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Harvard 1869 entrance exam

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Re: Harvard 1869 entrance exam

#71

Post by baddox »

dlss wrote: post #66
barrkel wrote: post #58

Earlier quoted context omitted.

I don't think that's the case. As baddox says, it's trivial to show it is true, especially using simplified definitions for exponentiation (i.e. sticking with integer or perhaps rational exponents), but demonstrating truth doesn't tell you about the "reason". Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.

Being able to explain proofs intuitively is a valuable way to check how deeply you know them. In this case, the reason that when different powers of the same quantity are multiplied together their exponents are added is because powers are short hand for a series of multiplications: 2^4 == 2 * 2 * 2 * 2 When you multiply 2^4 * 2^4, that is short hand for: 2^4 * 2^4 == 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 == 2^8 Of course you…

That works for positive integer bases and exponents, but try giving an "intuitive proof" with irrational exponents. Most things in math, even seemingly obvious things in arithmetic, require a lot of shared background knowledge (at least propositional logic, basic set theory, and a construction of the natural numbers) for two people to even converse formally.

Re: Harvard 1869 entrance exam

#72

Post by dpe82 »

seagaia wrote: post #57

does the requirement of Latin strike anyone as a bit elitist? or was Latin really that necessary in order to (presumably) study some Classics?

The European university tradition on which Harvard was created treated Latin and Greek as the languages of learning. At the time a college education included a rigorous treatment of the classics (Latin) and philosophy (Greek). The belief was probably that they couldn't be truly appreciated understood unless studied in the original language (which is arguably true). And of practical note, some scientific writing was still done in Latin at the time and the language was still in wide use in throughout Europe.

Heck, I learned Latin in high school in the late 90s. I hated it when I took it, but it certainly made learning other things a lot easier.

Re: Harvard 1869 entrance exam

#73

Post by the-cakeboss »

impendia wrote: post #69
fuzionmonkey wrote: post #36

Earlier quoted context omitted.

Do you actually find college students that don't know what an exponent is? I feel like anybody with a basic understanding of what an exponent represents should be able to explain why you add exponents.

They have a working knowledge of it, with very rare exceptions -- but I find that students are taught math as a bunch of rules and not all of them are comfortable giving explanations.

Yeah, that is what I perceive as the greateset failing of modern math education. More emphasis on memorization rather than a deep understanding.

Re: Harvard 1869 entrance exam

#74

Post by [deleted] »

baby wrote: post #55
BrandonM wrote: post #53

Earlier quoted context omitted.

> It's exactly like learning scheme. Seriously who fucking cares about scheme? Err... This website is written in Arc (a cousin of Scheme), which itself is implemented in MzScheme. So by extension, you care about scheme, and so does everyone else here.

I think this website has a pretty poor UI. And it's PHP also.

[deleted]

Re: Harvard 1869 entrance exam

#75

Post by pak »

The thing I noticed (sorry, LaTeX fan here) was how well they could typeset math and Latin all the way back in 1869! I have to say I'm impressed, and it makes me a bit sad that most of my college exams looked worse (typographically) than a paper produced over a century ago. Look at those goddamn gorgeously even margins, the ligatures, the kerning of the italics, and the protrusion of the hyphens.

Hell, this makes the SATs, with its ragged edges and sloppy Times New Roman straight out of MS Word, look like carelessly produced junk.

Re: Harvard 1869 entrance exam

#76

Post by BrandonM »

[deleted] wrote: post #74
baby wrote: post #55

Earlier quoted context omitted.

I think this website has a pretty poor UI. And it's PHP also.

[deleted]

I think people are more likely downvoting because "it's PHP also" is utterly incorrect, yet stated as if it's fact. As for the UI, many HN members (including pg?) have expressed the idea that the types of people that get hung up on UI shortcomings are a close approximation to the types of people we wouldn't want as members, anyway.

Re: Harvard 1869 entrance exam

#77

Post by wbhart »

baddox wrote: post #70
michaeldhopkins wrote: post #47

Earlier quoted context omitted.

That's the point -- you don't understand it well enough to explain it.

I would have to brush up a bit, but when I was in school I wouldn't have had much trouble proving it. But that's not the real issue. My main problem is the vagueness in wording (which might be attributable to the lack of formalization in mathematics in 1869). What does "reason" mean? Is it asking for a proof? And if so, what axioms and lemmas are you allowed to use? Are we talking about integer bases and exponents (t…

Raising a base to a power is a prescription for how many times to multiply by the base. If you first raise it to one exponent, m say, then to another exponent, n say, and then multiply, you have first multiplied by the base m times, then multiplied by the same base a further n times. In total you have multiplied by the base m + n times.

Back in those days they would have used slide rules and understood logarithms very well (which they used for multiplication by adding logarithms, essentially). So they may have just answered that to multiply values is to add their logarithms and exponentiate. If the logarithm is taken to the common base, the logarithms are given by the respective exponents.

Re: Harvard 1869 entrance exam

#78

Post by dan-k »

pflats wrote: post #25
william42 wrote: post #16

Earlier quoted context omitted.

I think that at that point calculus was a grad-school-level thing. At the very least, it wasn't being taught in high schools. In fact, I'm pretty sure it's only taught in high schools today under the aegis of Advanced Placement classes, which give you college credit.

Many, many schools today offer "College Prep" calculus, which is far less rigorous than AP Calc. It covers most of the same topics, but with less depth. It's unlikely a College Prep Calc class would spend much time, if any, memorizing Trig Identities for integration. Generally very little epsilon-delta work, etc.

I'd be surprised if they do more than mention that epsilon-delta stuff exists. Most AP calc classes have very little epsilon-delta work. We spent maybe two or three days on it in mine, certainly not enough time to write a full proof, and that was more than most people I've talked to who did it at different high schools. If my experience is at all representative, that rigorous of a development of calculus isn't normal until upper-level classes, when you're heading towards analysis.

Re: Harvard 1869 entrance exam

#79

Post by wolf550e »

barrkel wrote: post #60
ry0ohki wrote: post #6

The Latin and Greek parts aside, even the Geography portion is quite difficult, and I have a degree in Geography.

I don't think any of that is difficult ; it is, rather, a test of memorization, not ability. A lot of the mathematics section is also on the basis of memorization.

In those days, being an educated person meant memorizing a lot of things, because you couldn't haul your library with you everywhere you went, and in any case it didn't have full text search. Separate a modern educated person from google/wikipedia/etc. and see whether they can still converse at the same lavel. Most can't.
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