5 ms·

Wow, thanks for the plug and the kind words! Glad to hear the site is helping -- most of the time I miss the intricacies too, and end up adding things years lat

by kalid 12y ago

Wow, thanks for the plug and the kind words! Glad to hear the site is helping -- most of the time I miss the intricacies too, and end up adding things years later [after a re-read] or in response to a comment from the article. As a more general reply to the article, I was fortunate enough to work with Prof. Oakley on her Coursera Class (https://class.coursera.org/learning-001/lecture https://class.coursera.org/learning-001/lecture, I'm the last guest interview at the bottom) and I really, really like her learning strategy. Her article didn't use the exact phrase "deliberate practice" but I think that captures the essence of what she means by repetition. You need enough conceptual understanding to make sure you're following the path correctly, but then you want to practice -- at the edge of your comfort zone, with feedback, etc. -- to make sure it's really clicking. It's really easy to fool yourself into thinking "I've got this" when it's untested. In my own case, I realized I didn't "get" imaginary numbers and exponents when I couldn't estimate a^b [a raised to the bth power] in my head with equal fluency for all numbers a and b. I had never really tested every type of number in every type of exponent position (base and power). For example, 3^4 => this should be a positive real number greater than 1 3^(-4) => this should be a positive real number, very close to 0. 3^i => Hrm. i^i => Uh oh. I knew that unless I had fluency with all of these scenarios, I didn't truly "get" exponents or complex numbers. Sure, maybe I had a baby version where I could use them in well-defined ways, but I had a subconscious fear of i appearing as a base and/or exponent. I had to challenge myself and practice thinking through the various permutations before I recognized the gaps. Then I had to deepen my conceptual understanding, and practice again. (For the previous questions, 3^i should be a complex number on the unit circle, maybe around 50 or 60 degrees, but less than 90, and i^i should be a real number, greater than 0 but less than 1. I can estimate these without calculating them, see http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://betterexplained.com/articles/intuitive-understanding-... for more details.) Again, thanks for the mention!

shostack 12y ago
Very happy to give the plug--the quality of your content and your generosity for putting so much out there for free is something I wish more people did. I hope the business side of it is going well for you. Side note--have you considered trying to work with Khan Academy? I feel like Salman's style flows well, but sometimes substitutes learning tricks vs. internalizing the underlying logic. Khan Academy could benefit from a touch of BetterExplained.
- kalid 12y ago
  Thanks! The business side of things is coming along, I've just finished up a hectic few months (wedding, honeymoon) and am looking to crank on the site again. Long term I want the site to be an ever-growing part of my life. I'm very interested in collaborations, and helping people sprinkle an intuitive approach into whatever teaching they are doing. (I like Bret Victor's approach, where he provides guidance on what programming tools could look like, which might work its way into new programming languages like Swift, etc.). I'm in touch with an internet buddy at Khan Academy and have been meaning to work together, I'll be reaching out soon I think :).
kev6168 12y ago
The entries on complex number, exponential functions & e, and many other subjects at betterexplained.com are simply amazing! It helped to clear my decade-long confusions or fears when dealing with them, Thank you for the great work. In my opinion the three essential characteristics of a great teacher are expert level in the subject at hand, great communication skill and enthusiasm for teaching. Possessing two of them already makes a good teacher. You have all three.
- kalid 12y ago
  That is really humbling and gratifying to hear, thank you.
wfunction 12y ago
Is there really an intuition for i^i? I can give you the answer but I have nearly zero intuition for it.
- kalid 12y ago
  Yep, I think so. My philosophy, at least, is that any idea can eventually be intuitive, no matter how difficult at first, if we find the right analogies. I see any exponent like a^b as starting at 1.0, intending to apply a rate of change of (a), but modifying that rate by (b). For example, 3^2 is an initial rate of change of 3x, which is then applied for 2 units of time [leading to 9]. So, 1.0 would turn into 9.0. More here: http://betterexplained.com/articles/understanding-exponents-why-does-00-1/ http://betterexplained.com/articles/understanding-exponents-... So what's i^i? Well, our rate of change is "i", which means we plan on starting at 1.0 and having our rate of change be a rotation: 1.0 * i = 90 degrees on the unit circle. If we are using i^i, that means we are rotating our rate of change. That is, we intended to rotate around the unit circle at 90 degrees, but now I'm going to turn the "rocket boost" which is applying that rotation by another 90 degrees (the i as the exponent). This means the rocket is facing 180 degrees (backwards) and our growth is going to be an exponential decay. We'll be on the real number line, but shrinking. How about (i^i)^i? That's another 90-degree twist on the rocket, so it's pointing 270 (downward) and we'll rotate around the circle clockwise. It's probably a negative imaginary number, and (i^i)^i actually equals -i. This is a quick brain dump, and not very clear without diagrams, check out the articles above if you'd like!
  - wfunction 12y ago
    Your intuition is nice for figuring out the directions, but not so nice for explaining the numbers. By which I mean, I can understand why it's real, but I can't explain why i^i = e^(-pi/2)... I would expect it to be a nicer number instead (like 1/2, or 1/e, or something like that). I can't really explain why both pi and e end up int he formula.
    - kalid 12y ago
      Oh yeah, after getting the direction, figuring out the numbers is the next step :). Having i as a base means "we plan on rotating 90 degrees" which actually means pi/2 radians. e^rt models growth rate of r, for time of t. so e^(i · pi/2) creates a 90 degree turn (we intend on rotating, i, and do this enough to get a full 90-degree turn, pi/2). This is all a fancy way of saying: 90 degree turn = i = e^(i · pi/2) Now, with i^i, we're planning on modifying that growth rate (e^(i · pi/2)) that we just figured out! We're going to twist the "rate" from i (90 degrees) to i · i (180 degrees): i^i = e^(i · i · pi/2) = e^(-pi/2) which is the real number less than 1 (about 0.2). I'm summarizing on the fly, but the full explanation for i^i and i^i^i are here: http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://betterexplained.com/articles/intuitive-understanding-...

by kalid 12y ago

shostack 12y ago

Very happy to give the plug--the quality of your content and your generosity for putting so much out there for free is something I wish more people did. I hope the business side of it is going well for you. Side note--have you considered trying to work with Khan Academy? I feel like Salman's style flows well, but sometimes substitutes learning tricks vs. internalizing the underlying logic. Khan Academy could benefit from a touch of BetterExplained.

kalid 12y ago
Thanks! The business side of things is coming along, I've just finished up a hectic few months (wedding, honeymoon) and am looking to crank on the site again. Long term I want the site to be an ever-growing part of my life. I'm very interested in collaborations, and helping people sprinkle an intuitive approach into whatever teaching they are doing. (I like Bret Victor's approach, where he provides guidance on what programming tools could look like, which might work its way into new programming languages like Swift, etc.). I'm in touch with an internet buddy at Khan Academy and have been meaning to work together, I'll be reaching out soon I think :).

kev6168 12y ago

The entries on complex number, exponential functions & e, and many other subjects at betterexplained.com are simply amazing! It helped to clear my decade-long confusions or fears when dealing with them, Thank you for the great work. In my opinion the three essential characteristics of a great teacher are expert level in the subject at hand, great communication skill and enthusiasm for teaching. Possessing two of them already makes a good teacher. You have all three.

kalid 12y ago
That is really humbling and gratifying to hear, thank you.

wfunction 12y ago

Is there really an intuition for i^i? I can give you the answer but I have nearly zero intuition for it.

kalid 12y ago
Yep, I think so. My philosophy, at least, is that any idea can eventually be intuitive, no matter how difficult at first, if we find the right analogies. I see any exponent like a^b as starting at 1.0, intending to apply a rate of change of (a), but modifying that rate by (b). For example, 3^2 is an initial rate of change of 3x, which is then applied for 2 units of time [leading to 9]. So, 1.0 would turn into 9.0. More here: http://betterexplained.com/articles/understanding-exponents-why-does-00-1/ http://betterexplained.com/articles/understanding-exponents-... So what's i^i? Well, our rate of change is "i", which means we plan on starting at 1.0 and having our rate of change be a rotation: 1.0 * i = 90 degrees on the unit circle. If we are using i^i, that means we are rotating our rate of change. That is, we intended to rotate around the unit circle at 90 degrees, but now I'm going to turn the "rocket boost" which is applying that rotation by another 90 degrees (the i as the exponent). This means the rocket is facing 180 degrees (backwards) and our growth is going to be an exponential decay. We'll be on the real number line, but shrinking. How about (i^i)^i? That's another 90-degree twist on the rocket, so it's pointing 270 (downward) and we'll rotate around the circle clockwise. It's probably a negative imaginary number, and (i^i)^i actually equals -i. This is a quick brain dump, and not very clear without diagrams, check out the articles above if you'd like!
- wfunction 12y ago
  Your intuition is nice for figuring out the directions, but not so nice for explaining the numbers. By which I mean, I can understand why it's real, but I can't explain why i^i = e^(-pi/2)... I would expect it to be a nicer number instead (like 1/2, or 1/e, or something like that). I can't really explain why both pi and e end up int he formula.
  - kalid 12y ago
    Oh yeah, after getting the direction, figuring out the numbers is the next step :). Having i as a base means "we plan on rotating 90 degrees" which actually means pi/2 radians. e^rt models growth rate of r, for time of t. so e^(i · pi/2) creates a 90 degree turn (we intend on rotating, i, and do this enough to get a full 90-degree turn, pi/2). This is all a fancy way of saying: 90 degree turn = i = e^(i · pi/2) Now, with i^i, we're planning on modifying that growth rate (e^(i · pi/2)) that we just figured out! We're going to twist the "rate" from i (90 degrees) to i · i (180 degrees): i^i = e^(i · i · pi/2) = e^(-pi/2) which is the real number less than 1 (about 0.2). I'm summarizing on the fly, but the full explanation for i^i and i^i^i are here: http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://betterexplained.com/articles/intuitive-understanding-...