4 ms·

Actually, I think there's a really, really interesting point hidden in here: While technically true, it could also be argued that you should rederive

by Cushman 15y ago

Actually, I think there's a really, really interesting point hidden in here: While technically true, it could also be argued that you should rederive summation if asked to add all integers from one thousand to one million. However, math classes never seem to make such unreasonable requests. Do you remember the first moment, as a child, when you first grokked that two blocks and three blocks was the same number of blocks as four blocks and one block? I don't, but there must have been one. Some time where I looked at my hand, looked at the blocks, looked back at my hand, and suddenly got it. "Five." What was that, if not a derivation of summation? I propose that we are teaching physics (and most everything, really) much, much too late.

tonyarkles 15y ago
I found that statement interesting too, specifically because our math courses had us re-derive a PILE of stuff, and I felt much more enlightened after having done that. Summation, for instance, was derived early on. Same with integration (Fundamental Theorem of Calculus). One of my most memorable re-derivations was in my 4th calculus class (Sequences, Series, and Differential Equations, if I recall). We ended up taking the Taylor series approximation of sin(x), cos(x), and e^x. As the patterns started to emerge, the professor encouraged us to look at them carefully and start considering what happened when you substituted ix into the approximation for e^x. This ends up, of course, turning into Euler's formula, e^ix = cos(x) + isin(x). Seeing it with the Taylor series approximations, though, snapped all the pieces together. Up until then, it all seemed like magic (sure, you can talk about vectors and imaginary numbers, but you're just accepting it as true). That derivation fundamentally changed how I look at this stuff.
- tonyarkles 15y ago
  More on this topic :) Another really fascinating thing for me was first year "Intro to Electricity". That course was relatively formula based; for example, the voltage across a capacitor is given by Vcap = V0 * (1 - e^(-t/tau)). I had a hell of a time remembering that formula, and suffered for it. In second year, we started getting more in depth and had a few more tools at our disposal (e.g. the Laplace transform, Integration, light Differential Equations, etc). At that point, the formulas i = C dv/dt and v = L di/dt became fully engrained in my head. We took those, and derived the equations we'd been using in first year. From that point on, I solved every passive circuit using precisely these first principles, and would consistently get the correct answer faster than many of my classmates who were trying to remember big specific formulas. If I needed it for some reason, I could quickly do the integral for a specific circuit to get the closed-form voltages.
faucet 15y ago
1+99 = 100, 2+98=100 etc. So the sum of integers between 1 and 100 is 100*49 + 50. Is this what the author refers to? I believe he has misunderstood the point in that math problem.
- tonyarkles 15y ago
  I don't remember the derivation off the top of my head, but I suspect the article is referring to the derivation of the formula: > S(1..n) = (n^2 + n) / 2 This was, if I recall, the very first example of inductive proofs we did at my university.

by Cushman 15y ago

tonyarkles 15y ago

I found that statement interesting too, specifically because our math courses had us re-derive a PILE of stuff, and I felt much more enlightened after having done that. Summation, for instance, was derived early on. Same with integration (Fundamental Theorem of Calculus). One of my most memorable re-derivations was in my 4th calculus class (Sequences, Series, and Differential Equations, if I recall). We ended up taking the Taylor series approximation of sin(x), cos(x), and e^x. As the patterns started to emerge, the professor encouraged us to look at them carefully and start considering what happened when you substituted ix into the approximation for e^x. This ends up, of course, turning into Euler's formula, e^ix = cos(x) + isin(x). Seeing it with the Taylor series approximations, though, snapped all the pieces together. Up until then, it all seemed like magic (sure, you can talk about vectors and imaginary numbers, but you're just accepting it as true). That derivation fundamentally changed how I look at this stuff.

tonyarkles 15y ago
More on this topic :) Another really fascinating thing for me was first year "Intro to Electricity". That course was relatively formula based; for example, the voltage across a capacitor is given by Vcap = V0 * (1 - e^(-t/tau)). I had a hell of a time remembering that formula, and suffered for it. In second year, we started getting more in depth and had a few more tools at our disposal (e.g. the Laplace transform, Integration, light Differential Equations, etc). At that point, the formulas i = C dv/dt and v = L di/dt became fully engrained in my head. We took those, and derived the equations we'd been using in first year. From that point on, I solved every passive circuit using precisely these first principles, and would consistently get the correct answer faster than many of my classmates who were trying to remember big specific formulas. If I needed it for some reason, I could quickly do the integral for a specific circuit to get the closed-form voltages.

faucet 15y ago

1+99 = 100, 2+98=100 etc. So the sum of integers between 1 and 100 is 100*49 + 50. Is this what the author refers to? I believe he has misunderstood the point in that math problem.

tonyarkles 15y ago
I don't remember the derivation off the top of my head, but I suspect the article is referring to the derivation of the formula: > S(1..n) = (n^2 + n) / 2 This was, if I recall, the very first example of inductive proofs we did at my university.