Recent content by Icarus

Not bouncing or using extra dimensions. Merely a matter of direction.

Warr - You have accused hello3719 of something he did not do, and then you turn around and do the very thing yourself: This is what you are trying to show! To see that hello3719 did not commit this error, let me expand on the proof: -1*0 = 0 Anything times 0 = 0...

Every non-trivial additive subgroup of the integers is generated by its least positive element. So J = {ma | m in Z} for some a. Out of the 5 elements in the given set, you need to choose a set of 3 all of which are multiples of the same number, and such that the other two elements are not...

At the risk of injecting yet another towering out-of-control ego into a debate already over-run with them, I would like to directly address the concerns energia originally raised, while trying to avoid making unwarranted assumptions about why he raised them: Quite frankly, energia, I am...

I suppose it depends on what you see. I, like Fourier jr I suspect, see mathematics as a means of exploring new worlds of thought. Those definitions and lemmas were not dry facts of no worth outside of themselves, but rather clues to the solving of old mysteries, and the uncovering of new ones...

Perhaps you could elaborate: change what from general to transformational to mapping? These terms all have myriads of meanings in mathematics - what in particular are you trying to do?

One of the best things I ever did was spend a month learning calculus several months before I started the course. When the course started, I was always on top of things, unlike my compatriots. As a result, I had considerably more free time than they did, at a point when free time was an...

If temperature were the only thing affecting magnetic field strength, then yes you could conclude that the hotter something is, the weaker its field. However, in truth, magnetic field strength in stars and planets is generally stronger than it is outside, because flux is constant, and inside, it...

One (minor) nitpick with that: "c" is by definition the speed of light in a vacuum. The speed of light in other media is not properly denoted by "c". Also, referring to "c" as "the speed of light" is misleading. While the description is common, and accurate in a vacuum, it tends to make...

I've removed my posting of the solution so that anyone who wants to follow up on baffledMatt's excellent insight will have a chance to do so!

Differentiation has been studied for quaternions as well. The same definition that works for the Reals and Complex numbers still works for the Quaternions, and even for Cayley numbers (an 8-dimensional extension of the Reals which sacrifices full associativity - I'm guessing these are the...

No prejudice - simple scalpel: A "ghost measurement" is any quantity that cannot be actually measured - not just because no one is available to measure it, or the right equipment is unavailable, but because it is fundamentally unmeasurable. My best guess at what your comments meant suggested you...

I.e., yes. Godel's theorem says that "any consistent mathematical theory containing a model of the natural numbers is incomplete". (The conditions on the theory are a bit stronger than that, but the consistent requirement is part of the statement.) If you follow his proof, you can see...

Janitor is correct about (b). To see it, just consider what happens if C is the identity: 2I has 2 along the diagonal and 0 everywhere else, so the determinant is det(2I) = 8 = 23det(I).

Berkeley famously castigated differentials as "the ghosts of departed quantities". In his time, there was no solid mathematical basis for them. Today we have a number of justifications such as those mentioned above that work in various limited circumstances, but none that completely covers the...