Miscellany

[jack]

Router vs Modem

Did "router" always imply "including modem" or is that a change in usage that snuck up on me? It seems now the standard usage is to refer to a router, assuming it does both local routing with wireless and ethernet, and ADSL or cable modem-ing. Is that right? But I'm sure I remember modems and routers used to be separate?

Passover vs Easter

See my twitter for much theological discussion, including:

* Eat lamb as passover/easter?
* Is it heretical to believe Jesus was crucified with three nails rather than four; or pierced on the reverse side of his body by the spear? http://en.wikipedia.org/wiki/Triclavianism
* My old "dummy's guide to passover" posts were useful! Except not, because they were entertaining but possible not a sufficiently accurate guide :)

How do I solve 2^{x} = x?

I keep seeing questions like this, that give a technically correct answer in terms of "rearrnage it like this, then that's so-and-so-'s function [see wikipedia], then you get an answer".

But seem to miss out what I think is the conceptually important information which is "you might expect that all equations can be solved in terms of simple functions you're already familiar with, ie. arithmetic functions, maybe simple trig functions, etc. But actually no, lots of functions can't be simplified into those. All you can do is find the name other people have used for 'the function which is the answer to this equation', and hence find out what they know about it. But that's not 'solved' in the way you were hoping."

Comments

[simont]

For most of the time I've had NTL/Virgin cable broadband, I've had a cable modem with no built-in router, and my own router running behind that which did my NAT. And of course then I installed lots of other handy stuff on my router (notably secnet), and hence I was committed to keeping it once Virgin sent me an updated cable modem a couple of years ago with built-in router and wifi – so I configure that device into "modem mode", meaning turn off everything clever and go back to just passing through the results of what my router has done.

I have a vague idea that ADSL routers have a longer history of being an all-in-one router-and-modem device, so perhaps the confusion of terms is more widespread in that area? But I would say that 'router' and 'modem' are still clearly distinct technical functions, even if colloquial usage is heavily informed by a lot of devices doing both at once.

[andrewducker]

Yup, same here.

I think the terms got confused when ADSL became popular.

[simont]

All you can do is find the name other people have used for 'the function which is the answer to this equation', and hence find out what they know about it. But that's not 'solved' in the way you were hoping.

Well, somewhere in between, I think. Tim Gowers put it well on an old web page: "What is solved when you solve an equation?" The real point is that you want to introduce as few extra operations/functions/whatever as you can, and be able to express the largest possible set of useful things in terms of those functions.

Hence, for example, it's clear that to solve even easy quadratics like x²=2 you need to introduce the square root function, which is just boringly defined as 'whatever the solution to that quadratic is' – but the interesting thing is that, having done that, you can now solve any quadratic, even trickier ones, by rearranging them into a simple quadratic and using your existing new function, and you don't have to invent further extra functions. Even more so for complex numbers – you start by postulating that x²=−1 has a root, and next thing you know, every polynomial has a root, and the interesting thing is how little you have to introduce and how much you can get out of it once you do.

Similarly, it sounds like a cheat that the Lambert W function (which I'm guessing is the one you're alluding to in the 2^x=x case) is simply something we've defined to be the solution to equations of the form xe^x=y. But, again, the interesting thing is that you only have to define that one function and then you can rearrange about ten assorted other kinds of equation in ways that let you express their solutions too in terms of the same function – you didn't have to also define entirely separate Lambert X, Y, Z, W', W'', Z₀ and ω functions in order to be able to talk about solutions to all those other kinds.

(Actually, I notice that your equation form y^x=x is not in my list there of things you can solve using W, and it should be. I should add it to the comment :-)

[jack]

Oh, thank you. Yes, all the things you said! :) I should have remembered/guessed Tim Gowers would have a much better explanation.

Router Vs. Modem

[dreamatdrew]

Sadly, it really depends on the tier you're talking about. Consumer grade hardware loves to combine lots of actually different devices in one box, at which point the labels get... fuzzy.
TECHNICALLY, something which combines router and modem functionality should be called a 'gateway device', but getting your average person to use the correct label is a bit like pissing up a rope.

Re: Router Vs. Modem

[lavendersparkle]

After getting muddled about it many times I now use the catch all term 'internet box'.

Re: Router Vs. Modem

[jack]

Yes, that's actually better :)

[damerell]

I don't think it's quite that. It's not that "router" came to mean "includes modem", but that modems started to include routers and became known as "routers" even though "routers" without modems in still existed as non-consumer-grade hardware, and now we don't know what "router" means unless it has something to be a modem for in front of it like "ADSL router".