Hey, Kalid from BetterExplained here, thanks! My general philosophy is to be really, really honest with myself if I understood something. It's ok (really!) to admit when we haven't fully understood a concept.

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!

The percentages detail is very cool, never thought about it! I also think of sin/cos as percentages, but there's one small detail that can be easy to miss. While sine and cosine generate "percentages" between -1 and 1 they don't generate them uniformly, because their derivative is not a linear function! If we were to sample sine or cosine between 0 and PI at uniform intervals we would see that the percentages would "cluster" around the peaks and valleys because that's where the function "decelerates" (or the derivative decreases).

This bit me once!

Yes! That's a great point. Similar, if you ask people to pick a random point in a circle (random angle, random radius length) you don't get the random distribution you were expecting.

Kalid,

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!

Hey! Thanks for the compliment, it really means a lot when the site resonates.

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 :-).)

Thank you sir! That is revealing and informative.

Keep up the good work :)

Thanks, I appreciate it! :)

Euler's Identity is often explained in terms of mysterious language as well. Like, e^i*pi = -1 makes a lot more sense when you consider the exponential function as "proportional growth", i as "sideways", and pi as "halfway around a circle" - all Euler's Identity does is rephrase "going halfway around a circle" in terms of continuous and proportional sideways growth.

Exactly! I hate the Taylor Series explanation of Euler's formula. "Oh, just take the most analytic definitions of e^x, sin(x), cos(x), mix-n-match, and it works!" All symbols, no intuition.

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".)

What?!

I've wasted so much of my life.

Me too. I was even a math geek but didn't understand most of it. I knew all the rules for equations plus heuristics for how to apply most of them. I can only imagine how much more effective I'd have been if there was a constant, parallel learning process focusing on intuitive understanding of all foundational concepts in various math branches.

I think I left a palm-print on my forehead when I saw that.

The trig part was intuitively obvious for me, but the % part was wow!

I love this approach to learning. It's something I've been really working to take on in my studies (and my own projects).

Awesome. I think it's a general principle (blurry-to-sharp) that works for many fields.

> a% of b = b% of a?

wow this is mind-blowing stuff! nice explanation too!

Paid you a great compliment in my main comment. This...

"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? ;)

Just saw your earlier comment, thank you!

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.

the trig function trick is pretty neat! but i wonder if i only think that because i know trig already, and would've gotten more confused with more analogies

Yeah, to be honest, I'm writing for someone who saw the textbook definition of trig but didn't have it click. I'm not sure how it would work as a sole treatment. (However, I suspect the vast majority of people reading the article are learning trig in school alongside an existing lesson, vs. randomly reading about it with no context. The people who google "trig" for fun having never heard of it... are strange beasts.)

This is fascinating! I wanted to know more about the history of that function at id, which lead me to this beyond3d post (which was the topic of the reddit post that BetterExplained listed as a source):

https://web.archive.org/web/20070208003255/http://www.beyond...

A slight correction: the technique existed long before the paper, so I wouldn't exactly say that its the "original" paper.

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/

http://www.beyond3d.com/content/articles/8/

http://www.beyond3d.com/content/articles/15/

Here's the best explanation I've seen: http://h14s.p5r.org/2012/09/0x5f3759df.html

Why did you use the archive.org link?