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I made another math post. This one is a bit longer, but more elementary, it's the unbashful intro to infinity I wish I'd had in school.

If you read it all the way through, there's a surprise. Shhhh.
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Love these math posts you're doing. I had a great teacher in high school who would talk about mathematics this same candid way and although I was never very good at math, at least it kept me interested and curious for mathematical thinking.

It would be great to see math introduced like this instead of by rote memorisation.
These posts are incredible .. I wish my high school math teacher had a visualizer like the one you've developed. It just makes everything so much more understandable, and I can't think of a reason not to be interested in math when it's shown like this.
Love your stuff. I assigned your "How to Fold a Julia Fractal" to my complex analysis students to read. 
If anyone is interested in applying some similar concepts to middle school students there is two months left to make a submission for I entered recently for a health design challenge and it was a lot of fun
Wonderful. I have one minor technical issue though. When Chrome's browser zoom is active, constructed fractions get screwed up. I have seen this effect on other sites and traced it to a known bug in
MathJax. webkit-size-adjust settings can also mess up zooming.
As usual... Thanks for the show Steven :)
"Bam! No more iron anywhere."
Absolutely amazing! Being a fan of HTML5/WebGL I really enjoyed it. Moreover, your storytelling skills are terrific: that's how teachers should explain differential calculus to students.
Great article! Fantastic visualizations.

One thing, though, is that you mentioned a couple of times that the rationals weren't "continuous." I understand you are trying to convey the intuition that the the rationals don't form a continuum, but perhaps there is another word that doesn't conflict with the idiomatic usage of the term as referring to functions (i.e. it might be confusing when students eventually discover you can have continuous maps defined over the rationals). Perhaps you could say that the rationals aren't "complete" -- just to keep the terminology closer to the theory?
+Nathan Sorenson Yeah, it's tricky. I try not to get into nuances that the reader cannot be reasonably expected to understand based on what came before. I always say the posts are an exercise in carefully designed ambiguity rather than careful precision.

However, I do not feel it would be productive to replace "continuous" with "complete" in this instance. One of my biggest annoyances with mathematics is its insistence on redefining ordinary words with ordinary meanings to something very specific and counter-intuitive. Case in point, I was taught that left and right are the same "direction" mathematically. This is only going to make people look at mathematicians as if they're crazy. It also ruins simple definitions like "two vectors are equal if they have the same length and point in the same direction", which is perfectly clear using the ordinary meaning.

If I correctly understand the concept of a continuous function defined on Q, it is in no way intuitively 'continuous'. Rather, it satisfies the epsilon-delta concept of getting arbitrarily close, which is necessarily done in discrete steps. If you then go on to define functions that are continuous only in a finite set of points, you end up with what I can only call "troll-math". It's fun to think about, but it has no place in an introductory text. Even Wikipedia starts out by saying "the function is continuous if, roughly speaking, the graph is a single unbroken curve with no 'holes' or 'jumps'." A function defined on Q should not be called a curve, in my opinion.
I definitely agree that the subtleties of real analysis are outside the scope of what you're trying to convey. Perhaps because you italicized the word 'continuous' it struck me like it was presented as the term of art. Scare quotes around the word would be enough to satisfy me! :)
The joke at the end would’ve been even funnier had you said “staring off into the middle distance”.
I still feel sick from the rollercoaster ride! Amazing WebGL applications.
Very impressive.  Not quite sure I agree with the "Yet infinity is not just desirable, it is absolutely necessary."  Similarly with the hyperbole about hyperreals.  Extremely convenient, I'll agree, but necessary?  I think you'll find a few mathematicians who'd say that it isn't needed at all.
+Andrew Stacey Well, when a mathematician claims a certain concept is unnecessary, because they've come up with dozens of pages of proofs that show it can be derived from much simpler axioms, I nod, and slowly back away, and get back to getting real work done.

Optimized code is obfuscated code, and math reduced to its barest essentials is a distillate so pure it's toxic.
pleae keep posting these brilliant "lectures"!
+Steven Wittens (I didn't get notified of your comment, sorry.)

Sometimes, it's true. But this time not. In this case there's a whole branch of mathematics called finitary mathematics which denies the existence of the infinite. Some regard at as a much more accurate model of real arithmetic since in practical computations there is a limit to ones accuracy and thus one only ever deals with a finite set of numbers.
Really fantastic work. I don't know if you've read J.E.Gordon's books on materials science but they are probably the most approachable books to materials science I have ever read. Gordon tends to brush off a lot of the complicated engineering analysis and formulas in favour of teaching history, telling interesting stories and using familiar objects and situations to get the ideas and concepts across. The formulas are still there but they are not required to grasp the material.

The reason I mention him is because I've always been looking for similarly-written books covering other subjects such as chemistry and mathematics; your two posts (this one and the one on Julia sets and complex numbers) are about as close to Gordon's approach as I've ever found. Even if one doesn't truly understand the math, one still grasps the concept. Learning to visualise things in enlightening new ways is really very exciting. :)

I would suggest writing a book but it would be bereft of those wonderful visual tutorials. In any event, I look forward to whatever topic you chose to cover next.
Just went through To Infinity and Beyond, and How to Fold a Julia Fractal yesterday. These are amazingly good!
I wish calculus was taught to me this way in high school and college, because I would have understood it much more easily and quickly. You made what I call "the imperfect math" actually make sense with good analogies, and superb slideshows - the animation is exactly what calculus needs to be properly described, as it's based so much around rates of change.
Anyway, thank you for the enlightenment!
I really enjoyed this, but found it strange that you've used "therefor" to mean ∴ in everywhere but your opening quote. Therefor is an archaic word that means "for it", e.g. "I've supplied the binary and the source code therefor".
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