Thread: .999~=1!

Originally Posted by Xei

As a matter of fact I find it totally intuitive that 0.999... is equal to 1, as the number is infinitely close to 1 and hence equal; I do not understand anybody who finds the opposite intuitive.

I might jump off a cliff after I open up this can of worms with you again, but how does the number get (not that it is moving) infinitely close to 1? Apparently it does, but how? Infinity goes on forever. At any 9, you can move to the next 9, and 500 quadrillion googol octillion 9's later. The 9's extend forever. It has no end, so how is it 1? It just sounds so much like talking about the end of space. 1 is a very exact number.

0.999... is also an 'exact' number (being the same number, 1).

how?

Infinity goes on forever.

That's the answer really. Each consequtive 9 decreases the difference between the number and 1 by a factor of 10. You do this infinite times and you've decreased the difference by a factor of infinity; i.e. there is 0 difference.

I think you're asking at what point leading up to infinity is it 0; there is no point. That's the whole point of infinity, it isn't a number as such. It just represents what would happen if you did it again and again and again forever. You can't in practice, but it's just a concept.

The above is all intuitive however. The real answer to your question is simply the mathematical proofs which have been offered. Mathematics is abstract. Think what would happen if you tried to discuss the Banach–Tarski paradox in this way; it doesn't work.

I sort of see what you are saying, but it's that concept of infinity that is so baffling. Like, if you traveled an infinite distance in one direction, you would go through all of space in that direction. That makes sense in a way, and theoretically it is true, but it is still a big WTF since space does not end.

I remember proving this in Algebra 2. Fun.
inb4lock

Originally Posted by PhilosopherStoned

Taking it like a man It's a very understandable cause by the way. Math is hard enough without the fairly bad job they do of teaching it in school making the situation worse. Our brain isn't designed to handle it. You really do have to learn to just drop your intuition and go with the axioms, theorems and definitions. Trying to understand why your intuition is wrong, when it is, is also great for developing it. Over time, it becomes a friend, not a foe. The fact that you and khh were able to understand the proofs is fairly huge. Most people wouldn't be open minded enough, honestly.

Hah, I'm trying to be. I can't say I'm not a little worried, though. I'm supposed
to be going into calculus II after not having had any involvement with serious
mathematics for the past two years. I guess I know who to go to if I need that
little bit of help now, aye?

Originally Posted by Universal Mind

I sort of see what you are saying, but it's that concept of infinity that is so baffling. Like, if you traveled an infinite distance in one direction, you would go through all of space in that direction. That makes sense in a way, and theoretically it is true, but it is still a big WTF since space does not end.

I went back and edited my proof. It's much clearer now, I think. I had been up for about 36 hours when I first wrote it. I think it's the clearest proof of it on this thread because it's a 'why' proof that actually explains why it is the way it is as opposed to relying on calculus like the OP's proof, a formula like Kromoh's proof or algebraic trickery like xei's proof. xei's proof is the most elegant though without a doubt. My proof is essentially along the same lines as xei's informal argument that almost has you convinced. Making it formal might help. Unfortunately, it's the longest proof on the thread, but that's almost always the way it is with explanatory proofs vs. clever proofs.

Originally Posted by Invader

Hah, I'm trying to be. I can't say I'm not a little worried, though. I'm supposed
to be going into calculus II after not having had any involvement with serious
mathematics for the past two years. I guess I know who to go to if I need that
little bit of help now, aye?

I'll happy to help and I'm sure a lot of other people will as well. It's funny about calc II because I made a B in that class. First, I didn't do my homework which counted for 5% of the grade. Then, I didn't show up for class and that counted for like 2% or something. Then I missed the date on a test and got a zero for it. He dropped the lowest test grade so I was going to be fine. Then I forgot some formulae for some curves on the analytic geometry test because I was stoned and hadn't slept much. I was able to derive some of them in the allotted time but not enough to get above a 70%. Averaged in with my other perfect scores, I made a high B.
I should point out that that was not the first time that I had studied that material and was already studying toplogy and algebra in my free time so it was, at that time, pretty elementary stuff for me. I had to work hard to understand the material the first time around.

How does it work in the U.S. anyway?

I sometimes get the feeling that we do less in Britain. Most people who do any proper maths do Maths A-level which is a qualification which takes up two years (you start when you're 17), but I did another one callled Further Maths... it basically takes you up to euler's identity.

But I talked to somebody from the U.S. at my Cambridge interview and he'd done stuff like harmonic series and convergence which we hadn't even touched upon.

I'm not the best person to ask. I dropped out of highschool as soon as I was legally allowed to and then tested into calculs I (They wouldn't let anybody test any higher than that) when I started community college a few years later.

I know that the average "good" student starts with calculus one when the first get to college. We cover convergence of series and sequences in calculus II. We don't do partial derivatives until calculus III. We do the jacobian in calculus three, but only the determinant of it for change of variables in multiple integrals. We cover lagrange multipliers and a week form of stokes theorem that only relates the interior of subsets of R^3 to it's boundary.

After calculus II you can take linear algebra and differential equations. I think the only reason they make you wait till after calculus II for linear algebra is to weed people out. Logically speaking, I think linear algebra should come first because the language that you learn in it does a lot of good for calculus. If I had my way, we would skip single variable calculus all together and then go straight to multivariable calculus after linear algebra.

What age is this..?

I think 'college' means quite different things to us. In the UK it means a couple of years when you're 16 to 18 getting qualifications, usually to go onto university.

College is 18-22 generally. 16-18 is high school.

when you studied calculus, did you do the epsilon-delta formulation of limits, or did they just wave their hands over them? It was just hand-waving in the US

I'm 18 so basically I've just finished what you would call High School. University starts in October.

We've hardly done any limits at all. I can tell you what GPs converge to, I can tell you that harmonic series don't converge, and I can do some trigonometric series using complex numbers. Think that's it.

Originally Posted by PhilosopherStoned

I went back and edited my proof. It's much clearer now, I think. I had been up for about 36 hours when I first wrote it. I think it's the clearest proof of it on this thread because it's a 'why' proof that actually explains why it is the way it is as opposed to relying on calculus like the OP's proof, a formula like Kromoh's proof or algebraic trickery like xei's proof. xei's proof is the most elegant though without a doubt. My proof is essentially along the same lines as xei's informal argument that almost has you convinced. Making it formal might help. Unfortunately, it's the longest proof on the thread, but that's almost always the way it is with explanatory proofs vs. clever proofs.

Oh, I'm already convinced. I just still see paradox in it. I don't question the truth of it any more.

Neuroscience sucks. Xei do pure maths not applied maths.

I wouldn't say Further maths takes you up to euler identity. As hyperbolic functions is more advanced than euler identitiy. Sadly, I got a handout that had some basic complex analysis i.e. how to use complex numbers to integrate functions.

Xei results day is in like 14 days and some hours. I'm really nerous but I really shouldn't because I'm pretty sure I will get AAA. Unless, something crazy happens. Good luck Xei.

Yeah we had to do complex calculus although normally it was quicker to do it the simpler way. tbh hyperbolics in FM are very basic, it's essentially just the trig identities all over again...

Good luck to you too. I think I got 1,2 or lower in STEP (missed my offer) and AAAAA or AAAAB with the B in Biology. I'd say it's a toss up now if I get into ICL or Cambridge.