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Aug
15
awarded  Yearling
Aug
4
comment What is the accepted syntax for a negative number with an exponent?
@Beartech there's no contradiction in that picture you posted. Nowhere in there does the notation $-3^2$ appear.
Aug
4
answered How can the axiom of choice be called “axiom” if it is false in Cohen's model?
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@Mehrdad OK I think we understand each other. And in that case Ittay's comment provides the connection I think you're looking for.
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@Mehrdad well that's what I said a few comments back: the dot product is (for our purposes) defined by the projection. To be precise: given $\vec{A}$ and $\vec{B}$, and letting $A_B$ denote the projection of $\vec{A}$ on to $\vec{B}$, the dot product is (again, for our purposes) defined as the scalar value $A_B\lVert\vec{B}\rVert$. I think that makes it quite clear what the dot product has to do with projection. If you still don't think so, perhaps you can explain?
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@Mehrdad but you just said you accept that projection has to do with the cosine of the angle. And you know projection is related to the dot product. I don't see where the confusion comes up. (Maybe we should discuss in chat?)
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@Mehrdad it's often useful to take the dot product to be defined by the requirement that it gives something related to the length of a projection. But in that case, you're right: the fact that one can use $\sum_i a_i b_i$ to calculate the dot product is not obvious.
Jul
22
comment Definitions for complex numbers
This is the first really satisfying justification of complex multiplication I've seen. Very well written!
Jul
20
comment Algebra question about inequalities
At least, there are no positive integers $x$, $y$ satisfying $x^n + y^n = 1$. Did you mean positive real numbers?
Jul
20
comment Solve the equation. e and natural logs
@ColeJohnson no reason, but in the contexts in which one would be asked to solve this equation, it's often implicit that the domain of solution is taken to be the real numbers.
Jul
14
comment Set a new length for a vector?
@Pilpel where was that? I'm pretty sure they shouldn't have sent you here, since you're asking about implementations.
Jul
14
comment On the value of proofs vs counterexamples
Let us continue this discussion in chat.
Jul
14
comment On the value of proofs vs counterexamples
You're welcome :-) I guess this idea, that once existence is proven you can provide an instance, is not something I accept, at least not in general. In the usage I'm familiar with, merely showing that an example of something exists does not count as giving an example of that thing.
Jul
14
comment On the value of proofs vs counterexamples
Yes, it does. The second sentence says that you must give an example of a $T$ such that $P$ holds to prove the conjecture.
Jul
14
answered On the value of proofs vs counterexamples
Jul
14
comment On the value of proofs vs counterexamples
In some cases you can prove that there exists a $T$ such that $P$ holds without giving a specific example.
Jul
12
comment What makes a vector an object with both magnitude and direction?
I think you're conflating two different definitions of "vector": the physicist's definition, and the mathematician's definition. The vector space axioms constitute the mathematician's definition. In order to get a "physical vector", you need to add additional structure on top of that: the ability to calculate a norm, and the behavior under rotation and translation.
Jun
4
comment What does .9 with a line above the 9 mean?
@user1717828 totally random, I didn't feel like going to the trouble of looking up a "significant" fraction.
Jun
3
awarded  Nice Answer
Jun
3
answered What does .9 with a line above the 9 mean?