Amaton

Thanks. That is a rather odd notation. Also, for a data set containing only non-negative real numbers, doesn't [math]Q(x_1,...,x_n)[/math] hold the same four properties described above? Also considering the geometric construction, one wonders if the quadratic mean should be considered a Pythagorean mean, as it seems so closely related.

That I cannot answer on my own. I would've suspected it to be undefined. a matrix with both zero rows and columns is an empty matrix, as you may know.

From the Wikipedia article on Pythagorean means, a geometric construction... where [math]a[/math] and [math]b[/math] are two numbers represented as line segments. Their arithmetic, geometric, harmonic, and quadratic means are shown expectedly. Now, the harmonic mean is a sector of the line segment from endpoints of A and G. What is the significance of that angle, i.e. is it a right angle? Because otherwise I can see H taking on any other value. Also, the Wikipedia article also gives the following properties for the Pythagorean means: [math]M(n,n,...,n)=n[/math] [math]M(bx_1,...,bx_n)=bM(x_1,...,x_n)[/math] [math]M(...,x_i,...,x_j,...)=M(...,x_j,...,x_i,...)[/math] [math]\mbox{min}(x_1,...,x_n)\le M(x_1,...,x_n)\le \mbox{max}(x_1,...,x_n)[/math] It then introduces the quadratic mean for the ordering [math]\mbox{min}<H<G<A<Q<\mbox{max}[/math]. However, the four above properties also seem to hold for the quadratic mean. Is this correct?

John already explained it well, but just to add... Then they are greatly underestimating it. If they tease you, take a question from an old exam and tell them to work the problem. That'll shut 'em up I have yet to reach abstract algebra, but it seems to have a totally different feel to it than linear algebra, just by looking at some open-source material. I do know they are completely separate courses in university, and math majors will take abstract some time after linear.

Makes sense. Thanks for the lesson

Sounds reasonable. If we can define [math]0=\left\{\,\right\}[/math], is it not rigorously allowed to represent a "bag of apples" with zero by the empty set? Haha, I've had more than my share of arrogant statements, but thanks. You also make for well-discussion.

Or that -- which is much simpler and obvious, lol. But zero can be a natural number, as per [math]\mathbb{N}_0={0,1,2,3...}[/math]. It just depends on your preference, since there is no universal standard as to whether or not [math]\mathbb{N}[/math] includes zero. This is a bit above my level as a student, but why would zero make a difference in inductive reasoning? From Natural number - Peano axioms (Wikipedia):

Thanks a lot. I'm inclined to continue... 1. What are other common uses of superscripts on sets then? 2. For [math]S^n[/math], is it okay to think of [math]n[/math] as representing spatial dimensions? Or is it more subtle and/or complex than this?

You usually hear zero included in "non-positive" and "non-negative" subsets of [math]\mathbb{R}[/math], so I would guess neither. The Wikipedia page agrees. What the formal definition of 'sign' is, I don't know. I'd like to say that a number [math]n[/math] is either positive or negative if [math]n\ne -n[/math], where we consider the unary negation. So by this definition, zero is neither.

It's not good enough to just be the ratio of any two quantities. Those quantities in the ratio must be rational numbers themselves. [math]\pi[/math] is irrational by definition. It's impossible to express it as a ratio of any two integers, though actually proving this mathematically is a bit complicated. [math]C=2\pi r[/math], which is how we calculate the circumference in the first place. Since [math]\pi[/math] is irrational, [math]C[/math] can also be irrational, depending on what the radius is. You still raise a good point, since this still doesn't settle the question. Why is [math]C[/math] equal to [math]2\pi[/math] times the length of the radius? What intrinsic property of circles makes this fundamentally true (and is there a formal proof)? note: I think this is possible using a bit of calculus and series representations.

Fair. Yet you can only do so much with just counting numbers At least, without defining new kinds of numbers from them (i.e. their negatives for non-zero integers --> their quotients for non-zero rationals, etc.) Interesting. I like to follow the convention that any [math]2k,\,\, k\in\mathbb{Z}[/math] is even. Therefore, zero is obviously even. More so, [math]2k+1,\,\, k\in\mathbb{Z}[/math] is odd, meaning zero is most definitely not odd. But I'm not sure if this is 'good enough' to rest the issue.

I haven't formally studied the context of this yet, but I have some questions... 1. If I have a set [math]S[/math], does the superscript [math]S^n[/math] denote a Cartesian product so many times (e.g. [math]\mathbb{R}^2[/math])? Or is this something else? 2. Can't I think of the Cartesian plane as basically a graphical representation of the Cartesian product of [math]\mathbb{R}[/math] and itself?

Though generally considered a concept, it's not just any concept. I hate the notion that "infinity" is no more legitimate a number than "justice" or "love". Unlike the latter entities, [math]\infty[/math] actually has a defined, ordered relationship to the real numbers. Even still, there are number systems, besides the reals, where [math]\pm\infty[/math] are in fact numbers with some defined arithmetic relations. And they work completely well in their own respects. What contradictions? Arithmetic on zero only causes contradictions by assuming truth in certain statements which we conventionally regard as fallacies. Letting [math]\frac{1}{0}[/math] be undefined is the keystone example in this thread. Also, there is nothing wrong with zero itself (insofar as I see). It's only when you perform certain operations where things go erky. Do you also have a problem with the number 1? Because 1 and logarithms don't always go well together. Treating zero like infinity may essentially eliminate all trivial mathematical relations it has with the other numbers. Anyway, if you'd like to abolish zero as a number, go ahead. But you will have to reformalize your own rigorous framework of mathematics with many new rules and stipulations. Then somehow reconstruct the set of real numbers without zero in a way that makes math work sensibly (and the complex numbers also since they are so useful). And then continue to rework all of arithmetic, all of analysis, all of algebra, geometry, calculus, topology, etc. since they will obviously not be the same without zero (yet you never hear about geometry being broken because we use zero as a number).

"Why?" is a question that sits rather uncomfortably in mathematics, for any topic it may hold in question. We can define [math]\pi[/math], describe [math]\pi[/math], show identities, derive relations, etc. in order to answer 'why' on a more trivial basis. Maybe the most intuitive reasoning for [math]\pi[/math] is that it is the ratio between the circumference and diameter of a circle (a bit untechnical phrasing there). However, the fundamental "why" has a more philosophical flavor to it, and these kinds of questions are not so objectively answered.

Seriously. The switch 'goes up' when you 'turn on' the light, and there are children around. Whoever made this was either doing it for humor or had a bit too much wine at mass.

Thanks for the responses. Okay. So a "normal mixture" of boron, I'm guessing, would be approx. (80%) boron-11 and (20%) boron-10, as to reflect to their natural variation. Right? I see. So how is the natural composition of an element calculated?

P.S. - I don't know if Chemistry is the appropriate subforum for this thread. I'm not sure if it's physics, chemistry or, maybe best said, nuclear chemistry. Sorry if it seems out of place.

Sorry for my absence. In the while, I've gone through matrix solutions and other methods, but I don't feel for recalling those troublesome problems. Thanks anyway.

Hmm... Even though allowing operators and such is more of a semantic issue, allowing it gives room for some creative thinking. Consider the sequence of hyper-operations (succession, addition, multiplication, exponentiation, etc.) which continue using right-associativity. Let [math]a\cdot b[/math] denote the 999th such operation of [math]a[/math] by [math]b[/math]. Now take this for size... [math]9\cdot (9\cdot 9)[/math]. Of course, anyone can arbitrarily increase any of these parameters or introduce more operators / functions into the expression. Maybe it'd be a bit more challenging to make the restriction to unique operators / functions which are not defined by some boundless parameter. I don't know... factorial, double factorial, prime counting function.

Hello Might be here with another dumb question. So I'm looking at a list of the elements which is organized as a table of properties according to each element. These properties include... • mass density • phase at STP • standard atomic weight (or rather, mass) • number of naturally occurring isotopes • categorization (alkali, transition, etc.) • melting and boiling points • specific heat capacity • electronegativity • natural occurrence (as in primordial, in trace, or synthetic) Now, we know that each element exists as various isotopes, artificially or not. I'm wondering if some of the properties above differ among the given isotopes of a particular element (of course excluding those derived from or concerning the isotopes in general). Most lists make it seem as though these properties are tacitly identified with the element as a whole, rather than any particular isotope. For example, let's say we have the nuclides with 1 proton. Shouldn't any of the measurements of density, electronegativity, heat capacity, etc. differ among protium, deuterium, and tritium? If so, are these properties just identified with the most common or stable of the natural nuclides? I'm only taking high school chemistry as of now, so sorry if I'm making a trivial misconception. Just curious.

As the notion says, it's not so much about [math]e[/math] itself, but rather [math]e[/math] raised to a power. And from this, we see the obvious importance of the exponential function and the natural logarithm. This is also neat I think. Consider [math]e^k[/math] where [math]k[/math] can be any imaginary number. If we look at values of [math]e^k[/math] for all imaginary numbers, we see that these values form a unit circle in the complex plane. This is in my opinion one of the more beautiful and fundamental ideals regarding [math]e[/math].

Of course. I'll look through the work and see if I could a good example...

One day I was walking home... clear, blue sky. And strangely, I had an odd feeling when I was looking up. I guess I was more observant than usual that day, more insightful maybe. Usually, the sky's just there, and I don't really pay attention to it. But this time, I was gazing up, and I actually felt scared at the sheer vastness of it. It literally belittled me and made me afraid. It's hard to describe the feeling, but that's the only time I've ever experienced something like that. Anyway, later I got home, forgot about it, and life continued on as usual.

Wow! I know nothing about building or maintaining aquariums, but that was beautiful. I just love the idea of being able to observe an (imitation) ecosystem.

I'm doing work for math class (high school level), and we're currently on linear systems of equations. Simple linear systems like: ...as taken from my text. Some of several methods we're expected to use are elementary substitution and elimination (there are more efficient ways, but these are the focus for now). What's peculiar is that for some of the problems, it takes me 3 or 4 attempts to finally get a working answer when using these methods. I'm not sure why, but I often derive an answer which only solves one or two of the equations, rather than all the equations in the system. This is pretty strange, since whenever this happens, I can never pick out what I did wrong in the algebra. After the first one flops, I go at it again and derive a different answer. This time, however, it solves only some of the equations, just in a different combination/order... still not the entire system. Some of the problems take me quite a while to get a "global" solution, as I like to think of them in contrast to "local" ones which only satisfy part of the system. Why does this happen? And how can I work around the issue (still using these methods)?