<p>The Least-Upper Bound axion states that a nonempty set F of real number that is bounded above has a supremum (least upper bound).</p>
<p>For your problem, I will just say S is that nonempty set of real numbers that is bounded below. S = {s }</p>
<p>-S would then be a nonempty set of real numbers that is bounded above. According the the least upper bound axion, every nonempty set of real number has a supremum; therefore, -S has a supremum. </p>
<p>This means that there is a number B such that -s < or = B and no number less than B is a upper bound. </p>
<p>Therefore, there is a number -B such that s > or = -B and no number greater than -B is a lower bound. </p>
<p>Becuase s is any element of S, we have shown that the original set (the one with the lower bound) has a real number, which I called -B which satisfy the definition of great lower bound. </p>
<p>Is that a satisfactory proof of the first problem?</p>
<p>^^ newmassdad, that doesn't completely surprise me, as I had a very strong math background coming in, given that I was familiar with a lot of proofs.</p>
<p>A lot of students who take calc in high school do place out of 131/151, though. Maybe 40, 50 percent?</p>
<p>
[quote]
re you saying that I can screw up a problem like</p>
<p>Find Int[xdx,0,1]</p>
<p>answer the theoretical questions correctly and still place into honors analysis? Because if so that's totally awesome. Not that I don't know the stuff, but the idea of me placing into Hon. Analysis or not is depends on whether I make a calculation error is not very soothing, because I am known t make many dumb mistakes. Perhaps that is why I like proofs so much: you either know or it you don't. There is no such thing as a careless error (for the most part) with proofs.
[/quote]
</p>
<p>Yeah, I'm the same way. People look at me like I'm stupid when I tell them I got a 670 on the SAT Math. For one, multiple choice is distracting and hard to pay attention to, and secondly, the calculation errors I make are immense... which is why I probably failed my complex analysis midterm. This is what I get for taking a class under a probability theorist (I kid, I kid).</p>
<p>HenryJames: My mind is distorted from studying for the Honors Analysis midterm tomorrow (memorizing proofs is loads of fun, trust me), so I'm not even going to try to look at that proof. I'm sure someone else here can help you (particularly dleet).</p>
<p>I've never seen numbers regarding the distribution among the various math courses, only the number who place out. And to condition the numbers based on HS calc would take some serious access to serious data.</p>
<p>I would not at all be surprised if the majority of kids who had calc in HS place into the 160 sequence, because if someone did not do well in HS calc, they're not likely to be admitted to Chicago! (a selection bias if you will). But what percent of kids heading to schools like Chicago take Calc in HS? No idea.</p>
<p>HenryJames: I belive this is incorrecct. However, let me just make sure I get what you're saying.</p>
<p>You're saying, take any subset S of R bounded below, Then, take an arbtirary element s in S and consdier the singleton set {s}. Then, you proceeded to show that we can find an inf for the set {s}. You called in -B. While I believe the proof was okay until here, when you claimed that because s was arbitrary, -B was an inf for S. I believe this is incorrect because your inf for the singeleton set {s} will NOT BE THE SAME for every singleton set {s}, and therefore it does not make sense to conclude that -B is the inf for S because the -B is not unique.</p>
<p>Please clarify if I misunderstood.</p>
<p>The way I constructed the proof was to consider two L,U such that R=L(union)U and for every l in L and every u in U, l<u. Then, the idea is to show that supL=infU. We know that supL exists (I just took this as an axiom), and from here we show that:
1.) supL is a lower bound for U
2.) every other lower bound for U is less than or equal to supL</p>
<p>As phuriku said, at first the proof seems misealdingly simple. In the end, I think it took about half a page single spaced, perhaps even a bit longer.</p>
<p>Phuriku,
S is taking Complex now and nailing it. I asked why this was so different than MV and DiffEq, and he said "It's all proofs! No stupid calculation errors!"</p>
<p>Yeah, Complex Analysis differs per university and per professor. If you have a probability theorist teaching, then you're more likely to have to do lots of integration and working with numbers (as in my case). If you have an analyst teaching, well, you probably won't encounter many numbers besides the occasionally 2i*pi. Unfortunately, we use Serge Lang's Complex Analysis (supposedly a graduate text) in the class, and it's rather messy. Our tests are based off of Lang's awful homework problems, and only about 1 in 5 questions involve a proof. I think the hardest thing I've done all quarter is prove that every odd Bernoulli number is zero.</p>
<p>Unlike real analysis (at the graduate level), complex analysis has a lot of applications in engineering and physics, so you have to cope with the fact that your teacher could be more on the applied side than pure.</p>
<p>The hubble prime lens was ground to the wrong curvature (wrong focal point) because the contractor made a simple mistake: (from wikipedia0
[quote]
Working backwards from images of point sources, astronomers determined that the conic constant of the mirror was −1.01324, instead of the intended −1.00230.[46] The same number was also derived by analyzing the null correctors (instruments which accurately measure the curvature of a polished surface) used by Perkin-Elmer to figure the mirror, as well as by analyzing interferograms obtained during ground testing of the mirror.</p>
<p>A commission headed by Lew Allen, director of the Jet Propulsion Laboratory, was established to determine how the error could have arisen. The Allen Commission found that the null corrector used by Perkin-Elmer had been incorrectly assembledits field lens had then been wrongly spaced by 1.3 mm.[47] During the polishing of the mirror, Perkin-Elmer had analyzed its surface with two other null correctors, both of which (correctly) indicated that the mirror was suffering from spherical aberration. The company ignored these test results as it believed that the two null correctors were less accurate than the primary device which was reporting that the mirror was perfectly figured.
<p>
[quote]
Yeah, I'm the same way. People look at me like I'm stupid when I tell them I got a 670 on the SAT Math. For one, multiple choice is distracting and hard to pay attention to, and secondly, the calculation errors I make are immense... which is why I probably failed my complex analysis midterm. This is what I get for taking a class under a probability theorist (I kid, I kid).
[/quote]
</p>
<p>I'm the same way, also. I got around a 670 on the SAT Math, and felt like a total sham. However, at university, I consistently got high (sometimes top) marks on all midterms which involved proofs (vs. repetitive calculations). It took me a long time to recover from low math scores and understand where my weaknesses/strengths lay.</p>
<p>Anyhow, I'll be taking the Calc entrance exam in a couple months, and I'm not quite sure what I'll be doing as a transfer. I've taken Calc I & II at my current university, but we did nothing like what I hear is covered in Honors Calc at Chicago.</p>
<p>I have no idea what most of you are talking, but I am googling it...lol.</p>
<p>If you've aced AP Calculus BC and retain the knowledge of it throughout summer, what course would you most likely get into, if you do well on the placement test based on your knowledge of calc BC? Could I possibly get out of the math requirement?</p>
<p>Less than 10% place out of calc most years, usually less than 8%. To place out, you need excellent retention of the BC5 stuff and the ability to do proofs, among other things. </p>
<p>But why worry about it? There is no badge of honor, no article in the Maroon, and no recognition at graduation that you placed out of calc!</p>
<p>bohbeep, a 5 on BC fulfills the math requirement (it gives you credit for 2 quarters of calc.) But you might be able to get more credit and place out of calc. It's possible even knowing only BC material-you would be placed in 199 if that happens. The key to that last step up would probably be good knowledge of definitions and things like convergence tests. If you don't care about taking more math, though, a 5 on BC means you're done with math whatever happens on placement.</p>
<p>Probably, but I don't know. But if you get that score you should be able to do well enough on the placement test to get that same credit. Also, if you get a 4 on BC you get credit for 151.</p>
<p>if i plan to major in economics, and got a 5 on bc, is it wise to opt out of calc first year? or take the math exam to try to test into high level math course?</p>
<p>BC 5 gives you 2 quarters of calc. Econ requires 3 quarters of calc, plus other higher level math (either analysis or math methods in sosc. sci. and linear algebra, maybe more). So definitely take the placement test to see where you place. You can start at 153 (third quarter calc), or take 160s all year, or if you do really well you might be able to start with the other math.</p>
<p>Just curious, unalove and others who placed out of calc or into challenging analysis-- what background did you have in math? A really good high school education? Experience at another college? A lot of self-study in math?</p>
<p>We didn't even touch proofs at my high school, though I learned something about them on my own. Now that I've had a year of college I'm able to run with ideas much more easily, but I'm wondering if there are really high schools out there that prepare students so well with theoretical calculus. That would be so different from my high school experience.</p>
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Just curious, unalove and others who placed out of calc or into challenging analysis-- what background did you have in math? A really good high school education? Experience at another college? A lot of self-study in math?
[/quote]
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<p>I took linear algebra, differential equations, multivariable calculus, and discrete math in high school. None of them were on a theoretical background. The summer before I took the test, I studied lots of material from Rudin's Principles of Mathematical Analysis, and subsequently tested into (and stayed in) Honors Analysis. </p>
<p>
[quote]
We didn't even touch proofs at my high school, though I learned something about them on my own. Now that I've had a year of college I'm able to run with ideas much more easily, but I'm wondering if there are really high schools out there that prepare students so well with theoretical calculus. That would be so different from my high school experience.
[/quote]
</p>
<p>Hmm. I don't think there are many, if any, high schools that offer theoretical calculus. The people who test into Analysis probably have had a bit of independent study or, more likely, college experience that treated subjects like Calculus 3 or Linear Algebra rather rigorously.</p>