Comment by kalid
9 years ago
Much of this resonated with me.
One of the biggest lessons of my Ivy-league education was, paradoxically, that it wasn't sufficient (or even necessary). At least not for the goals I wanted.
If you want access to the elite Wall St. or consulting jobs, then yes, you need it. Once you're in, it's almost impossible to be kicked out. Finish, interview, and you're on the path.
But I had longstanding doubts about the state of my actual education. Despite numerous engineering classes, I couldn't, for example, work out i^i in my head instantly.
Imagine someone asked you "Roughly, what do you think 2^512 to be?"
"I don't know exactly, but it's an enormous positive number, probably more than the atoms in the universe."
Ok. You should spit that out instantly. Now imagine someone asks "Roughly, what do you think i^i is?"
Is it positive? Negative? Real? Imaginary? Close to 1.0? Microscopic? Even if you eventually work it out after a minute, does your long delay mean you understood exponents or imaginary numbers? Is this something that should be solidified before, say, studying the Fourier Transform?
Despite scoring well in classes taught by famous professors, I couldn't answer such basic questions. This was literally middle school operations (exponents) used with high-school parameters (imaginary numbers). I was stuck -- and it didn't seem like anyone thought this a reasonable thing to understand.
It led me on a journey of self-education, but I needed to see firsthand that even in an Ivy-league environment, the basics could still be missed. It ended the fantasy that things were transformatively different in some other place -- it was up to me to fill the gaps.
ps. If you want to work out i^i in your head: https://betterexplained.com/articles/intuitive-understanding...
The link you provided seems to omit the fact that i^i is multi-valued, which is a consequence of a fundamental property of the logarithm.
That's true, let's just go with the principal value for now :).
(I hadn't studied complex analysis when I wrote that post, learning it now. The various possibilities for the multi-valued solution follow a pattern however.)
A very strange use for the word "interest" in there. Any reason why?
People are often introduced to exponential growth through compounding interest (of a monetary investment).
Exactly. One model I like is e^x represents 100% interest, perfectly compounded for x units of time. a^b is really just e^[ln(a) * b]. Then you can think about "ln(a) * b" being the interest earned scaled by time.
Exponentiation isn't a middle school operation. Most probably don't know what real exponentiation is until they do the famous problem in chapter 1 of baby Rudin defining it (or equivalent problem in another book), and bright students are unlikely to study this before mid high school. Normal students sometime in college probably.
> "Roughly, what do you think 2^512 to be?"
Instantly I spit out "a 4 followed by 153 zeros"
Now... looking it up it's actually more like a 13 followed by 153 zeros.