What the Tortoise Said to Achilles

[Nev]

This is AWESOME. Especially if you were ever a "mathlete" or similar.

http://www.lewiscarroll.org/achilles.html

I haven't even read the whole thing yet, but I remember in calculus class thinking it was somewhat neat that an infinite sum can nevertheless converge to a noninfinite number. I can't really write the math notation here...unless...

∞
<font size=+2>Σ</font><font size=+1> ½n</font>
n=1

Hope that works! Those of you/us who took good math courses in high school will probably know that this is 1. For those who can't read sum notation, it's the sum of all numbers in the sequence that starts at one-half and cuts itself in half every time - i.e. one-half, one-fourth, one-eighth, one-sixteenth, and so on.

The link I posted starts off with a verbal description of this that's quite amusing.

BTW, Sine and anyone else, I'll get to the drama thread soon enough...no reason to spoil my morning quite yet...

EDIT: woohoo! it works!

[Kupek]

I don't know if you meant

• ∞
  <font size=+2>Σ</font><font size=+1> (1/2)n</font>
  n=1

or

• ∞
  <font size=+2>Σ</font><font size=+1> 1/(2n)</font>
  n=1

The top series does not converge to anything, so I don't think that's what you meant. The bottom series isn't quite what you described. It expands to

• <font size=+1>1/2 + 1/4 + 1/6 + 1/8 + 1/10 + 1/12 + 1/14 + ...</font>

If you want the series

• <font size=+1>1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + ...</font>

Then in sum notation, it's

• ∞
  <font size=+2>Σ</font><font size=+1> 1/(2ⁿ)</font>
  n=1

[Nev]

Errr, yes, you're quite right. (ahem)

I think the one I posted would end up being infinte after all.

(Mental turns slightly pink)

Baaaaaa! (sheepish)

Thanks for the correction. I still think the thing I linked to is cool though.

[SineSwiper]

I'm always amused at the entire concept of Calculus. When it comes down to it, the entire field exists because we're too stupid to solve X/0. You graph it out, and you KNOW what that hole in the graph is supposed to be, but we're just dumb enough to not know what it is, and we dance around this hole to come up with the answer, or "what is it coming close to" without directly answering the question.

I remember writing up an HS essay on trying to prove that our number system is looped, using stuff like y=tan(x) as proof. It had a circular number line with 0/inf on the left/right and 1/-1 on the top/bottom. (Like zero, inf was signless.) It's impossible to actually "prove" mathematically, even though you can look at it logically and say "look, it's RIGHT THERE". I had some rules on the properties of infinity (as an actual number, instead of a concept), though I've already forgotten them. However, it did help me figure out if a calculus problem was impossible to solve or not.

Anyway, there was another weird avenue similar to that, involving the sqrt of -1 times pi equals log(-1) or something like that. (E to the iy equals pi?) I always wanted to tackle that as another project. Oh, yeah: e to the i*pi = -1. It's weird because if you solve for i, you actually get the answer to "what is the square root of a negative number?"

It would seem that the "imaginary number" system is not exactly imaginary at all, but another numeric system attached to ours. It seems like when we teach the 2D numeric system, it limits our ability to solve all of the problems, because we keep running into this holes where we go "Oh, well, we can't actually solve that". It's not that the answer doesn't exist or isn't possible, we're just using the wrong number system entirely.

Think about when we didn't have zero. One minus one didn't actually mean anything. When we got zero, we didn't actually mull over mathematical proofs to go "okay, now we can use this as an actual number". We just said fuck it, and stole it from the Arabians. The concept made sense, so we didn't actually argue about it. It seems like we don't do that any more, and try to put these problems on another plane of existance, instead of acknowledging that it's still IN our numeric system, but we have to intrepret our entire numeric system differently. (Like saying "the square root of -1 actually goes sideways on our number 'line'".)

[Nev]

I dunno whether to experience a pang of nostalgia at seeing one of Sine's old, weird, bizarre pseudoscientific posts or try to correct what I see as (again) bizarre and nearly incomprehensible oversimplification and/or misunderstanding.

Sine, calculus is a Good Thing, at least in my book. I really don't see what derivatives/integrals (which I consider the primary features of calculus) have to do with ∞ (in the conventional real number system), other than the fact that you can definitely GET infinity as an answer to doing actual computations of derivatives or integrals on certain functions with certain ranges of values...there are a LOT of things in modern science which without the concepts of calculus would be completely buttfuck impossible - I'm pretty sure that most of quantum physics falls into this category...and as an aside, I don't think you can simplify the similarity between positive and negative ∞ in the conventional real number system down to "real numbers are a loop" or anything close to it. If that were true, the sign on ∞ would be irrelevant in all equations, which it isn't...I'm trying to think of a counterexample offhand, but nothing comes easily to mind...

I have to say there's something a little bit comforting about seeing Sine go off again, though...I didn't know I'd particularly missed seeing bizarre political statements (the Arabs and 0 - I do know the concept of 0 was first codified by Arabian mathematicians, but the rest of what Sine said seems spectacularly dubious in its history) mixed up in something like equally bizarre mathematical statements...

[Nev]

I'm wrong! - at least, if the Wikipedia is right.

"The late Olmec had already begun to use a true zero (a shell glyph) several centuries before Ptolemy in the New World (possibly by the fourth century BC but certainly by 40 BC), which became an integral part of Maya numerals.

By 130, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero is the earliest known documented use of zero as a number in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70).

Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol. When division produced zero as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.

The earliest known decimal digit zero is documented as having been introduced by Indian mathematicians about 300.

An early documented use of the zero by Brahmagupta dates to 628. He treated zero as a number and discussed operations involving this number. By this time (7th century) the concept had clearly reached Cambodia, and documentation shows the idea later spreading to China and the Islamic world, from where it is recorded to have reached Europe in the 12th century."

[SineSwiper]

Well, think about it. Every problem you ever had in calculus couldn't actually be solved normally. If you did, you'd ended up with a division by zero. You graph the function, and either there would be a hole in the graph or it would reach infinity.

See X/0 = infinity, at least in the purely logical sense. Just do it in your calculator. Not the exact problem "1/0", but try "1/0.0000000001", and you get a very large number. Thus, it's logical to deduce that if you divide by nothing, you get infinity. And if you try "1/-0.0000000001", you get a large NEGATIVE number. Therefore, infinity is connected to both sides of the number line, and thus, the number line is actually looped. You can clearly see this in a y = tan(x) graph. Why nobody acknowledges this simple concept is beyond me.

This is why calculus is directly related to infinity. All problems involving it are trying to get around X/0, or infinity. Now I'm starting to remember the rules. See, 0/0 is basically two forces pulling each other apart. Think of zero and infinity as gravitational forces. 0/0 is basically zero times infinity, which cancels each other out to every single number on the number system (think Infinate Improbability), or what is officially called an "indeterminate form". THIS is where calculus comes in. You need to find out WHICH number it is. Now, on problems where you get something other than zero over zero, like 5/0, then the solution really is infinity, and you don't even need to bother trying to derive it.

I'm sure this is completely confusing you. It's hard to conceptualize without a few graphs to detail the ideas. This isn't really me just ranting random nonsense, but it's somewhat hard to convey. I'll have to dig out that essay, because I had a few graphs and a picture of the number loop.

Anyway, what I was talking about with zero was the fact that we didn't try to figure out a way to PROVE zero existed. In order for something to enter mathematics, we have to write some huge and long mathematical proof to prove that it is real and a part of our numeric system. You can't mathematically PROVE infinity (as a number) just like you can't mathematically PROVE zero, if zero wasn't in the number system currently.

The best you can do is just logically explain it and then change the number system entirely. We did this with fractions. We did this with zero. We did this with negative numbers. We did this with irrationals. And then, we stopped doing this because we thought that our numeric system was "complete". But, it's never complete because we learn more stuff beyond that.

Now, everything we know is forced around the mathematical proofs to our unchanging number system. Instead of trying to change the number system itself, we invent ways to go AROUND the problems we can't solve and put it inside our own system. Yeah, sure, we could solve some problems within our own real number system like that, but we aren't really exploring any new terrority, ie. we still don't know what the square root of -1 really is. (And i doesn't count...it's just a placeholder.)

[SineSwiper]

Okay, back to that e to the i times pi deal. Yes, this connects to what I'm saying because it's yet another illustration of mathematics dancing around what it doesn't know to solve problems. Somehow, some way, this guy named Euler invented this formula with some interesting properties:

e^iPI = -1

Okay, let's solve for i:

ln e^iPI = ln -1

At this point, you realize that ln -1 is also "unsolvable", so you have to create another placeholder for that (like i is a placeholder for sqrt -1):

ln e^iPI = j
iPI = j
i = j / PI = (ln -1) / PI

So, now you have alternate solutions to i. You know that the square root of -1 is equal to the natural log of -1 divided by pi. You also know that visa-versa is true, ie. the natural log of -1 is equal to pi times the square root of -1. But, what about pi? Why is a real number part of a solution to an "imaginary" number? Consider each of the "solve for" equations:

i = j / PI
j = i * PI
PI = j / i

So, you divide two imaginary numbers and get a real one? Maybe it's because the numbers aren't "imaginary" to begin with. It's just that when you try to do sqrt(-1) or ln(-1), they "fall off our radar", so to speak. They are still real numbers, but our old and ancient numeric system can't visually see it. And then we cross two imaginary numbers, and it blimps back into our real number system. (This isn't the only case: i^i equals a real number, too.)

It seems like the numeric system is not a number line, but a sphere. Imaginary numbers are the z-axis. They aren't "imaginary" at all, but a real value, just like infinity is an actual number.

[Kupek]

I'm always amused at the entire concept of Calculus. When it comes down to it, the entire field exists because we're too stupid to solve X/0. You graph it out, and you KNOW what that hole in the graph is supposed to be, but we're just dumb enough to not know what it is, and we dance around this hole to come up with the answer, or "what is it coming close to" without directly answering the question.

We went through this several years ago, and you still don't know what you're talking about. If you're really interested in the subject, pick up a calculus textbook or take a course.

[Andrew, Killer Bee]

When it comes down to it, the entire field exists because we're too stupid to solve X/0.

NOOOOOOOOO

[Nev]

I'm always amused at the entire concept of Calculus. When it comes down to it, the entire field exists because we're too stupid to solve X/0. You graph it out, and you KNOW what that hole in the graph is supposed to be, but we're just dumb enough to not know what it is, and we dance around this hole to come up with the answer, or "what is it coming close to" without directly answering the question.

We went through this several years ago, and you still don't know what you're talking about. If you're really interested in the subject, pick up a calculus textbook or take a course.

ROTFL. OMG that is funny.

[Don]

You don't get to just question fundamental things about math because you think it doesn't work right. i is defined as the square root of -1 and unless you've a good reason on why the square root of -1 shouldn't be i, that's what it is.

Though you're right about how Calculus exists to deal with infinities that you could just say 1/0 = infinity instead. But my understanding is that saying 1/0 = infinity gets you into trouble when you go deeper in mathematics so you can only say 1/x approaches infinity in the limit x goes to 0.

[Nev]

Yep. 1/0 isn't really infinity, I would say...it's just an idea that doesn't make much sense. How many nothings fit into something? You could argue that you have an infinite amount of nothing in anything, but I think the better way to define it is just as a silly concept (formally, "undefined").

However, 1/x - as x gets closer and closer to 0 - will get larger and larger without bound (in other words, "infinity"), as Don said.

[Ishamael]

Im always amused at the entire concept of Calculus. When it comes down to it, the entire field exists because we're too stupid to solve X/0. You graph it out, and you KNOW what that hole in the graph is supposed to be, but we're just dumb enough to not know what it is, and we dance around this hole to come up with the answer, or "what is it coming close to" without directly answering the question.

We use the concept of infinity because it's not something that can be used in calculations. Everything approaching infinity can be used in calculations just fine though.

[Ishamael]

Now, everything we know is forced around the mathematical proofs to our unchanging number system. Instead of trying to change the number system itself, we invent ways to go AROUND the problems we can't solve and put it inside our own system. Yeah, sure, we could solve some problems within our own real number system like that, but we aren't really exploring any new terrority, ie. we still don't know what the square root of -1 really is. (And i doesn't count...it's just a placeholder.)

Well...that's kinda right. We don't know what -1 "really is". At least we can't describe it with the real number system since, as we all know, the square root is the inverse of squaring which can't result in a negative number (i.e., there's no number you can square to get -1). So somebody hammered on the concept of imaginary numbers to get around this. Sometimes you have to improvise.

So imaginary numbers are a notational convenience to describe something that can't exist on the real number line....or that's the best I can describe it.

However, I do wonder what the history of "i" is. Never did learn that.

[SineSwiper]

We went through this several years ago, and you still don't know what you're talking about. If you're really interested in the subject, pick up a calculus textbook or take a course.

I've already taken calculus before, or else I wouldn't even know about what the concepts are. (It's been ages since I've taken a derivative or integral, but I know what they do.) I understand complex numbers and why both complex numbers and calculus are important to a lot of fields. I'm not saying that they are useless, but that the way they are represented should be changed.

You don't get to just question fundamental things about math because you think it doesn't work right. i is defined as the square root of -1 and unless you've a good reason on why the square root of -1 shouldn't be i, that's what it is.

Again, i is just a placeholder. All we are doing is multiplying or dividing negative square roots until we get sqrt(-1) and then replacing that with i. We are entering another number line/sphere/whatever here, or a sideways angle into mathematics.

We could create a number line based on multiples of i, but what about j, which seems to be located in the same realm as i. E is equal to an actual number on the number line, and so is pi. We know the real-number difference between the two. But, with i and j, there is no number line to place them at. It would seem just as valid to make the imaginary number system based on multiples of j as it would multiples of i, but really, they should be based on how they connect to real numbers.

We have found the quark of mathematics and we refuse to change the laws to accommodate it. Think about what happened when somebody discovered the quark:

S1: "Hey, check this out! I just found this subparticle that is actually going faster than the speed of light!"
S2: "What?! That's impossible! It goes against the laws of physics!"
S1: "Just LOOK AT IT! It's breaking the laws of physics!"
S2: "Uhhh...errr...hmmm...shit! We'll need to change the laws of physics to work with this subparticle."

I understand why you can't take a square root of a negative number. Yet, we have problems that do just that, and they either end up as a real number, or a number into this "imaginary" realm. You have real problems, real numbers, bending the laws of mathematics. It's a mistake to have problems that "cannot be an answer", because you'll run across a situation where it will crop up. For another example, what about before we had fractions:

S: "What is 5 divided by 2?"
T: "Somewhere between 2 and 3."
S: "But, what is the actual number? What is it exactly?"
T: "It's not an actual number. The answer doesn't exist."
S: "So, if I had 5 apples to give to 2 friends, how would I be able to give an equal amount to both?"
T: "Well, you could give 2 apples to each, and split one in half for each of them."
S: "So, 2 apples and one half. Isn't that some sort of number?"
T: "No."

(Honestly, I don't know if we have fractions before/after we had division, but you get the idea.) Again, we need to change the system so that we can map out the "imaginary number system" and connect them to the real number system, because they do actually connect. If the real number root of a real number equals an imaginary number, then there is a connection. (You know, I've never tried to look at a y = x root of -1 graph before.)

Though you're right about how Calculus exists to deal with infinities that you could just say 1/0 = infinity instead. But my understanding is that saying 1/0 = infinity gets you into trouble when you go deeper in mathematics so you can only say 1/x approaches infinity in the limit x goes to 0.

I'd be curious to know where in mathematics that would get you into trouble. Granted, rules with infinity would require exceptions with zero, because instead of one absolute in our number system, we would have two, and when they collide, it becomes an indeterminate form. But, every other occasion where a non-zero number is multipled by infinity, it becomes infinity.

Infinity + 1 equals a very large negative number. The actual value of that number is currently immeasurable, but it still exists. If infinity were on a finite number system, we could measure it just fine. (For example, this happens all the time on computer systems. You add + 1 to the largest positive number, and it becomes the largest negative number. Though, on computers, there is no concept of the opposite of zero, so it doesn't actually hit infinity before hitting the other end of the spectrum.)