BetterExplained: Math Lessons That Explain Concepts
betterexplained.com
betterexplained.com
An article I really enjoyed was his explanation of Quake's inverse square root method: https://web.archive.org/web/20150530232103/http://betterexpl...
What I also appreciate is that he admits when he doesn't understand something and doesn't pretend to give an incomplete, vague explanation and just straight up says he does't fully get it (like why the specific 'magic' number in the Quake algorithm). Thankfully he linked the original paper and I read up on it myself. But his initial explanation provided a solid background for the paper.
https://web.archive.org/web/20070208003255/http://www.beyond...
Want a fun example? How about percentages. Yeah, that thing we mastered in 4th grade or whatever. That thing we use every day.
Well, did you know that
a% of b = b% of a?
Let's say you want 16% of 25. Ugh. Ok, let's multiply it out... divide by 100... no!
How about 25% of 16? Well, that's 4. Easy. But are they the same thing?
a% of b = a/100 * b
b% of a = b/100 * a
Either way, it's ab/100. Now every percentage problem has a 50% chance of being expressed more easily. How did we miss this? (Argh!!!)
Math is full of insights like that. Trig functions (sine, cosine, tan, etc.) are actually themselves percentages. A sine of .95 means you are at 95% of your maximum height (where the max is the hypotenuse). Sine and cosine are unitless numbers, and that's why the can be each other's derivatives (the percentage change of a percentage change...). So many things click! What else have we overlooked?
Anyway, really appreciate the note!
I've wasted so much of my life.
If we see each concept individually (continuous growth + rotation) we can deduce that we get something like "continuous rotation" or a circular orbit.
And if we use a complex number (a + bi, not purely imaginary i) we get a spiral pattern. Euler's Formula becomes "obvious" dare I say ("obvious after the greatest mathematician figured it out for us".)
"a% of b = b% of a?"
Seems like a bit of a cheat as it's too obvious. I did 25% of a 100 to keep it simple as we use quarters and dollars a lot. Also can visualize it as a rectangle or stack of boxes that I take one chunk out of. Maybe use word "whole" or phrase "all of" for 100% to make 2nd part more obvious. I took a chunk out of all of that stack. All of this chunk equals the chunk I took. See? Too obvious. Stay on the harder shit like e, trig, etc.
"Trig functions (sine, cosine, tan, etc.) are actually themselves percentages. A sine of .95 means you are at 95% of your maximum height (where the max is the hypotenuse)."
Boom! Excellent example. I understood it almost entirely in equations back when I did it. Outside of some examples with trees and stuff we rarely got to sit on what the terms mean. So, let's see if I follow that.
So, a sine of 0.95 is like putting a protractor on a picture of a right triangle and marking a chunk of it that goes 95% to the top of that? If I did it visually, that is. Looking up the other two's definitions I found someone already did the visual thing I was attempting although not quite there yet in presentation (see pictures w/ angles):
http://www.mathsisfun.com/sine-cosine-tangent.html
SO, if we do it visually on those pics, does the percentage the sine represents start as a line coming from bottom-right to hypoteneuse? And where do the other two start? Or is my intuition screwing with me? ;)
1) Yep, the regular percentage formula is pretty basic. Mostly, I like it because we've overlooked something that's been under our noses for years or decades. What else have we been missing?
2) For trig, check out:
http://betterexplained.com/static/articles/intuitive-trigono...
The traditional way of showing sine/cosine/tan (as on that page) leaves out the surrounding context where the percentage comes to life. Let me know if that link above clears thins up.
I have a question, but first a compliment :) I am enamored with this site. It validates a lot of my learning approaches and is helping me now when I was asking some of these 'how to learn' questions. Thank you!
The question: In my experience, 'generalized overview to specific' is too much required attention span for some listeners when trying to explain; Do you have a recommendation for being able to convey ideas to people that are tired/low attention in high stress situations? Example: office work and project work.
Referenced here: http://betterexplained.com/static/articles/adept-method/ "Start with a rough analogy and sharpen it until you’re covering the technical details."
Again, this site is amazing. If you don't get to the question, I am still happy to have found your site and hear how passionate you are about it. Thank you!
I'm not super experienced with lectures (mostly writing), but I think it works in office settings. Check out this talk from Simon Sinek:
https://www.youtube.com/watch?v=u4ZoJKF_VuA
People are usually motivated by the why, the mission, the story. Then you get into the what and how it's accomplished. With math, it can be similar: the "why" sets the stage. Humans in general prefer a narrative to a list of facts (Hacker News readers excepted :-).)
Keep up the good work :)
wow this is mind-blowing stuff! nice explanation too!
This bit me once!
The paper was written by Chris Lomont in 2003, when the gamedev.net community first started investigating the code (I thought Beyond3D did their investigation earlier, but it looks like they did it in 2004). There is a different paper that published a similar idea in 1997 (Floating Point Tricks), but its not the one linked. The code first appeared in the wild on comp.graphics.algorithms around 2002, apparently.
I remember this because I followed the discussion on Gamedev.net when Chris Lomont wrote his paper. Some years after, I found the Beyond3D articles where they traced the origin back to Ardent Computer.
https://www.gamedev.net/topic/139956-carmacks-sqrt/
That is exactly how I like to be introduced to any new concept, in special when I'm a complete newcomer to the field and can't relate it to previous ideas in it. Judging by the reactions to the BetterExplained site, other people agree with that.
For instance, learning about Gödel's theorems without having the Hilbert back-story explained. Learning about Leibniz's and Newton's calculus without learning about infinitesimals. And so on.
There's generally a reason _why_ concepts are born _when_ they are born. If you think about your maths classes, sometimes you're instructed to learn a method because it is useful and because it has real-world applications but it I don't think anybody is ever first taught algebraic geometry properly, if I may use that word. I don't think kids are taught the geometry is one thing and algebra is another and that different spaces can have different metrics. Am I making sense here? Do people see what I'm trying to get at?
Am I arguing for HADEPT? :) (Historical context, Analogy, Diagram, Example, Plain English explanation, and just then Technical definition)
(Yet we require students to internalize it without issue in a single lecture.)
Historical context is huge. I think I'm now stuck with this ADEPT name but maybe it fits into the Plain English portion :).
I did not know that! This makes total sense. I'd like to know more about that. When you think about it, only whole positive numbers make sense from a quantitative perspective. One thing, two things, three things, and so on. What's half-a-thing? Right? a half-a-thing is still just one thing, if you know what I mean. And how can no thing (nothing) be a number? And how can negative numbers be "numbers". It has always struck me that imaginary numbers are really badly named. Zero and the negative numbers are just as 'imaginary', equally unintuitive from a certain perspective.
I applaud what you're doing. I think there is a metric-tonne of dogma and bad naming schemes in the standard maths curriculum. Remember in software engineering they say that naming things is one of the hardest parts of the task? I think the same applies to maths, perhaps more so.
Ugh, tell me about the naming. "Imaginary numbers?" How about "rotated numbers". Nobody complains "Hey, when will I ever use the second dimension?". But "imaginary numbers" are setup to be eye-rolled.
----- > I'm no chemistry expert, but I can see a relationship to the primes. Chemical elements have properties based on their location in the periodic table of the elements:
Atoms in group 8A (Neon, Argon) are the noble gases. They don't react and won't blow up in your face. Atoms in group 4A (Carbon, Silicon) bond well. They're great building blocks for other elements. Atoms in group 1 (Sodium, Potassium, etc.) are very reactive. Drop 'em in water and see them explode. And in organic chemistry there's an idea of a functional group: several atoms can determine the class of the entire molecule. For example:
Alcohols are a certain carbon-hydrogen chain with an OH group at the end. Methanol, ethanol, and other alcohols share similar properties because of this OH functional group. Those are the basics, if I didn't mess it up. Now let's see what happens when we treat numbers like chemicals.
First Example: Guessing Evenness In general, an organic chemical contains carbon (not quite, but it's a good starting point). No matter what elements you mix together, if you never add any carbon then you can't create an organic compound. -----
Anyway, a single example doesn't negate your point. I love analogies in learning, but one has to be careful to pick analogies from a level of understanding (way) below what you're trying to explain. I guess kids are introduced to primes and chemistry at roughly the same age, but I'd have picked a non-academic analogy to explain an academic concept. But even then, it's tricky. For example I've been confused by my fair share of 'sports analogies' in secondary school books, for sports I happened not to have ever tried or knew the rules for. But really, the analogy should be completely supplemental, and if possible marked off in a side box that people can, but not should, read for better understanding if it helps them. I find many school books do this really well, but I haven't seen it translated to web content as much somehow. For example, on Evenness he'll continue by explaining how if you have a factor of 2 in your number (e.g. 24 = 2^3 * 3), then no matter what, the number is even, likening it to an organic chemical which contains carbon no matter what (though, noting a caveat without going into it). I don't think that analogy is very strong, it's confusing if you don't know chemistry, and it's pretty redundant if you do. In fact I'd personally be better of without it, and understood Kalid's normal explanation without issue. Yet I had to read through something about Atoms in group 4A and their properties, unsure whether I could just skip it or whether it was important to grasp some larger point. Anyway I was already familiar with primes but my 12 year old self probably would've been confused with the chemistry analogy.
For this specific example, I was writing to a high-school version of myself who wanted to really get an intuition for primes. What can we deduce from a prime factorization, are there other ways to think about it? (Number theory is studied later, even though numbers are introduced early.)
For a younger child, I'd probably use Lego or Minecraft to show how numbers can have "building blocks". And if you didn't use any Redstone as a building block, there won't be any Redstone in the result. (I.e., a number which never had the "2" building block added, will never be even.)
Thanks for the comment!
>>>>> Please donate to the Archive while you are at it! :-)
Google cache: http://webcache.googleusercontent.com/search?q=cache:http://...
And fwiw, I finally learned an intuitive understanding of radians by reading the Tau Manifesto (not on BetterExplained). It would be awesome if BetterExplained used tau instead of pi in the lesson on radians, but that's a minor nitpick for a very helpful set of lessons.
Why should BetterExplained use tau? Tau is a fairly niche idea that will only confuse way more than explain. That's a bit like saying BetterExplained should be written in Esperanto.
Personally while I agree that historically pi would have been better off if defined as 6.28... instead of 3.14... (And even told my students this long before tau manifesto came out) I don't feel that a factor of 2 warrants using up a whole other Greek letter.
(I'd be much more supportive of "tau" if instead of the Greek letter Tau Hartl chose a unique new symbol, much like physics uses hbar instead of h as he Planck Constant for radians instead of cycles).
I think it's way, way too easy for people on the terminal end of education to forget just how easy it is to get tripped up by very, very little things. Especially if they were themselves "good at math" and really aren't bothered by extra factors of 2 flying around. This is not the normal experience.
What confuses me is not the relevant issue. I've got $BOATLOADS of higher ed. I'm not the interesting case.
My actual only personal argument against Tao as Hartl proposed it is the use of a standard Greek letter. Really would have preferred he picked something that was more "backwards compatible", exactly as physicists did with hbar.
If you're for reducing confusion, do you think adding a new fundamental constant to the body of mathematics that is a Greek letter already used for countless other variables in history, will this cause confusion between old and new mathematical science texts and papers?
Asking students of science for any texts and papers they read whether tau is 2pi or some other variable, and keeping track across them, seems more confusing to me than just consistently using pi and extra factors of 2.
Not necessarily. Facts (in the sense of "anecdata") are simple observations.
The only make any sense against or in favor of something when we combine them with a theory why it is so.
OK, students faring better on exams when they are taught with tau is an argument in favor of tau only if we theorize (assume) that it was the tau/pi thing that made the difference.
Whereas it could just be that the second group of students was just better naturally (e.g. a good school vs a mediocre one), or just a chance outcome, due to the small sample we picked, or the teacher going for tau was also better at explaining and would have fared just as good if he also was the one to teach the pi group.
I'm not obligated to stop using my brain just because we have no SCIENCE! papers to work with. In fact, you can't. You have no choice. You must do your best with the data you have, because we do not have "science" for all the questions we encounter every day.
Given how standardized education is nowadays, as I complained in another comment, it isn't exactly a far-out theory that when the same person, in the same context, switches to using tau, and sees a difference in results, that it may have something to do with using the more mathematically-sensible concept for teaching.
In fact, I think quite the opposite. Its one of the great errors that people assume peer-reviewed papers are always correct, a lot of the time they're not.
We should be sceptical of people's claims when not supported by evidence. That doesn't mean we should ignore said claims, just that we shouldn't blindly accept them. This is the point where we should go and get some evidence, try to get a few teachers to use tau from the beginning and see if that helps.
Though I do agree that we shouldn't be scared of doing things just because we don't categorically know that it is right.
> Given how standardized education is nowadays, as I complained in another comment, it isn't exactly a far-out theory that when the same person, in the same context, switches to using tau, and sees a difference in results, that it may have something to do with using the more mathematically-sensible concept for teaching.
Or thinking about the same concept in different ways help you to understand the concept better. Its certainly something I believe has helped me a lot previously.
Note that this is just my opinion, but I have kids and this gives me an alternative viewpoint when they have questions. I wonder what my more math oriented friends will think about Tau
The question is whether you improve things by abandoning a universal standard with hundreds of years of support, momentum, and ginormous corpus of math/science literature all to save a 'simple' factor of two.
My primary gripe with Tau is that its symbolic representation is not "backwards compatible". Ie, Hartl proposed to use yet another Greek letter which is widely used elsewhere in all sorts of formulae and constants, instead of something that would cause far less ambiguity. For example, a new symbol, as physicists did by introducing hbar instead of h (Planck constant in radians instead of cycles).
If you were baking a pie (pi?) and someone told you how much of an ingredient to add per half pie, that would be less intuitive.
And no, I do not get confused by concepts like 90 degrees, but this is about inspiring intuition, not the ability to do calculations without error.
People don't instinctively understand why on earth they would use a measurement unit where '1' of the unit has no obvious purpose. What can you do with a 1 radian angle? It's a bit less than the angle in an equilateral triangle. It's too big for usefully measuring things - most angles you come across will be somewhere between 0 and 3 radians, which is just weird.
What's that, you don't use radians like that? You use 'fractions of pi' radians? But my calculator can only approximate pi to 9 decimal places, so how does that help me?
It's incredibly easy for people to intuit that a ninety degree angle is the same as a 30 degree angle plus a 60 degree angle. But it takes deeper intuition about fractions to figure out that a pi/2 radian angle is the same as a pi/6 radian angle plus a pi/3 radian angle. People are not used to units of measure that are typically denominated with rational fractions of irrational numbers.
My original point is that IMHO, mainstream radian educational materials should NOT use the lesser-known and niche 'tau' in lieu of the globally understood 'pi' (other than a possible footnote or sidebar). Ie - it can cause more confusion to the student, especially when connecting it back to the greater scope of their studies.
I suggest using pi, but with the number 2 before it. /s
I'm not even gonna touch the Tau vs. Pi debate because it's like a sports team rivalry, and not a rational discussion.
Nitpicking, but that is a self-defeating argument. If everybody used Tau, there would be no problem in teaching it.
Not that I disagree that doing the bootstrapping its usage and reaching there would be a problem, but all by itself is not a reason to not try changing it.
I'm a fan of Tau as an interesting simplifying concept but don't think mentions of pi should necessary be removed. My intuition is that radians are "distance traveled" as you go along a circle. Pi is the "neutral to max to neutral" distance and Tau is the full cycle (neutral, max, neutral, min, neutral). Ideally we just think about "distance" and context determines whether we want the journey to be "there and back" or just "there".
Two Questions:
1) What are your thoughts on interactive content like ExplainedVisually? I've been thinking about doing something similar for data structures / algorithm topics. How much of learning math concepts is exploratory vs learn-by-doing?
2) Are you running BetterExplained as a side business, or full-time? And if you're willing to share techniques and numbers for the entrepreneurial HN community, what are some things you've done to market it, monetize, etc and what were the results?
Thanks!
1) In general I like any efforts to explain things in new ways, and ExplainedVisually is great. My philosophy is that teaching is like humor. You want to make something funny, present it well, but not overexplain it. Let's people enjoy the joke. If you do too much handholding, you ruin the surprise and it's not fun to be told "Ok, the punchline is coming up...". It's not an exact science, but you get a nose for when something is truly illuminating vs. trying to chew your food for you.
2) BetterExplained is a side business. Happy to share numbers, etc. I have some earlier posts about ebook sales and techniques:
https://news.ycombinator.com/item?id=870015
I don't think I should be taken as a model of learning though (I write very infrequently) but thankfully math is evergreen. Many of my most popular articles are 5-7 years old.
I do want to dedicate more time to it. I realize I was afraid of ruining my love of learning by turning it into a profession, but I'm slowly coming to grips with it. That's one of the hardest parts for me actually, feeling I'll kill the golden goose by squeezing too hard.
I'll probably do a blog post / postmortem on marketing, numbers, etc. so keep an eye out :).
2) Thanks for the link! Definitely looking forward to your blog post/postmortem (don't let BetterExplained die!). Have you found learning to be less enjoyable by writing about it, or do you end up discovering a dozen other tidbits of math magic you want to share with everyone?
Two more questions if you don't mind :)
3) What are your thoughts on word problems?
For example, in most linear algebra textbooks, you are given matrix and are asked you to process it. Rarely are you given a word problem and are asked to think through the entire process (data and operations matrices setup --> processing --> meaningful end result).
4) For inspiration, what are you experiences explaining concepts in a cross-disciplinary manner?
When I was a student, I never understood why a concept is important. Homework problems were abstracted out of all real-world context to train for mechanical problem solving. Only now, after exploring data and writing algorithms in health, journalism and finance, have I finally been able to answer the question I always had as a kid: "why is this stuff useful?"
1) Exactly. It's like spoiling a movie.
2) Hah, the postmortem is more about the Reddit interaction (write up about went well / things I'd change). I'm planning on working on the site as long as I can. It's a life mission at this point.
Learning has stayed enjoyable, I tend to write insights that really strike me and get excited to share. (Which leads to me studying it more and figuring out new insights.)
When learning is drudgery (this happens often), I tend to let the topic sit a bit, and I don't write publicly about it. The articles on the blog are what genuinely get me excited about the topic. I do think there's usually a way to see a topic that makes it come alive.
As an example, I'm working on quaternions. I have a large list of notes here: http://aha.betterexplained.com/t/quaternion/267 and I'm slowly getting an intuition that I'll then work into an article.
3) I like word problems because they force us to ask the uncomfortable question of whether we can think with the material (vs. follow the steps). That said, this check of whether you're thinking or following steps can be accomplished with other types of questions too. For me the method isn't as important as the outcome.
4) Good question. So far, my audience is typically people who are self-motivated (i.e., they have a test, are curious, need homework help, etc.) vs. giving a talk to a potentially uninterested audience. (Not intentionally uninterested, but a volunteer audience.)
The primary motivations to learn are probably:
* practicality
* curiosity
* beauty / awe
* sense of accomplishment
Depending on your audience you'd have to tailor it. But I think beauty/awe is more powerful than we think. Even for a technical talk, I'd get people see the aha! moment. It's the sugar that helps the medicine go down.
http://betterexplained.com/articles/intuition-isnt-optional/
I really liked the explanation of sine as something that makes things 'circly'.
http://betterexplained.com/articles/intuitive-understanding-...
Does the site take external contributions?
:)
This guy's site should get more attention and probably an award. I'll be experimenting with less-math-inclined people to see how effective it is.
Oddly enough, I find betterexplained much more useful when I have already grasped the core concepts, because it does a great job of connecting to other concepts.
http://www.feynmanlectures.caltech.edu
(Put online by Michael Hartl, of TauDay fame among others.)
I haven't gone through them extensively but Feynman was a master of making the complex simple.
For starting from a clean slate, you might like:
http://betterexplained.com/static/articles/rethinking-arithm...
Once you can visualize the basic operations (add, subtract, multiply, divide), every new math operation becomes a lot easier (complex numbers = rotations, exponents = growth, combining them = orbiting a circle...).
Multiplication by 2 means "1 (starting point) times 2 (scaling)"
If we "do the opposite" we can take the inverse of the starting point or the scaling:
-1 (additive inverse of starting point) times 2 = -2
or reverse the operation
1 (same starting point) * 1/2 (scaling inverse) = 1/2
Of course, we assume the scaling term is what's being inverted, but it's important to think about the meaning. There's a hidden parameter for these operations and sometimes making it explicit can be helpful. (I.e., euler's formula, e^ix, is better seen as 1.0 * e^ix. That is, you are starting at 1.0 then doing a rotation.)
If you support innovation in education anymore, your first goal now is to get the central bureaucracy out of it. Until that happens, nothing else can change.
Many of us grew up in school districts prior to the Bush and Obama centralization regimes and quite literally were taught that Evolution is a Satanic lie by public (non-private non-religious) schools.
Besides, are we not discussing exactly a situation in which the central bureaucracy has overspecified an inferior education? Nothing about "panning science, attacking evolution, teaching the bible and religion" has anything to do with whether or not there is a central organization; one stray election and you'll get all those things coming out of the central organization too.
The very topic of this discussion is that an inferior education technique is being mandated right now. The casual default presumption that the centrally-mandated education is perfect is already falsified by the evidence.
If it really is a problem, an obvious solution is to just supply prepared materials that are based on this to the schools, and that will do a lot to make it available, if indeed it does work as well or better than the current materials.
A lot of people love learning new things. It's the mode of learning them that is offputting to so many.
Yes, ideally, every student would be motivated to learn for its own sake, but for an individual say, 8th grade, teacher to get a class of kids to that point is an enormous task. And yes, perhaps the entire education system should be revamped so that kids never lose motivation, but how to do that is hardly a solved problem.
Which means "Here's the deeper principles that motivate this problem" is going to have a huge uphill battle against "just tell us how to do the problems that are going to be on the test", or worse, "this has no relevancy to my life, so I'm going to tune out this entire class".
If they were taught to care about deeper principles from day one, I guarantee they would care about deeper principles by year 8.
I always see this excuse as a failure of the teacher, rather than the students. I believe students want to learn, and the non-motivation is usually a result of something that isn't so hard that the teacher can't get around it. I used to feel that way before and only gear my lessons towards the motivated ones (why should I waste my time on kids who dont want to learn?), but I realized that it was I who was not motivated enough to get through to those kids. The movie "Stand and Deliver" portrays what I'm trying to say in a really fun and useful way.
I think teachers need to be held to a higher standard and blaming their lack of success on students should be the last resort, after everything has been tried. Sure you'll get some really pathological cases where the student is absolutely unreachable, but I think that's so rare that it's not worth talking about.
I think it depends on overall context (socio-economic status of parents, what they see everyday, what other teachers do, what the policy of the school is in general, what their society at large perceives as success, etc.).
Empirical observation however -- and I've taught 2 different secondary schools myself although just for a couple of years -- tells us that some students are motivated and others are not. The teacher can try and nudge them towards the subject, but it wont do that much with most of the unmotivated students. My experience has been in what in the US you'd call "inner city" schools btw.
I'm not saying that this is an absolute rule, so individual counter-examples don't really negate this, unless they do indicate a reverse GENERAL trend. Sure, you could get a greatly motivated student even in a crappy school with crappy teachers, the question is how often would that happen.
I also don't agree that the teacher should be "held to a higher standard" (at least when meant to an extreme). Sure, there are indeed crappy teachers.
But students should come into school willing to learn and respecting the environment, something that's not always the case. It shouldn't be up to the teacher to do some special stunts to get the students basic attention -- instead of, say, playing mobile games on their smartphones, talking to each other loudly, even listening to music on headphones.
Also, you seem to be arguing from the perspective of what is rather than what ought to be. I'm arguing for a shift in perspective where teacher competence and enthusiasm and high expectations of students isn't something special or extra, but rather the norm.
Students should have a respect for school (and the role of the teacher) before any other kind of respect can be earned by the teacher as an individual.
Or, to put it another way, earning the students respect as a teacher is OK.
But having to earn the attention, and having to fight against students making noise, playing, ignoring the lesson etc, should not be the case.
>I think that when I show how deeply I love the topic I'm teaching, it rubs off on the students and they go along with it.
As I said, assuming the teacher is capable and passionate, it still depends on the students. Depending on the school/area/class/etc some students wouldn't care even if Alan Kay taught them programming and Richard Feynman did physics.
The idea that students will be captivated by a passionate and eloquent teacher doesn't always pan out in reality. A lot of times it's more like: https://www.youtube.com/watch?v=Bdf_XdDwc-o
>Also, you seem to be arguing from the perspective of what is rather than what ought to be.
Well, to get things to where it "ought to be" you should first tackle and work with "what is".
Article:
http://betterexplained.com/static/articles/a-quick-intuition...
Video a teacher made:
https://www.youtube.com/watch?v=zBYs6ppuMSk
I don't know anything about educational policy, classroom management, etc. I want to provide the best ingredients I can and let teachers make the best meal they can.
If you ever have a paid subscription, I'm buying :-)
No real paid subscriptions yet but it's a great idea =).
I asked what a "dot product" _meant_ once and was just told how it could be used to determine these other values. I later learnt what it actually was by reading some renderer library code.
I had a similar "aha" moment when I read how you described sine/cosine/etc as percentage values in relation to positions around the circle. I hate how I was taught maths since secondary school (when I was so confused at trig/pythag lessons but learnt the concepts very quickly when I was coding around with positions and angles in a game engine mod to get a position x units in front of a character instead of doing homework)
I really think this site will help me in the coming year, thank you!
1 * x^2 = 9
"What transformation (“times x”), when applied twice, will turn 1 into 9?"
That's such a great way of thinking about algebra. The "understood" 1 is equivalent to, e.g., an understood "YOU" in the English language.
e^ix
is really
1 * e^ix
That is, we're starting with 1.0 and doing a transformation (rotation) on it.
Static links to some favorites:
• Imaginary numbers: http://webcache.googleusercontent.com/search?q=cache:http://...
• Understanding e: http://webcache.googleusercontent.com/search?q=cache:http://...
• Intuitive Trig: http://webcache.googleusercontent.com/search?q=cache:http://...
• Calculus intro: http://webcache.googleusercontent.com/search?q=cache:http://...
• Sine waves: http://webcache.googleusercontent.com/search?q=cache:http://...
• Euler's Formula: http://webcache.googleusercontent.com/search?q=cache:http://...
• Linear Algebra: http://webcache.googleusercontent.com/search?q=cache:http://...
How do they know he encrypts everything?