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Harvard 1869 entrance exam

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Re: Harvard 1869 entrance exam

#101

Post by jules »

baddox wrote: post #49
cousin_it wrote: post #44

Earlier quoted context omitted.

Yeah, and I'd be even more interested in a similarly intuitive explanation for the case of complex exponents ;-)

That would be difficult, since a formal construction of even the real numbers is a somewhat advanced (3rd or 4th year college mathematics) topic. I forget the details, but I believe a^n for real a and complex n is formally defined using the exponential function ( e^x ).

At least in the Netherlands, construction of the reals (for example from rational Cauchy sequences) is standard 1st semester stuff. Understanding reals is required or provides a good source of examples for virtually all mathematics courses, so I can't imagine how some universities teach mathematics without it.

Re: Harvard 1869 entrance exam

#102

Post by estevez »

mortenjorck wrote: post #22

In a way, this reinforces my hypothesis that Latin in traditional Western higher education was never quite so much about Latin itself as it was about gaining a deeper understanding and greater praxis of your native tongue by reading its source code.

You might also consider the role that Latin played as the lingua franca of scholarship.

Re: Harvard 1869 entrance exam

#103

Post by jules »

derleth wrote: post #99
veyron wrote: post #19

Earlier quoted context omitted.

Plane geometry is woefully lacking in education nowadays. The situation is so sad that some of these questions are very similar to the geometry questions in the USAMO/IMO

Plane geometry was, at the time, perhaps the premier formal system to be studied as such; now, we get the same concepts across using calculus (which doesn't seem to be mentioned on the exam) and set theory (which would only really take shape in the 1870s). (The history of mathematics doesn't get so much as a mention, either, but I don't expect it to.)

This is an interesting "book" that reduces geometry to automated symbol pushing.

http://www.math.rutgers.edu/~zeilberg/GT.html

Re: Harvard 1869 entrance exam

#104

Post by nitrogen »

Peaker wrote: post #95
wbhart wrote: post #77

Earlier quoted context omitted.

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 ver…

That explanation is somewhat problematic with fractional exponents. The generalization of exponentiation makes the "multiply N times" explanation fail.

Not really; a fractional exponent n yields the quantity one would have to multiply 1/n times to return the original value. Multiplication an integer number of times could be seen as a special case of a broader concept of "fractional" multiplication (much like the gamma function (Γ(n)) extends the discrete factorial to a continuous domain).

Re: Harvard 1869 entrance exam

#105

Post by FelixH »

mortenjorck wrote: post #22

In a way, this reinforces my hypothesis that Latin in traditional Western higher education was never quite so much about Latin itself as it was about gaining a deeper understanding and greater praxis of your native tongue by reading its source code.

How is Latin the 'source code' of English? Sure, a large number of Latin derived words made it into English (mostly via Norman French), but that's only half the picture

Re: Harvard 1869 entrance exam

#106

Post by Peaker »

nitrogen wrote: post #104
Peaker wrote: post #95

Earlier quoted context omitted.

That explanation is somewhat problematic with fractional exponents. The generalization of exponentiation makes the "multiply N times" explanation fail.

Not really; a fractional exponent n yields the quantity one would have to multiply 1/ n times to return the original value. Multiplication an integer number of times could be seen as a special case of a broader concept of "fractional" multiplication (much like the gamma function (Γ(n)) extends the discrete factorial to a continuous domain).

1/n times may not necessarily be whole either.

How do you explain irrational exponents this way, for example? What about complex exponents?

Indeed you can extend the special case to the continuous domain -- but then the definition is expanded as well.

I still think "multiply N times" is just a special-case, and as such, not usable as a definition -- let alone an explanation of why we can add exponents in the general case.

Re: Harvard 1869 entrance exam

#107

Post by 32ftpersecond »

CamperBob wrote: post #97
32ftpersecond wrote: post #52

Earlier quoted context omitted.

But we've strayed away from the old knowledge (e.g. classics) -- the stuff that keeps repeating itself...

If history repeats itself, then why not study newer, more relevant iterations?

The Classics are timeless and thus always relevant, that's why they're still read today. Can anyone really argue that reading Jimmy Carter over Cicero or Marcus Aurelius is preferable because of temporal relevancy factors?

Re: Harvard 1869 entrance exam

#108

Post by samsol »

The good thing about entrance (seemingly thought-provoking) exams is that , they used to concentrate on the real sciences/ social sciences . And used to test aptitude in these disciplines to ensure that people who pursue them have enough passion to go thru end. And they let leadership skills emerge after acquiring those analytical/ philosophical skills.

Rather than in the current education system where very very few people want to proceed working in these pure sciences and majority of them want to become leaders and thanks to Univs of US in which leaders are "annointed" by dishing out MBA's based on GMAT / CAT scores.

Re: Harvard 1869 entrance exam

#109

Post by lobo_tuerto »

gbaygon wrote: post #91
dlss wrote: post #66

Earlier quoted context omitted.

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…

Well explained, but 2^4 * 2^4 doesn't take into account the "different powers" part.

2^3 == 2 * 2 * 2

2^2 == 2 * 2

2^3 * 2^2 == 2 * 2 * 2 * 2 * 2 == 2^5

Re: Harvard 1869 entrance exam

#110

Post by officemonkey »

barrkel wrote: post #58
michaeldhopkins wrote: post #47

Earlier quoted context omitted.

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

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.

I think "reason" here is implying "show me how you think (that is, your reasoning).

In fact, I was impressed at how open-ended these questions were. For instance: "Leonidas, Pausanias, Lysander" is about as open-ended as you can get.

I for one, would just create a list of interesting anagrams. (eg: Paranoia Saleslady Snide Sun) I figure it would show my moxie.

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