the tree

Yes they can.

Very much so. Say you needed to calculate x(y-z) or xy-xz, then one method will give a more accurate answer than the other, especially if y and z are close by. With x=1.111, y=1.533, z=1.521, and a given accuracy of 4 significant figures for every FLOP (yeah okay that's a little contrived, but anyway) then you'll get relative errors of about 0.02% and 2.5% respectively. One of many rules of thumb is to avoid subtracting numbers that are close by, as that always results in a significant loss of accuracy.

A function is smooth if it has a derivative of all orders, so you simply have to find its derivatives for all orders. You should be able to prove that by induction.

I think we're all focussing a bit too much on Khaled's terrible choice of approximation. There is a valid point to be found in the fact that given finite precision, the accuracy of a calculation does depend on the way that calculation is performed as well as the inputs themselves.

I'm pretty divided on that one. On one hand the idea of not using technology that would allow more people to eat doesn't sit right, on the other hand if that technology is damaging in the long term then I can't really be for it either. So sometimes organic is better than the alternative, but obviously not always. I do try to go for locally grown, which tends to not be mass produced but isn't organic either - since local farmers still use chemicals that are well established and known to be safe. I would much rather - if instead of a standard for 'organic' there was a well known standard for environmentally friendly production. Or at least if less people assumed they were they the same thing.

I would imagine that the whole thing probably made more sense if you looked at it from another perspective. Maybe the second assistant was more senior and had the ability the process all of that herself, whereas the first one couldn't authorise the exchange herself. What I really don't get, is why the first assistant didn't just let you use the shop's landline rather than suggesting that you pay out for a public phone.

If you're using the bike for one short trip every day for nearly an entire year, then it works out cheap - and presumably the bikes are pretty good as well. Hiring bikes in Amsterdam is very popular, especially for tourists and that's done on the basis of one or a few days. If you were visiting Amsterdam for a short holiday, or a buisiness trip then you wouldn't want to take your bike with you and though you could hire a car, it's one of those cities where walking is faster than driving. I don't think that's such a bad thing, if that's what people want to do. I remember being fascinated in Chicago by the stick-your-bike-onto-the-bus thing. Over here we can take bikes onto the buses and use the wheelchair space - assuming no-one who actually needs the wheelchair spaces gets on the bus, this means that a bus can hold at least two bikes, but it should be difficult to build buses that can carry many bikes. Who taught you all of that? My parents taught me to ride and how to change a tire, but I picked up a lot of bicycle maintenance to earn a badge at cub-scouts. It seems that in some cases it would be worth teaching in schools, especially since there's no minimum age to ride a bike it would present a lot independence and empowerment to a lot of kids even if their family couldn't afford a car.

It's never the same from all perspectives, no.

Hmm... was it by sheer coincidence that you actually got fairly close that time? I mean, it's not as good an approximation as the much more obvious [imath]\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots+\frac{1}{8!}[/imath] but at least this time you have actually given an approximation rather than just a very poorly written completely random number.

If your educational establishment doesn't give you access to the journal you need by default, you should request it from their library who should be happy to pay for it for you. We're not to cool with copyright violations here so we're not going to hand it over to you.

Well, crudely speaking: [math]3m \cdot 5m = (3\cdot 5) (m \cdot m) = \underbrace{(5 + 5 + 5)}_{ 3 \; \mbox{times} } \underbrace{(m + m \dots + m)}_{ m \; \mbox{times} } = 15m^2 [/math] Like Dave said, it's mostly just a case of doing the multiplication right.

If it's an element (I'm assuming this is what you mean by 'an atom'), then what is it's atomic number? From there, a good portion of it's properties can be derived - though what makes that particular element so special? If it is something as big as an atom, and exists in such abundance, and has these super special qualities, how is it not easily detectable?

Dark energy is possible, sure. In fact, there's even evidence to suggest that it actually exists.

It varies from browser to browser, buy 'copy image URL' or 'copy image address' or even 'open image in new window/tab' and copy the URL from the address bar there.

It's fairly apparent that [imath]A\bar{B}=\bar{B} \implies B\bar{A}=\bar{A}[/imath] if you draw a venn diagram. edit: worth showing that AB=0

Well then good luck with your exam.

Sure we can. Sorry to push at the rules (I'm assuming the deadline has been passed) but the answers as calculated according to the definitions as I said in my previous post: [hide]displacement: x(t)|t=2 = 8.sin(61.pi/6) = 4 velocity: d2x(t)/dt2|t=2 = 40.pi.cos(61.pi/6) = 20.sqrt(3).pi acceleration: d3x(t)/dt3|t=2 = -200.pi2.sin(61.pi/6) = -100.pi2[/hide] Clearly, there's no need to know what the period of the function is, that it is oscillating, that it is un-dampened, what it's amplitude is - or even what any of those words mean in order to answer the question as presented.

There is no minimum level of intelligence for any part of SFN - the restrictions, as has already been covered, are to weed out poor intent. There are very, very rarely restrictions on the support board of any online forum - for fairly obvious reasons.

He didn't. At all. There was no accepted theory that the world was flat.Columbus thought that there was another route to India, there wasn't, he found something else. (which as it happens, had been found a handful of times before). If there was an accepted theory that the world was flat, and he challenged that without evidence then he would never have gotten funding for the voyage. In order to get the funding and do get people to come along with him, he had to work with known observations about the winds and tides involved and present a well formed hypothesis about the feasibility of the route. Should he have scorned evidence and rigour then he'd have gotten nowhere.

Yes. A number is irrational if it cannot be expressed as a ratio of two integers - nothing to do with the base used. It is a consequence of being irrational that it cannot be expressed as a series of integer powers of any natural base. There are a handful of well studied properties that are base dependent, but none with much application.

Why not just pick a completely random number? Would that not be less computationally intensive for the same degree of accuracy?

I think you're reading a little too much into a joke. England let in a equaliser that maybe they could have prevented, thus they get mocked for it, that's how football works.

The common and colloquial definitions of words, should not be falsely equivocated with their definitions in a specific context (be that science, philosophy, literary analysis, stage design, music, sports... ).

Since you already have a formula for displacement, you can just evaluate it at t=2. Recall that velocity and acceleration are the first and second derivatives of displacement, respectively. From there you will have formulae which again, simply need to be evaluated at t=2 What on earth does that have to do with anything?

Urm, G is approximately 6.7x10-11 where as [imath]\frac{(1+\frac{1}{2}+\frac{1}{16}+\frac{1}{32})^{3} (3\sqrt{2} )}{2^8}[/imath] is a little over 1x10-2.