A Billion.

[Petra]

I always thought that a billion was a million million, but it seems that that standard now is a thousand million.


Anyway....


I read a snippet that put "a billion" into perspective:

The next time you hear a politician use the word "billion," casually, think about whether you want the politician spending your tax money.

A billion is a difficult number to comprehend, but one advertising agency did a good job of putting that figure into perspective in one of its releases.

a.. A billion seconds ago it was 1959.
b.. A billion minutes ago Jesus was alive.
c.. A billion hours ago our ancestors were living in the Stone Age.
d.. A billion days ago no-one walked on two feet on earth.
e.. A billion dollars ago was only 8 hours and 20 minutes, at the rate our government spends it.

Snopes naturally tweaks it in order to bring it up to date and check it for truth and accuracy. (http://www.snopes.com/inboxer/trivia/billions.asp)

[JoeP]

Hi Luna. When the British invented the billion, it was twice as many zeroes as a million: 1,000,000,000,000. And a trillion, three times as many: 1,000,000,000,000,000,000. Mono=one, bi=two, tri=three etc, each a count of groups of 6 zeroes.

Then the Americans were born. They didn't understand this, but they did understand that having to say "a thousand million" a whole bunch of times was a pain, so they just called that number a billion. And ease of use took over the world.

That friggin wikipedia bullshit (http://en.wikipedia.org/wiki/Billion) blames "a minority of Italian and French scientists" but that's nonsense. It was obviously America.

Aside: American and British English differences (http://en.wikipedia.org/wiki/American_and_British_English_differences) makes fascinating reading.

[Petra]

Hi Joe. :kiss:


Your links, especially the second one, made me book some 'residential' therapy. I'll return as soon after the operation :brains: as is possible.

:pancakebunny:

[justaman]

That friggin wikipedia bullshit (http://en.wikipedia.org/wiki/Billion)...
Don't diss the wikid. :nojustno:

[JoeP]

That friggin wikipedia bullshit (http://en.wikipedia.org/wiki/Billion)...
Don't diss the wikid. :nojustno:
Sorry! I sometimes assume that everyone remembers every post ever made on FF. "friggin wikipedia bullshit" is a phrase used by a certain other member not that long ago, in jest I hope, and I have this urge to quote it from time to time.

I love wikipedia.

[justaman]

I've seen that quote, and I don't like that individual because of it. He's a bastard, do you understand?

All hail the wikid. :worship: :worship: :worship:

[JoeP]

I prefer to assume he was angry at the time. Like I'm angry with you for that row of oven-mitt-waving smilies.

[Shake]

I dunno, but I'd pay a billion millionths of a cent to see a larger version of luna's avatar!

[MooseIBe]

It's nuns sitting on bar stools that have legs painted on :). I've seen that before somewhere but can't remember where.

I dunno if OFFICIALLY we have switched to the American way of saying billion in this country, but unofficially we certainly have.

[viscousmemories]

It's nuns sitting on bar stools that have legs painted on :).
Ohhh... it's obvious now that you mention it. That's pretty funny. :yup:

[Skep]

Carl Sagan invented the word "billion". :whup:

[Dingfod]

Billion and billions of times.

[Shake]

It's nuns sitting on bar stools that have legs painted on :). I've seen that before somewhere but can't remember where.
Ahh, thanks, Moosey!

Now that you mention it, I think I've seen it somewhere before, too. Hmmm. :chin:

[JoeP]

Carl Sagan invented the word "billion". :whup:
Only a new pronunciation of it.

[JoeP]

It's nuns sitting on bar stools that have legs painted on :). I've seen that before somewhere but can't remember where.

When liv raised the limit on search results recently, I came across one of my earliest posts. I don't know what luna's av was then but my comment is strangely appropriate to the current one:

http://img.photobucket.com/albums/v437/SecretCode/lunamoonav.gif

[viscousmemories]

:laugh:

[livius drusus]

:glare: legacy postbit :glare:

[Skep]

:glare: legacy postbit :glare:
:tease:

[John Carter]

To our senses, and in the physical world, a billion (either version) is a significantly large numner. But as a Natural Number, it's pretty small potatoes. After all, you can add as many zeroes as you like. Think how big a 1 with a billion zeroes after it is! And even that is dwarfed by other numbers, since you can tack on that many zeroes again to make a truly impressive number. But what is really cool, and unimaginably huge, are some of the surreal numbers. There are surreal numbers that are so big they are larger than any natural number, and are thus infinitely large!

[Ensign Steve]

There are surreal numbers that are so big they are larger than any natural number, and are thus infinitely large!

There are? Like what? You mean like 1/0 or something cooler?

[Ymir's blood]

When liv raised the limit on search results recently, I came across one of my earliest posts. I don't know what luna's av was then but my comment is strangely appropriate to the current one:
I'm pretty sure it was an image of a woman's back, stopping just above the ass.

[John Carter]

There are surreal numbers that are so big they are larger than any natural number, and are thus infinitely large!

There are? Like what? You mean like 1/0 or something cooler?

Much cooler. 1/0 is simply undefined, as is x/0 for any nonzero x; there is no such number (0/0 is a different story, it's not undefined, but indeterminate). It's not infinite, even though some physicists are sloppy enough with their math to call it so.

[Ensign Steve]

Much cooler.

Do tell. :popcorn:

[John Carter]

I'm not sure how to answer that. How much do you know about how the real numbers are constructed? First, you develop the rationals (all numbers that can be written as a ratio of two integers, i.e. fractions), whcih is fairly straightforward. Then, you use these to generate the reals. One common methd is by looking at all Cauchy sequences of rationals. The other, and historically the first rigorous construction of the reals, is by using what are called Dedkind Cuts, which involvews partitioning the set of rational numbers.

We use a method similar to the second, i.e. creating a partition of existing numbers, to create the surreals. The construction rule is: If L and R are sets of surreal numbers, and no member of R is less than or equal to any member of R, then {L|R} is also a surreal number. It's a recursuive definition, so we generate more numbers with each step using induction. We start with 0, as a partition of the empty set {{}|{}}. {0|{}} is 1, and -1 is {{}|0} and so on. If we confine ourselves to regular old finite induction, the set of numbers we can generate in this manner is limited, as we can only consider finite sets. We start by inductively defining Sn as follows:

S0 = {0}
Si+1 is Si along with all of the surreals that can be generated from subsets of Si.

What we get is this:

S0 = {0}
S1 = {-1, 0, 1}
S2 = {-2, -1, -1/2, 0, 1/2, 1, 2}
S3 = {-3, -2, -1 1/2, -1, -3/4, -1/2, -1/4, 0, 1/4, 1/2, 3/4, 1, 1 1/2, 2, 3}

The only numbers that can be generated are of the form a/2b where a, b are integers and b > 0.


Next, we extend this induction to what is known transfinite induction, and now we can use infinite sets and things get really interesting (and we can generate all of the reals, not just the special rationals mentioned above). For example, s = {{0}|{... 1/16 ,1/8, 1/4, 1/2, 1}} is a number such that 0 < s < x for all real numbers x, and therefore 1/s is larger than any real number x.

[Ensign Steve]

The construction rule is: If L and R are sets of surreal numbers, and no member of R is less than or equal to any member of R, then {L|R} is also a surreal number.

Did you mean to say "no member of L is less than or equal to any member of R"? If so, then YAY I am asking intelligent questions. If not, then I am in way over my head.

Thanks for the explanation. I am still trying to wrap my brain around it. Stand by.

[Ensign Steve]

If it gives you any indication of how interested I am in this, I have spent the last hour exhausting my math friend via IM asking for help with this. I am aware that I lack a necessary fundamental understanding of induction, and that is something I have promised myself I will work on. In the mean time:

Is there any way you can define (even vaguely) surreal numbers, without appealing to them? My math friend admonishes me that induction is circular by nature, but even just a little bone (along the lines of "an imaginary number is the square root of a netagive number") would be happily accepted.

If it bores you to explain it to me as though I were a child, I totally understand. But it interests me to no end, so know that your efforts are appreciated. :bow:

[John Carter]

The construction rule is: If L and R are sets of surreal numbers, and no member of R is less than or equal to any member of R, then {L|R} is also a surreal number.

Did you mean to say "no member of L is less than or equal to any member of R"? If so, then YAY I am asking intelligent questions. If not, then I am in way over my head.

Thanks for the explanation. I am still trying to wrap my brain around it. Stand by.


Ummm... damn. Sorry about the typo. This stuff can be confusing enough without mistakes like that! No, I meant to say no member of R is less than or equal to any member of L. IOW, every member of R is strictly larger than every member of L.

[Ensign Steve]

Sorry, I didn't need to point out the boo-boo when it became obvious in context what you meant. But maybe for the sake of that lurker ...

Anywho (in case we crossposted) scroll up and you will find a slightly less stupid (I hope) question.

[John Carter]

Sorry, I didn't need to point out the boo-boo when it became obvious in context what you meant. But maybe for the sake of that lurker ...


Don't apologize. I'm glad you pointed it out.

If it gives you any indication of how interested I am in this, I have spent the last hour exhausting my math friend via IM asking for help with this. I am aware that I lack a necessary fundamental understanding of induction, and that is something I have promised myself I will work on. In the mean time:


Good. I was afraid I'd scared you and everyone else off.


Is there any way you can define (even vaguely) surreal numbers, without appealing to them? My math friend admonishes me that induction is circular by nature, but even just a little bone (along the lines of "an imaginary number is the square root of a netagive number") would be happily accepted.


Unfortunately, no, there isn't. I realize it seems circular, but keep in mind that even though the definition says "two sets of surreal numbers..." the empty set is specifically allowed, and thus we begin with (and define our first surreal number in terms of) the empty set on both sides to get 0, and we can combine 0 with the empty set to get 1, and then we're off to the races.

IOW, it's an iterative process. You define the first surreal in terms of the empty set, and then define the next numbers in terms of what has been previously defined.


If it bores you to explain it to me as though I were a child, I totally understand. But it interests me to no end, so know that your efforts are appreciated. :bow:

Not at all. It's hard to know how much detail to give. I've left out a lot of important points, such as equivalence classes, in order make things simpler.

[Ensign Steve]

Unfortunately, no, there isn't. I realize it seems circular, but keep in mind that even though the definition says "two sets of surreal numbers..." the empty set is specifically allowed, and thus we begin with (and define our first surreal number in terms of) the empty set on both sides to get 0, and we can combine 0 with the empty set to get 1, and then we're off to the races.

IOW, it's an iterative process. You define the first surreal in terms of the empty set, and then define the next numbers in terms of what has been previously defined.

Okay, I will take that nugget and run with it! My IM friend has promised me a lesson in induction, but he's got that crappy "work" thing to contend with right now, so I may try and wiki it in the meantime. If I am successful, I will return with better questions!

[Ensign Steve]

Wow, it was painful and ugly, but I have been properly schooled on equivalence classes. Or at least why X/~ is the set of all possible equivalences, to put it crudely. Not sure I'm any closer to "getting" surreal numbers, but it was fun anyway. Isn't there a quote in the random generator along the lines of, "Math is like sex. Sure there are practical results, but that's not why we do it."? Anywho, off to wiki induction now. :)

[Ensign Steve]

Okay, okay, I think I'm onto something.

I totally understand why (for example) S? = { 0 | ..., 1/16, 1/8, 1/4, 1/2, 1 } is infinitesimally small, so I guess the reciprocal of that would yield an infinitely large number. Is that what you meant by this:
For example, s = {{0}|{... 1/16 ,1/8, 1/4, 1/2, 1}} is a number such that 0 < s < x for all real numbers x, and therefore 1/s is larger than any real number x.

ETA: oops, I don't know how to make the symbol that is supposed to go where the ? is, but it looks like a sideways trident, or the letter E.

Edit: Edit in progress, please stand by.

ETA: Okay, hell, I don't know what that sideways trident is supposed to represent, anyway. I was thinking at first when I read it on wiki, that it was that natural logarithm constant "e", but at the time that I read it and it made sense to me, I was thinking it's value was somewhere between zero and one. But would my above thingie still hold, for any value ? such that 0 < ? < 1

Or am I still out in left field?

(intentionally leaving punctuation off of that second-to-last interrogative sentence)

[John Carter]

Okay, okay, I think I'm onto something.

I totally understand why (for example) S? = { 0 | ..., 1/16, 1/8, 1/4, 1/2, 1 } is infinitesimally small, so I guess the reciprocal of that would yield an infinitely large number. Is that what you meant by this:
For example, s = {{0}|{... 1/16 ,1/8, 1/4, 1/2, 1}} is a number such that 0 < s < x for all real numbers x, and therefore 1/s is larger than any real number x.

Yup.


ETA: oops, I don't know how to make the symbol that is supposed to go where the ? is, but it looks like a sideways trident, or the letter E.

Edit: Edit in progress, please stand by.

ETA: Okay, hell, I don't know what that sideways trident is supposed to represent, anyway. I was thinking at first when I read it on wiki, that it was that natural logarithm constant "e", but at the time that I read it and it made sense to me, I was thinking it's value was somewhere between zero and one. But would my above thingie still hold, for any value ? such that 0 < ? < 1

Or am I still out in left field?

(intentionally leaving punctuation off of that second-to-last interrogative sentence)

Sounds like you are describing a greek letter: lower case epsilon. This is often used to represent an arbitrarily small number, so yes, you were probably on the right track.

[Ensign Steve]

:woohoo: On the right track, baby!

Thanks for the excellent discussion, JC! I feel like I have grown as a person today. :yup:

[JoeP]

:glare: legacy postbit :glare:
I'm just testing the option! What does it do anyway?