@Nobody: Thanks for the comment. I totally agree – many subjects come together, and are even fun when understood at an intuitive level.
Nice post! One problem is that as a math teacher, one’s instinct is to only say true things… which ironically can get you into real trouble with exposition.
Case in point: me. I read your definition of a circle as the most symmetric 2D shape possible, and immediately started thinking, “but wait, a set of concentric circles is just as symmetric. As is a point. And hey, the entire plane has even more symmetries.”
Then I realized, wow, I shouldn’t be a jerk. Pedantry like mine is exactly the problem you’re complaining about! Thanks for making me take a look at myself.
@anonymous: Thanks for the insightful comment! Yes, sometimes the nitty gritty is useful to focus on, but often it can be a hurdle to beginners.
In this context, word “shape” means something along the lines of “a smooth, unbroken convex curve in 2d” which should hopefully eliminate the plane itself and a single point :).
Thanks for the comment.
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Hi Kalid,
Thank you kindly for your clear and unobstructed definition of e and ln. Learning basic principles is often very frustrating for maths inept individuals such as myself. I certainly benefit from a simple yet useful explication of what are, at first, abstract topics. Website such as yours have inspired and enabled me to teach myself the very topics from which I once shyed away. Heck, I’m even finding derivatives of complex functions! haha
Thanks again, and please continue to add to your website.
J
@J: Thanks for the note! Glad you found it helpful and have moved onto doing crazy things like differentiating complex functions :). I’ll try to keep the articles coming.
hello.im from iran but im human an i have some idea in mathemtic symetry and i play with some formula but i cant now relate theme to a thing please send for me some article about all basic in math please.because i want to study phyzics in the best university with all instrooment for this situation that i want it i most improve very well me math because the mathematics is the base of physics.i read in the petroliom ingeneering in iran in scine and research university but i dont like it im in the end of third year.i want read math in the best university and then study quantom physics in the best univerrsity .here when we say an idea, they laugh.thanks alot and goodbye
simple object,great idea!
Great articles you have written. They help me a lot in my quest for understandning mathematics not just “doing it”. Keep up the good work!
@Seyed: Glad to see you’re interested in education – the articles on the site are what I have now, but Wikipedia and other resources are good jumping off points for more details and references, especially for an area like quantum physics. Good luck.
@Saman: Thanks! Really glad they were useful to you.
I sumbled upon your site not too long ago but the superb comments made about your posts forced me to read through your posts. You definitely explain things really we’ll. Many people can understand things but cannot explain them well. You understand things very well and you also explain those things very well!
@Frank: Thank you! I try to explain things as I wish they were taught to me when I was first learning. I’m happy you’re finding the site useful :).
Sorrym but there are some bad grammar errors in the text, which makes it hard to understand certain things, because first, you have to guess what you could have meant. One example is »We then say that “e” is the number that takes exactly 1 unit of time to grow to.«. To what? The sentence makes no sense.
There are many more such “bugs”. I think your article is pretty nice, but please proofread your texts before posting them. 
Sorry, but there are some bad grammar errors in the text, which makes it hard to understand certain things, because first, you have to guess what you could have meant. One example is »We then say that “e” is the number that takes exactly 1 unit of time to grow to.«. To what? The sentence makes no sense.
There are many more such “bugs”. I think your article is pretty nice, but please proofread your texts before posting them.
See, that’s the difference between proofreading, and not proofreading. 
Thank you very much for creating this site and for sharing your ideas. I find them very-very helpful. Being a Radio Designer I have to use math/calculus to solve problems every day. Dry/rigorous approach never worked for me. I was “lucky” enough to go through university in very competitive class and our teachers seemed to abuse the idea to explain as little as possible to “make you think” let alone trying to come up with intuitive explanation. The only way to succeed was to come up with my own “theory”. But I have to admit that yours are way better. That’s why I find your articles invaluable. And yes, e bothered me forever, but not anymore :-).
@RF_Guy: Thanks for the note and encouragement! I’m working through my notes & would love to make a calculus book one day. I’m going back and revisiting concepts I thought I understood, and seeing more and more insights that I completely missed the first time around.
I really agree with you about the need for rigor – it has its place, but more important is to nurture an enthusiasm/enjoyment for the subject. I’ll definitely keep contributing to the site!
I totally agree with your POV as a struggling math undergrad.
In particular, in the area of Analysis, pedagogy is lacking in the majority of texts.
Hi kalid,
This is a bit long and you may find it amusing or funny but I wrote it straight as i was thinking; I hope you would pardon me and please pardon me!!!
If you recall I had posted a comment earlier mentioning that I go blank while trying to find relationships etc.
Same thing happened again with higher concepts, but one day I recalled that I was doing same thing long time ago while I was in primary school.
(Pardon me coz this may sound childish)
I had rote learned this particular formula :
(a+b)2=a2 + b2 + 2ab (a2 as in a square).
I thought about going to basic and start from top down… i.e. from formula to history using my imagination.
First que: How does it help? if i know 3 square = 9 & 4 square equals 16… will this help in getting to know 7 square?
Sec que: How did they arrive at it in first place?
all i could envision (gaphically) was two square boxes. one of length 3 & other of length 4.
Now I moved on to bigger box of length 7 & now I was thinking of relating it to formula & I divided it into two lengths of 3 & 4.
Within this bigger box i could draw area of 3 square & 4 square with there edges meeting at a point inside the 7 square box.
left over areas were two recangles, one had length 3 & breadth 4 while the other had opposite figures… so i could think of graphicall division in terms of area in a plane.
But now I want some input regarding how am I doing and if my directin is correct at all I would to know how ancient engineers could have used these equations to solve what kind of problem?
(BTW i don’t how to post my drawings here so i had to explain them sorry for inconvenience)
Thanks & Regards
hitendra
@hitendra: Thanks for writing! Yes, I like visualizing the multiplication with the diagram – I think I know what you mean. You can take a square of side 7 (3 + 4) and break it into a 3x3, a 4x4, and two 3x4 rectangles.
In terms of use, it’s a neat algebraic property… some people might think that (3+4)^2 = 3^2 + 4^2 but clearly that is not the case. So when solving any problem with area, it’s important to know the correct result of that equation. In terms of everyday use, let’s see… if you are marking off a square plot of land, it’s better to make one big square (7x7) instead of two smaller ones (3x3 and 4x4), so the king can collect more taxes :). Or, if arranging soldiers in a field, and you see an army approaching (7x7) you know your two squadrons (4x4 and 3x3) will not be enough to fight them. I’m stretching here, but algebra is general is pretty useful :).