heard of mathematicians who lost their eyesight (Even at a very early age) and constructed some beautiful geometry, supposedly being able to visualize things outside of the real world with less distraction, but do you have any examples of mathematicians who were born blind (and thus never had the chance to experience vision) and contributed in geometry?
Mar
17
comment How do you find two functions $f$ and $g$ such that $f(x) \cdot g(x)=f(x)-g(x)$?
As for the identity, $$\tan^2x\sin^2x=\tan^2x-\sin^2x$$ is just $$\sin^2x=1-\cos^2x$$ multiplied by $\tan^2x$, and the latter is the Pythagorean theorem which is indeed very useful and well known.
Mar
16
comment Recursively Enumerable Languages and Turing Machines
I just looked at Wikipedia for Turing Machine, and the informal definition looks pretty good, and I would cite the Church-Turing Thesis as my "justification" for just writing the algorithm in words (precisely constructing an actual Turing machine can be long and painful). Unfortunately I don't know any more comprehensive online resources, but I recall having studied from a book called "Introduction to the Theory of Computation" by Michael Sipser.
Mar
16
comment How to detect an asymptote
You could try run the user input through some symbolic analysis program and get the "real" asymptotes, but for your program it will probably be better for you to apply some heuristic such as calling an asymptote anywhere you find the "derivative" ($\frac{\Delta y}{\Delta x}$) is larger than some threshold and the nearby values seem large.
Mar
16
comment Recursively Enumerable Languages and Turing Machines
In this case (I suppose) you can just write the algorithm out in words and ignore the inner workings of a Turing machine. The method I described above allows you to run $M$ for any finite number of steps on all possible inputs; that means that you'll eventually be able to pick out any input for which $M$ terminates in a finite number of steps. Therefore, if there are $637$ of these, the machine will find them in a finite time and be able to accept $M$.
Mar
16
comment Recursively Enumerable Languages and Turing Machines
It is, but I'm not sure you got my explanation right; to show that $L_1$ is r.e. you have to construct a Turing machine that accepts it, that is: Given a machine $M$, construct a Turing machine (that can use $M$ and the method I described above) that checks whether $M$ terminates on at least $637$ inputs.
Mar
16
answered Recursively Enumerable Languages and Turing Machines
Mar
16
revised $\mathbb Q(\sqrt2 + \sqrt3)$ is what set?
converted to latex
Mar
12
answered How do I calculate the number of members in a limited Fibonacci series?
Mar
6
comment What is larger x and what is smaller x?
Can you copy the exact context? This sounds like $x$ should be large/smaller than something (or as large/small as possible) for something to hold.
Mar
5
comment $\sum$ over disjoint union of sets
In fact it's not really defined without order of summation for general $a_k\in\mathbb R$. Are you sure this is not a definition of $$\sum_{k\in A}a_k$$ as some sort of short notation for RHS where the $A_n$ are finite or something?
Feb
28
comment Fastest way to multiply numbers mentally?
Well then you can try: $43=44-1=4\cdot11-1$. Therefore $64\cdot43=64\cdot(4\cdot11-1)=704\cdot4-64=2816-64=2752$ - just work with whatever tricks you can compute quickly and split the numbers. Learning the first powers of $2$ is recommended either way, though.
Feb
28
answered Fastest way to multiply numbers mentally?
Feb
27
comment What does $(\mathbb C\backslash\{0\})\times\mathbb R$ mean?
And in the latter case it would "look" like a full 3D space with a single line missing.
Feb
27
comment “Looping” equation
Relevant: what-if.xkcd.com/43
Feb
27
reviewed Approve suggested edit on If $F\subseteq\mathrm{Mat}_n(\mathbb{Q})$, then $[F:\mathbb Q]\leq n$?