h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

hi

in your post (276 in the 'Are We Alone in the Universe thread') you put up some heavy math (for me ) so i came over to your intro to see if you might have some math info there i could use to understand what you said... and i noticed this: (2^0+8)x(4+6-4)=42

dont you mean [-(2^0)+8] x (4+6-4) = 42 ? or am i wrong i could be wrong i am more frequently, than not, wrong.

btw i am reviewing my math stuff so i can understand what you said... its so interesting!

The relation 6x9=42 is allowed because the Hitchhiker's Guide to the Galaxy says so It is mentioned in A530560.

Heavy maths? I don't know if they are heavy or if I was just not very clear (mostly, I couldn't do a picture ). Was it the bit about the open half-line being "equivalent" (in some sense) to the full line?

Eer, do you know about logarithms or graphs of functions? That might help, but by no means would it be compulsory. I don't have much time just now but I can come back to it this afternoon. If you post before I do, try to pinpoint what was troubling you.

aha!! you see, i was right!! about being almost usually wrong!! hehe

oh i just need to review my sliderule stuff tho i didnt really ever use one... and the logarithms and graphs of functions maybe draw it out for myself.. thanks for the words for what to look up i sensed that you were very clear... i agree we need pictures.

i did pretty good in Calculus but that was about 15 years ago. this will be fun thanks

i am 10 hours behind you and probably will be asleep when you get back but we can hook up some other time

Ten hours? That must be at least the West Coast of the United States, then?

Good night

Calculus? That's fine. Let's try some explanation first. The logarithm induces a continuous bijection (one-to-one correspondence) between the open half-line t>0 and the real line. Let's explain a bit.

Open half-line: just a half-line, i.e. whatever is one one side of a given point ("t=0"), but excluding the point. I called it "t>0" because when you do calculus, you often have two axes, the horizontal one being called "the x-axis" or "the t-axis", because physicists like to have time (t) as a parameter. "t>0" just means I take all positive parameters, but not 0.

Real line: just a line, extending indefinitely to the left and to the right.

Bijection: invertible transformation. You can go from the half-line to the full real line with logarithm, you can go back with the exponential. Let's not be too formal about this.

Continuous: if you take two close points to start with, they end up close too.

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As for time after Big Bang, the idea is that the closer you are to BB, the more difficult it is to understand what happens in the universe. Say that t=1 is one second after BB. It would require as much effort understanding what is going on between t=1 and t=0.1 than between t=0.1 and t=0.01, than between t=0.01 and t=0.001, etc. The time span between these little intervals do not grow linearly (i.e. adding the same amount of time at every step - like in t=1, t=2, t=3, t=4, ...) but exponentially (i.e. *multiplying* by the same amount at every step - like in 1,2,4,8,16... or 1, 0.1, 0.01, 0.001, ...).

But since we don't really like (especially when we draw pictures) having intervals which get smaller and smaller, we use a logarithmic scale: what happens between 1 and 0.1 is drawn as if it happened between 0 and -1, between 0.1 and 0.01 as if it was -1 and -2, 0.000..01 with 42 zero's and 0.000...01 with 43 zero's as if it was -42 and -43, etc.

Every tiny wee little interval near t=0 is represented by a length one interval, but since they all have the same length you cannot cram them all to the right of some point - the closer to t=0 you go, the more towards minus infinity your logarithmic representation will go. Of course you should never "reach" t=0, because you won't "reach" minus infinity either.

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Another way to visualise the open half-line as equivalent to the real line is via the graph of the logarithm (or various other functions, such that x->1/x - let's not bother about that).

You start with your x and y axes (or t and y, but when there is a y axis we like the other one to be called x). Draw the graph of the function y=log(x): you get a big bended line L: lo and behold you have constructed a continuous bijection between the half-line x>0 and the bended line L.

Indeed, every point x>0 corresponds to the point of L lying just above it (of coordinates (x,y) = (x,log(x)) in the plane, actually); and you can go back from L to the half-line x>0 by projecting vertically on the x-axis.

But now, the bended line L does resemble a straight line - all you have to do is unbend it. So it seems reasonable that L is equivalent to the real line - thus, the half-line x>0 is equivalent to L is equivalent to the real line.

If you don't like bending things, you can imagine someone grabbing the half-line x>0 around the origin and pulling it indefinitely to the left (a bit like chewing gum).

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In all this constructions it is important to exclude the origin of the half-line we start with. Otherwise, this means that it is a space on which you can go in one direction (decreasing t's) long enough until you meet a wall (t=0) - then you cannot move any further.

If you allow only t>0, then you can move along in the direction of decreasing t's and never meet a wall - you can always move a little bit further. Even if you are at t=0.000...01 with 42 zero's, you can still make another step (a very small one I grant you) and reach t=0.00...01 with 43 zero's, a bit closer to the left.

In the real line, it is obvious that no matter how far you go to the left there will always be some space to move further.

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There, enough for today. If you have other questions, feel free to ask

Also, you can go and check out my entry A1134091 - a fascinating subject I have to say.

wow!! very impressive! let me chew on all that for a while. So far, i can say that for the first time i see what you guys are visualizing when you say you can't get TO the actual zero event.

and i peeked in at The Topology of Two-dimensional Spaces and am really excited about reading it! i have always loved moibus strips. And how fascinating that the surface of an object is actually two-dimensional, not three-dimensional!! It is intuitive in Calculus but for some reason never came up, per se, or i was asleep, from doing the homework the night before, when it did!

btw: i have had all the introductory hard science courses (science major courses) ...except physics... organic and inorganic chem, botany and zoology, geology, meteorology, astronomy and the maths thru calc II ...and had a 3.5 gpa. i have been wanting to take that physics course lately.... soooooo maybe i will now. you have my interest piqued! you are a good teacher!


>>>Continuous: if you take two close points to start with, they end up close too.<<<

question #1. what do you mean by "they end up close too." ?

oh

question #2: is it possible that minus infinity could be time?

Question 1: what do I mean by "they end up close too." ?

This would be formally answered using limits, which I will try to avoid. Here, "close" would mean in a numerical sense, i.e. the difference is almost 0. On a picture (drawing graphs again), it means you get a continuous curve (no jumps).

It means that if you take an approximation y of a given number x (but y different than x), then log(y) should be a reasonable approximation of log(x) - and the closer y is to x, the closer log(y) is of log(x).

An example of discontinuous function: the sign function - f(t)=1 if t>0, f(t)=-1 if t<0 and, say, f(0)=0. The function is discontinuous at 0: you can see it because the graph jumps at 0. Also, if you take an approximation y of 0 - a very small number, like 0.001 - then f(y)=1 is far away from 0. Worse, no matter how close you choose y to be from 0 (and provided y itself is nonzero), f(y) will always be "far" from f(0), actually 1 unit away.

I could expand a bit on the importance of continuity if you wish, but that would lead to more topic drift

Question 2: is it possible that minus infinity could be time?

I'm not sure I understand this question.

Minus infinity should really be considered as a symbol to say "we allow going to the left as far as we please". It doesn't really describe (in any easy way, anyway) a moment or something concrete.

Have to now. Glad you like the explanations!