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 Aug4 comment How many positive and even factors does $2013!$ have? Your last few questions revolve around the number 2013, looks like someone is posting competition questions Oct21 comment Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$ Took me a minute Oct5 comment Common Einstein Notation Identities @NieldeBeaudrap they weren't meant to be complex examples. For instance working out $A(u)\times A(v)$ where $R$ is 3-by-3 matrix takes a few moments and is something I've encountered a few times. Sep9 comment Finding a function's domain from the function's formula @jorki, you are right, its a little late here so I confused the example in my head with the actual example given in the OP. Jul6 comment You see a route 14 bus on the moon. What is the most likely number of bus routes on the moon? As a school prank you release 3 greased pigs into the school with the numbers 1,2, and 4 painted in bold letters on their backs. What is the most likely number of pigs from the point of view of the school administrator? Jul6 comment Prove in full detail that the set is a vector space @Jyrki I agree and I believe that most reasonable instructors would agree as well. However if this ambiguous wording had come up on an exam I would certainly ask the professor if we are allowed to use any theorem presented in the course. After all, there are much quicker and easier ways to prove that it is a vector space than even the subspace theorem (See Bill Dubuque's answer) Jul6 comment Prove in full detail that the set is a vector space @Virtuoso the key phrase Prove in Full Detail suggests to me that the test wants you to show all the vector space axioms, you might be marked down on an exam if you prove by showing its a subspace. Jul6 comment Prove in full detail that the set is a vector space @Virtuoso Typically you are not allowed to use theorems and concepts that have not been covered up to that point in class. If you are unfamiliar (i.e. it hasn't been covered yet) with the concept of a subspace then you should show all the axioms. Since a subspace is a vector space in its own right, you only need to prove that this set constitutes a subspace of $\mathbb{R}^2$ - it contains 0, closed under addition, and closed under scalar multiplication. Jul1 comment Definition of Ring Vs Rng @Pete the Deskins book (well the dover reprint, I obsessively buy dover reprints for dirt cheap at used book stores) is the one I remember seeing, I was trying to track it down again but @Brian got it before me. Jun30 comment Definition of Ring Vs Rng Didn't really expect to start such debate. I haven't studied enough algebra to see the benefit of using one definition over the other. I didn't realize it was such a contentious issue. Jun30 comment Definition of Ring Vs Rng @Pete I've also encountered some textbooks that state $1\neq0$ as a ring axiom specifically to exclude the trivial ring. Why they do that is beyond me. Jun28 comment Examples of results failing in higher dimensions Proof by pictures still exists in higher math, for example see: en.wikipedia.org/wiki/Commutative_diagram . Although I agree that at the very least there is a rigor behind them. Jun28 comment Finding $\csc$ with $\cot$ @Adam trig functions represent relations between the sides of a right triangle. Excepting some corner cases (like $\theta = \frac{\pi}{2}$) you can always construct a right triangle that "represents" the trig relation. Jun28 comment Finding $\csc$ with $\cot$ @Adam as I pointed out in my comment, even though you checked 12 times you missed the clear arithmetic error Jun28 comment Finding $\csc$ with $\cot$ You did everthing right, but 9+16 = 25 not 27 Jun27 comment The mathematics behind Clebsch-Gordan Coefficients @Theo thanks for that, its what I get for working in a lab with a bunch of Russians - "gord-on" Jun24 comment Number of occurrences of the digit 1 in the numbers from 0 to n While this is a correct way to approach this particular question, the actual problem in question (which is a project Euler problem) requires the sum total of all numbers that satisfy this condition for all digits 1-10. Not so easy (or fast) do with this brute force method (where do you set the upper search limit for instance?). Jun24 comment Number of occurrences of the digit 1 in the numbers from 0 to n This is a not-so-veiled attempt to elicit a solution to a project euler problem. Jun4 comment Reason why these two probabilities are equal (Picking Balls in exact order without Replacement) Intuitively there is a degeneracy between "sublevels" of the same ordering (i.e there are 12 different ways to pick AAABB but they are equivalent). Thus there is nothing special about one ordering versus another, therefore you would expect results that only differ in ordering to have the same probability. May31 comment Throwing All Numbers From 2 to 12 With Two Dice What do you mean by "expected". In other words, at what probability threshold do you draw the line - the probability of all numbers appearing approaches, but never hits 1.