Recent content by Russell Berty

:approve: Which, of course, we have D(x times) = (1 times), so... D(x + x + ... + x (x times)) = (1 + 1 + ... + 1) (x times) + (x + x + ... + x) D(x times)= (1 + 1 + ... + 1) (x times) + (x + x + ... + x)(1 times)= x + (x) = 2x

That is a cute one. :-p But... (1 + 1 + ... + 1) "x times"? The right hand side is only defined for non-negative integer values of x. How can you write the symbol 1 "x times" when x is 1.5, or square root of 2, or pi? How can you add 1 to itself "x times" when x is pi? If you cannot...

One will cost you $.25. Fifty will cost you $.50. One hundred will cost you $.75. What is for sale?

:smile: Perhaps a weaker axiom would suffice: 0\approx1 or 0\in0

We can assert individually that “I exist” as well as “thinking exists.” But our assertions may not be valid. “Thinking exists” is an extremely reasonable assumption. However, “thinking exists” is not known to be true beyond all possible doubt. Playing the skeptic game is very useful...

We are dealing with antiderivatives. So, given F(x) and G(x) that are antiderivatives of 1/x, it is true that F(x) = G(x) + C, for some constant C. For example, the second to last line you have can be written as ln(x) + C = 1 + ln(x) + D for some constants C, and D. We do not then...

Hints: For part a: Note that the range of M is a subspace of Rp with dimension Rank(M). Likewise, the range of L is a subspace of Rm with dimension Rank(L). For the composition ML, notice that after L is applied, the range of L is not necessarily all of Rm. Moreover, when you next apply...

Choosing an arbitrary element from a set is a random process. Choose an element from the set {1,2,3,4,5}. What is the probability that the element is the number 4? 1/5. My “choice” is random in the probabilistic sense. If it was not, I cannot conclude the probability of choosing 4 is 1/5...

I only said that it sounds like a random walk problem. That is, it has that same flavor. I am sorry to have mentioned random walks. I was primarily trying to simplify the original question to “what is the probability that a curve intersects itself?” In fact, I am still uncertain what the...

When looking at the case where the gods speak English (Yes or No responses), here is another hint. Hint: The first question has the following form. “Is it true that (p and q) or (w and v)?” Where p, q, w, v are simple statements. From the response to this question, you should be able...

I am assuming that you do not necessarily mean “graph of a function” when saying “curve”. So a curve can cross itself. So an easier question to ask is “What is the average number of times a curve intersects itself?” This sounds like a random walk problem. So, one could try to answer the...

Speaking of the square root of two… Here is a fun little nonconstructive existence proof. Prove that there exists irrational numbers a and b such that a^b is rational. Proof – Note that \sqrt{2}^{\sqrt{2}} is either rational or irrational. We will argue by cases. Case 1) If...

I believe, historically, i was introduced to name a solution to x^2 = -1. I don't know if this is what you want, but here is a “real world” interpretation of i. In the real world, we have the idea of marking a central point on a line (call it 0.) Then we can mark evenly spaced units along...

Label the following unknown angles: a = BDF b = FDC c = DEF d = FEC e = BFD f = DFE g = EFC Then, write 7 different equations involving them. For example, a + e + 140 = 180. Once you have 7 equations with 7 unknowns, it can be solved. Though, it is messy. If you know...

In keeping with the original question of this post, I believe there are some other concepts that need clarifying (not merely “infinity”.) What is a constant? What is a variable? These also need to be understood from context. From the question it sounds like the underlying context is the set...