Wow, you are amazing! You explain things a way nobody has ever done (around me I mean). It is unbelievable: it’s like you enter my head, grab the questions inside and then explain each of them with a 100% chance of understanding!
Very nice and useful article, explains what I used to consider trivial, wow!
I love the site, please keep on publishing such wonderful articles!
Thanks a lot.
@nschoe: Thanks, really glad it helped! Yeah, I really feel it’s important to answer those “huh?” questions we get in our heads as they come up, instead of pretending that learning is one smooth process from A to B. We need to explain why we don’t want to go to C and D :).
Appreciate the support!
Hi Kalid, thanks for all these articles, I’m finding them illuminating.
I’m still struggling with this one a bit, though. I’m confused by the use of the phrase “growing instantly”. E.g. “3 is the end result of growing instantly (using e) at a rate of ln(3), i.e. 3 = e^ln(3)”
If something grows instantly, then its rate of growth is infinite, isn’t it? I don’t understand how something can grow “instantly” at a finite rate.
And then “3^4 is the same as growing to 3, but then growing for 4x as long” - but if the growth was instant before, then “4x as long” is also instant. Four times zero is zero.
And what does “using e” mean in this instance? Starting with a value of e? Why are we doing that? I thought e was a sort of fundamental growth rate, not a starting value? In your Expand-o-tron analogy in another article, the base is the desired growth rate per unit time, and the exponent is the number of units of time to grow for. But here, the exponent seems to be the rate, while the base is the starting value.
@PaulH: Thanks, great question (I need to setup a FAQ for exponents since there are so many parts that I still have to remind myself also).
You’re right – perfectly instantaneous growth would indeed be infinite. Getting at this idea of “infinitely small” is one of the problems of Calculus actually. A better phrasing may be “Growing at a compounding interval that appears perfectly smooth to us”, i.e. we cannot see the sudden, punctuated growth that pops up with compounding on an interval (like making a staircase so fine-grained that it looks like a smooth curve). For example’s sake, we can imagine compounding every nanosecond (billionth of a second).
“Using e” means figuring out the net effect of compounding as fast as possible (every nanosecond, let’s say). So taking the examples you mentioned:
>> E.g. “3 is the end result of growing instantly (using e) at a rate of ln(3), i.e. 3 = e^ln(3)”
We can think of 3 in two ways. The first is the traditional way – at the end of the year, we have 3 times as much (we start with a dollar, get 3 dollars at the end). The other way is to say “3 is the result of starting at 1.0 and growing instantly (compounding as quickly as possible) at some rate.”
If our rate is 100% and we compound as quickly as possible, we end up with 2.718… (e). But we want to get to “3”, so we need to grow a bit faster than 100% per year. The actual number is ln(3) = 109.8%, so we can say:
“We can get to 3.0 at the end of the year if we start at 1.0 and grow at 109.8% per year, compounding as fast as possible”. e is the way we work out what super-fast compounding gives us, so we can write it
e^1.098 = 3
So, we do start at 1.0, but e^x gives us the impact of starting at 1.0 and compounding at x% return as fast as possible.
In other words, e isn’t the starting point because we want to start at 2.718. We start at e because we want to compound 1.0 as fast as possible, and e is the shortcut for that (i.e., the price after shipping, handling, and taxes if you get my drift).
Hope this helps!
How would you go about solving this problem, i was asked to put ln( pi*i -ln (-e^w)) into the for a + bi, and the question states that w is a complex number. Need help for a final tomorrow.
Fantastic!!! You should do more explanations like that, you’re as good (or perhaps even better) than Sal Khan from Khan Academy!
@Rafay: I can’t really comment on specific homework problems, but break it into groups: you can see that ln(e^w) should just be “w” (since ln and e are inverses) so you get something like ln(pi * i - w). From the rules of logarithms, this equals ln(pi * i) / ln(w). The natural log of i is explained above – that should help get you started.
@Flo: Thanks! I plan to keep creating more :).
Nicely done. Another internet applied math whiz is discovered…
Wow, the last time my mind was blown this hard was when I did LSD. Only this time I can remember what it was that had caused the mind-blow (lol?) the next day.
Kalid,
I’m not sure, but I think your description of 3=e^ln(e) is a very interesting way of converting discrete mathematics to continuous. It is like saying, there is a finite rate with which to generate a number (3), you grow so much per unit time. You can use this method to compute new numbers, by “growing them”. Brilliant.
^ I meant 3=e^ln(3) :).
@Mark: Glad you liked it! Yes, it’s interesting to see regular whole numbers (like 3, 4, etc.) as just “some amount” of growth that e takes you to.
You are simply brilliant !
Please keep updating your web with new and interesting problems.
Thanks
It deserves to be called a “phenomenal” article.
Hope, to see Fourier Transform/Laplace Transform/Z-transform articles. If that happens, thousands of students would change the way they learn (I think they will like it more).
@Nasser: Thanks! I’m planning on doing the Fourier transform soon – the others, I still need to learn about :).
I’m a math major studying topology, and really really sympathize with the attitude that math is about the underlying structure, the geometry and simplicity of the problem; with the analytic formulas being a good book-keeping device, but unenlightening unless you have a picture in your head. I can’t say enough how well-written I found this article (and the one about “e”), I have never enjoyed complex numbers because it always feels like a bunch of tedious algebraic laws, but this is a wonderful explanation of euler’s formula; and after reading the small bit in "Visual Complex Analysis; I picked up a copy and its awesome.
Keep up the good work; and I also look forward to the fourier transform/laplace transform installment!
@Steve: Thanks! I really like Visual Complex Analysis too – even though I wasn’t a math major (maybe in another life :)) I have a severe interest in it and want to study it deeper. Appreciate the encouragement!
Just a comment I always considered e^ix identical to e^x, except the mapping of e^ix to the 3D complex plane created a helix that turned the runaway exponential to spiral around the x axis, but still always increasing in size.
The projection on the real and imaginary axis being the cosine and sine made it look like a simple rotation, but looking at the complete 3D picture this is not so.
This may be the most insightful article I have ever read. Finally someone care about making formulas intuitive! Very VERY well explained. Genius!
@C: Thank you – really made my day! 