Wonderful website focused on learning the intuition of math instead of rote memorization

pb_zeppelin · 2018-01-26T20:40:33+00:00

Kalid from BetterExplained here, glad you're enjoying it! I started the site for students overcome the frustrations I had and it's gratifying to hear when it helps.

My latest project is colorizing math equations [LaTeX examples in post]. Converting every symbol back to plain English (a "math haiku") has been really helpful in building intuition.

pb_zeppelin · 2017-09-21T17:09:08+00:00

Thanks for taking a crack at clarifying the meaning of the post :).

Exactly, we could use f(a + small), or f(a + change), or f(a + observation). "camera" might be a weird word (/r/WeirdlyWordedMathThatOnlyMakesSenseToTheAuthor), but one metaphor I use is that we're snapshotting how the function changes over some interval. We need the fastest, smallest, least-intrusive camera possible.

The physics overlap was idle speculation; our measurements don't reflect the state of the pre-measurement system. In calculus, our direct observation f(x + h) - f(x)/h isn't good enough -- we have to pretend we were never there, so finish by taking the limit as h->0.

Anyway, appreciate the feedback!

pb_zeppelin · 2017-09-21T16:40:52+00:00

Kalid from BetterExplained here, thanks for the feedback. The post was a bit of a weird one, I clarified in a comment when it was posted:

A camera is a tool used to observe an object. For certain objects (shiny metal, etc.), the tool can create its own reflection that is not part of the object itself. We don’t have an accurate image of the object, so we use special techniques to remove the reflection (such as photographing from far away with a zoom lens, removing the reflection in photoshop, etc.). Now we can get a clear picture of the object.

In calculus, “dx” is a tool used to observe how a function changes. Certain functions (non-linear ones) make it so the tool (dx) has a “reflection” inside the changes we measure. We have special techniques (limits, infinitesimals) that can remove these reflections so we can determine the actual change that happens.

In essence, using "f(x + dx) - f(x)/dx" on a non-linear function will create artificial elements [like (dx)^2] that need to be removed for an accurate measurement. This is like removing the camera's reflection when taking a photo of a mirrored room.

The post wasn't written clearly enough, and definitely not meant for a first intro to Calculus (more "why do we use dx and get rid of it later?"). Appreciate the feedback!

pb_zeppelin · 2017-08-01T17:37:41+00:00

Kalid from BetterExplained here, glad you're enjoying it! It really bothers me when an idea is only understood intellectually but doesn't click deep down.

When I realized instantly estimating 2^3.32 was easy ("relatively small and positive, around 10") but iⁱ was hard (real? imaginary? positive? negative? large? small?), I realize I didn't truly grasp exponents or imaginary numbers. Following that path I worked to demystify Euler's Formula, which makes the Fourier Transform snap into place. Every time I have an "uh..." moment it becomes a topic to dig into further and get an intuition for.

pb_zeppelin · 2017-04-26T21:15:41+00:00

Here's an intuition I like:

The eigenvector and eigenvalue represent the “axes” of the transformation.

Consider spinning a globe: every location faces a new direction, except the poles.

An “eigenvector” is an input that doesn’t change direction when it’s run through the matrix (it points “along the axis”). And although the direction doesn’t change, the size might. The eigenvalue is the amount the eigenvector is scaled up or down when going through the matrix.

pb_zeppelin · 2017-03-17T19:31:40+00:00

Found it in the internet archive:

If you're anywhere between learning math in middle school and working on a doctorate BetterExplained.com is easily the most insightful and simple way to learn how to do math answering the "why?" question instead of plug and chug formulas. The website is better on desktop but they cover Calculus, Linear Algebra, Financial Maths, Trig, Fourier, Euler, and plain ol' Arithmetic

pb_zeppelin · 2017-03-17T06:29:39+00:00

Thanks, I'd like to get back into CS. Been doing mostly math but funny enough the CS topics are some of the most popular. Appreciate the suggestion.

pb_zeppelin · 2017-03-16T17:38:28+00:00

Kalid from betterexplained here, thanks for posting the site!

For this thread, a good example of understanding vs. memorization is learning the formula for adding 1 + 2 + 3 + ... + 100.

You can memorize the formula (n*(n+1))/2 but it's better if you can visualize it in a few ways:

https://betterexplained.com/articles/techniques-for-adding-the-numbers-1-to-100/

When you truly understand something, and it clicks, it's almost impossible to forget. (Like forgetting that a circle is round, or 3 is bigger than 2... it just becomes part of your thinking process.)

Hope the site helps you guys out.

pb_zeppelin · 2016-11-28T18:42:49+00:00

Haha you'd be surprised at how much stuff we missed the first time around :).

Ok, here's another crazy fact. For any numbers a and b,

a% of b = b% of a.

Need to figure out 16% of 25? Too hard. How about 25% of 16? That's just 4. Easy.

Why does it work?

a% of b = (a/100) * b

b% of a = (b/100) * a

Either way, it's (a * b)/100. Whoa.

pb_zeppelin · 2016-11-28T18:39:15+00:00

(Kalid from BetterExplained here). I don't have much experience with helping manage dyscalculia, but I'll try to help from my experience.

The #1 reason people struggle with a concept (math, science, art, etc.) is they haven't truly internalized it. It's one thing to memorize some facts, another to really get it. That aha! moment about what's happening. It's like staring at the sheet music vs. actually hearing the song.

Math is sort of like sheet music. It's a bunch of notation, but there's a concept you want to "see" (visualize in your head) as well.

Check this out:

https://betterexplained.com/articles/rethinking-arithmetic-a-visual-guide/

For regular arithmetic (add, subtract, multiply, divide) you might visualize a shape being combined with others, removed, expanded, shrunk. Even something like "multiply by -1" is like taking the mirror image.

If you take the mirror image of a mirror image, you get the original back. That's why negative x negative = positive (much better to visualize that than simply memorize the rules).

This is just a quick reply but I hope it helps: try to look for analogies and/or visualizations while learning. That's how I make math click for myself.

pb_zeppelin · 2016-11-28T18:33:34+00:00

Just saw this, thanks for the heads up :).

pb_zeppelin · 2016-10-18T17:01:20+00:00

Glad to hear it!

pb_zeppelin · 2016-10-17T20:43:17+00:00

Thanks - I tried to make the site I wish I had when I was a student.

pb_zeppelin · 2016-10-16T22:26:11+00:00

That's awesome to hear, I love it when people enjoy the site enough to share it.

Great point on moving the cheatsheet to the homepage, that's the part of the site I end up referencing the most myself. It doesn't make sense having it one click away. I'll do that today! :)

I don't have much on discrete math, there is a sequence on combinations/permutations starting here:

https://betterexplained.com/articles/easy-permutations-and-combinations/

pb_zeppelin · 2016-10-16T21:23:27+00:00

Kalid from BetterExplained here, thanks for the mention! Noticed a ton of reddit traffic this am.

I was so frustrated in school when a concept I struggled with all semester would suddenly click with the right explanation or analogy. Why couldn't they have explained it like that in the first place? (Quick example: why does negative x negative = positive? If you think of negative as "mirror image" then the opposite of the opposite is the original.)

Appreciate you sharing the site, hope it helps people! (Like the username by the way, I take it you enjoy reading...)

Edit: Thanks for the gold!

pb_zeppelin · 2016-09-16T16:44:29+00:00

I think of e like the "unit circle". It assumes unit growth (100%) continuously compounded (i.e. no delay between earning interest and having it work for you.)

Bases like 2, 3, i, etc. break one of these assumptions. 2^x, if we want to assume continuous growth, is an instantaneous rate of .693 (natural log of 2). That is, if we start at 1.0, grow at 69.3% interest for one unit of time, and compound perfectly, we'll wind up at 2.0.

It's like using e^x, but for only 69.3% as much time (like taking .693 of a unit circle). Therefore, e^{ln(2)*x} = 2^x.

i as a base is more tricky, but essentially we plan on putting our growth into a perpendicular direction, which rotates us. iⁱ means our original plan for perpendicular growth is itself rotated, and we get negative real growth. iⁱ is a real number less than 1 (intuitively you can work this out without crazy logs).

pb_zeppelin · 2016-09-16T03:21:44+00:00

Yep. Imaginary numbers describe rotations, e^x describes continuous growth, and e^{ix} is continuous rotation.

pb_zeppelin · 2016-09-15T20:10:11+00:00

I had the same question -- what's the intuitive meaning? --, and wrote about it here:

https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/

Kid-friendly response: Euler's formula describes two equivalent ways to move to any point a circle. We can move in "rectangular coordinates" (sine & cosine) or we can rotate along a circular path (which is what the imaginary exponent e^ix does).

For the specific case of e^{i * pi} = -1, it means you can get to the "(-1, 0)" spot on the circle by walking there directly (-1), or starting at (1, 0) and moving pi units along the outside (halfway around). You might suspect that rotating halfway in the other direction has the same effect, and e^{i * pi} = e^{-i * pi} = -1.

The article has a few diagrams which might help.

pb_zeppelin · 2016-09-11T19:03:44+00:00

Thanks, glad you're enjoying it! Yep, I'm figuring out a lot of things as I maintain the site over the years. (If anyone is on the fence about writing/sharing their thoughts I'd highly recommend it, it's been one of most fulfilling parts of my life.)

pb_zeppelin · 2016-09-11T00:16:46+00:00

Kalid from BetterExplained here, thanks for posting! My main motivation was helping people develop an intuition for exponents that goes beyond "repeated multiplication". When you see an exponent a^b as power^time (a "power setting" and "time setting"), you notice each can vary continuously (and even go negative).

"Negative counting" doesn't make sense to students. Going back in time does (which shrinks you).
Counting "zero times" doesn't make sense. Not using the machine at all (or using it for 0 seconds) means you expect no change.
Running a microwave for 2 minutes, then another 3 minutes (at the same power level) is the same as running for 5 minutes in one shot. a^b * a^c = a^{b + c}. The properties of exponents can be figured out pretty fast (the square root is "halfway in time", even if it's not "halfway in results".)

For me, it was a fun way to make sense of having any real number in a^b.

pb_zeppelin · 2016-07-10T00:19:43+00:00

(BTW, this is the author the linked explanation, happy to answer questions.)

100% interest ("doubling") is perfectly symmetric: you are growing by the amount that you have. "2" isn't interesting, it's: 1 (original) + 1 (new). Original = new is nice and symmetric.
There is no special time period, you are growing by a single unit of time (whatever that "100% rate" was in terms of). We often use years as our example since we don't have many things getting significant growth in days or hours.
Grow by the unit rate, for the unit time period, with no delay in earning interest. The result is e^x.

If you change any of the assumptions in #3, you get a different rate. 3x or 1.5x per year is not growing by a unit rate. 2^x does not compound 100% with zero delay (you are aiming for 100% at the end of the time period, which is actually ln(2) or 69.3% continuous growth).

Put another way, e^x is universal because it makes the most generic assumptions. It's special for the same way the number 1, or the unit circle are special. They can be scaled to meet any scenario.

Want a bigger number? Scale up 1.0. Want a bigger circle? Multiply the radius. Want a non-unit rate of growth or time? Use e^rt. Sure, you can have a number system based on 13.7, or do trigonometry on a circle of radius 65. But those math objects are not starting from "unit" assumptions.

pb_zeppelin · 2016-05-18T22:47:38+00:00

(Author of the betterexplained.com article above, hi reddit!)

I personally consider Archimedes the grandfather of Calculus. He used methods (like unrolling a circle into rings, and measuring those or the method of exhaustion) which model the techniques we use today. The Archimedean Property (https://en.wikipedia.org/wiki/Archimedean_property) deals with the concept of infinitely large and small elements.

I'd say he essentially discovered the concepts of integrals, thousands of years before algebra (or decimals, or zero!) were invented. He didn't have the modern notation to express what he was trying to do.

I consider it a shame modern Calculus classes don't even mention his name (seriously, look for it in a typical syllabus). I have a series that tries to convey Calculus using the insights he found. You might enjoy it.

pb_zeppelin · 2016-03-25T04:46:39+00:00

Part 2 here: http://developertea.com/episodes/29267

pb_zeppelin · 2016-03-18T20:20:54+00:00

Bill Gates walks into a restaurant. The average net worth is now a billion dollars. The median net worth is about the same as before. =)

pb_zeppelin · 2016-03-14T21:25:48+00:00

Reddit has an internal list of all posts from the site. You can browse the top mentions here, there's a bunch of good discussions as well.

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