4 ms·

Kalid from BetterExplained here, it's awesome to hear the approach is working for you. I had very similar reservations, I could derive many results (like a robo

by kalid 10y ago

Kalid from BetterExplained here, it's awesome to hear the approach is working for you. I had very similar reservations, I could derive many results (like a robot) but had little intuition behind e, i, pi, radians, etc. (let alone how they all came together in something like Euler's Formula). My litmus test became: If I can't intuitively describe i^i (an imaginary number to an imaginary power) I don't understand it. I don't care if I can derive the equation 15 different ways. If I couldn't spit out some properties of i^i (positive or negative? Real or imaginary? Big or small?) after a glance then I knew I didn't know it. (Why can I spit out properties of 2^3 or 3^(-4) in a few seconds, but not i^i?) Code is an excellent way to practice these ideas; the bugs in your logic correspond to the bugs in your thinking, and you see (very explicitly) where to correct them.

Bharath1234 10y ago
Beautifully explained :)
jacobolus 10y ago
Kalid: you should really check out geometric algebra (a.k.a. Clifford algebra). It will give you a much deeper understanding of what i is and what the exponential function is, and how they generalize to higher dimensions and more complicated models, and it will help stitch together the weird inconsistent little fragments of understanding provided by imaginary numbers, quaternions, matrix algebra, Lie theory, differential forms, etc. into a more unified/cohesive model. This is the model all high school and college students will be taught in 100 years, or perhaps even in 50 years, and it will prevent an enormous amount of confusion and misunderstanding. It’s already becoming the practical tool of choice in many geometric computing problems, and among niche groups of physicists. ”Reforming the Mathematical Language of Physics”, http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf “Grassmann’s Vision” http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf “Imaginary Numbers are not Real” http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/Im... “Geometric Algebra” http://arxiv.org/pdf/1205.5935v1.pdf http://arxiv.org/pdf/1205.5935v1.pdf (this one is a good place to go if you get stuck in another source). (Or books might be better sources for going in depth. Search for New Foundations for Classical Mechanics, Geometric Algebra for Computer Science, Geometric Algebra for Physicists) You’d also probably enjoy Hestenes’s work on modeling in physics teaching, e.g. http://modeling.asu.edu/R&E/ModelingThryPhysics.pdf http://modeling.asu.edu/R&E/ModelingThryPhysics.pdf http://worrydream.com/refs/Hestenes%20-%20Modeling%20games%20in%20the%20Newtonian%20World.pdf http://worrydream.com/refs/Hestenes%20-%20Modeling%20games%2... http://modeling.asu.edu/R&E/Notes_on_Modeling_Theory.pdf http://modeling.asu.edu/R&E/Notes_on_Modeling_Theory.pdf http://modeling.asu.edu/R&E/Hestenes-ModelingTheory2007.pdf http://modeling.asu.edu/R&E/Hestenes-ModelingTheory2007.pdf

by kalid 10y ago

Bharath1234 10y ago

Beautifully explained :)

jacobolus 10y ago

Kalid: you should really check out geometric algebra (a.k.a. Clifford algebra). It will give you a much deeper understanding of what i is and what the exponential function is, and how they generalize to higher dimensions and more complicated models, and it will help stitch together the weird inconsistent little fragments of understanding provided by imaginary numbers, quaternions, matrix algebra, Lie theory, differential forms, etc. into a more unified/cohesive model. This is the model all high school and college students will be taught in 100 years, or perhaps even in 50 years, and it will prevent an enormous amount of confusion and misunderstanding. It’s already becoming the practical tool of choice in many geometric computing problems, and among niche groups of physicists. ”Reforming the Mathematical Language of Physics”, http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf “Grassmann’s Vision” http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf “Imaginary Numbers are not Real” http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/Im... “Geometric Algebra” http://arxiv.org/pdf/1205.5935v1.pdf http://arxiv.org/pdf/1205.5935v1.pdf (this one is a good place to go if you get stuck in another source). (Or books might be better sources for going in depth. Search for New Foundations for Classical Mechanics, Geometric Algebra for Computer Science, Geometric Algebra for Physicists) You’d also probably enjoy Hestenes’s work on modeling in physics teaching, e.g. http://modeling.asu.edu/R&E/ModelingThryPhysics.pdf http://modeling.asu.edu/R&E/ModelingThryPhysics.pdf http://worrydream.com/refs/Hestenes%20-%20Modeling%20games%20in%20the%20Newtonian%20World.pdf http://worrydream.com/refs/Hestenes%20-%20Modeling%20games%2... http://modeling.asu.edu/R&E/Notes_on_Modeling_Theory.pdf http://modeling.asu.edu/R&E/Notes_on_Modeling_Theory.pdf http://modeling.asu.edu/R&E/Hestenes-ModelingTheory2007.pdf http://modeling.asu.edu/R&E/Hestenes-ModelingTheory2007.pdf