YSK about betterexplained.com. It is a site that explains confusing mathematical ideas in a very intuitive way instead of "it just works"

there should be a bot that does this. linking url's that are mentioned in a thread title.

Thanks!

Vector calculus checks out

If the former algebra teacher / football coach at my high school ever made a math based website, this would be the underlying theme. The man could only do the math, but had no idea why it worked the way it did, only that it just works. He skipped around in the text book and cherry picked the stuff he knew how to do, and told us to "just do it like this, and you'll have the right answers."

If we had further questions, (and we all did), he invited us to join him on the football field sideline and he'd try to answer our questions while he marched up and down yelling at players on the field. – Uh...no thanks.

Everything I ever learned about math I learned outside of that class.

Wikipedia is very good with that.

rad

It's kind of like this with calculus, or atleast it was with me. We started with derivatives, when really, they're not the 'first step' of Calculus.

I got a little taste of that in the Diff Eq class I had to take. Learned where formulas came from that I have been using for years. Was pretty cool actually!

Some people, like vihart and Donald Knuth, have said that people don't hate, can't hate, math, just the way it was taught to them. While I think it may be a bit absolutist, I mostly believe it.

I was on this exact situation most of my life. I was atrocious at math throughout mandatory schooling (grade school and etc) and thought I was just bad at it and hated it. Then, in my second semester of college I started teaching myself as well as being educated by people who knew what they were doing. Now, with the addition of a few years after some time off for medical problems, I'm a math major and doing pretty well.

No really, I got to the state's Mathlympics last year yet couldn't progress any further due to the need to study for calc (which I did pass).

I told all of my friends "it's because Mathlympics is logical math, and school math is mindfuck". Only did I understood calculus after they changed 1 variable (can't remember which) that just made sense not to have.

This is one of the things that makes computer science so interesting to me, a massive chunk of the relevant maths and underlying workings were discovered in the last century, which means if you want to know how something was discovered or what problem was trying to be solved with it, you can probably find it straight away.

I have always made sure to figure or find out why concepts and equations work and I testify that it makes math so much more enjoyable.

(Kalid from BetterExplained here)

Thanks for the suggestion -- I have a giant backlog of topics I'd like to cover, group theory is definitely on there. I was a engineering/CS major so didn't get too much of the theoretical math track beyond the engineering courses. Hoping to correct that down the line.

One on group theory would be very useful for me right now.

Nothing on Lagrange. :(

Solving this calls for my boy Lagrange, but all in due time, all in due time: enjoy the gradient for now.
- Vector Calculus: Understanding the Gradient

He says all in due time, but the time never comes.

It takes a quarter of a semester to install the idea of rate of change on people's head when they take calculus 8 minutes isnt that bad.

Yeah, the video walks through the entire article (10 mins) but the 1-min part is the very first section. Here's the 1-minute version:

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We usually take shapes and formulas at face value, as a single pattern. Calculus gives us two superpowers to dig deeper:

• X-Ray Vision: You see the hidden pieces inside a pattern. You don't just see the tree, you know it's made of rings, with another growing as we speak.

• Time-Lapse Vision: You see the future path of an object laid out before you (cool, right?). "Hey, there's the moon. In the next few days it'll be rising and changing to a nice red color. I'll wait 6 days and take the perfect photo then."

So why is Calculus useful? Well, just imagine having X-Ray or Time-Lapse vision to use at will. That object over there, how was it put together? What will happen to it?

[For me, that would be way more useful than the abstract 'rate of change', which is essentially X-raying a path and seeing how big a single slice is.]

Argh, that attitude kills me. The funny thing is a "proof" is more a demonstration that something is correct, but not necessarily an explanation why. Most people want the why.

(For example, there have been proofs constructed by computers which take thousands of pages to show something is correct. It's true, but nothing was really understood.)

As an algebra teacher who moves at a quick pace just to cover half the material I am supposed to by the exams, which are not at the end of the year but in April for some reason... It's sad but true. There just isn't time for most of the fun stuff. I still try my best to make it fun and interesting, though.

For example, I always (and I mean 92% of the time ALWAYS) use Bob as the star of my examples. After a while, we started calling him my imaginary friend, but when we got to imaginary numbers, I explained that Bob was, in fact real... It was his evil twin Borg that ruled the land of imaginary numbers, which could only be accessed through his portal (you can use the key, which looked like the square root of -1).

Anyways, I'll stop there, but my point is, I want to be more fun but it's difficult... :'(

Oh man, I'm really sorry if that was the impression. I see myself as an excited/slightly strange friend who wants to chat about math over coffee.

Let me see if I can clarify. Over the years, we've had to constantly evolve our understanding of what a number is.

They started off with counting numbers (1, 2, 3, 4). Then we figured out that a number could represent nothing (zero -- pretty groundbreaking). And that numbers could be between numbers (fractions, decimals). And numbers could represent things less than nothing.

At each step, we objected because we were fixed in our understanding of what a number had to be ("I can't hold negative apples in my hand! Negatives don't exist!").

But with the right analogy (negatives are going backwards, or moving underground, or debt), we were able to wrap our heads around it.

For imaginaries, we have that same objection. I want to "pretend i exists" in the sense that "Ok, show me this hidden property of numbers I've been missing all along."

In this case, that property is direction: numbers can point in other directions instead of left/right on the number line (up/down, at other angles, etc). In essence, numbers can go 2d if we need them. Crazy concept right?

Sure, 99% of the time regular numbers are fine. But if you're doing geometry, or graphics, having 2d numbers might be really useful.

Appreciate the honest feedback. That video was my first ever and unfortunately shot in one take -- way too hard to fix up typos! The best you can do is little annotations.

Read the article and the context will make more sense. The main confusion I didn't make clear is that 2^x typically represents binary growth doubling [integer values of x] and e^x represents continuous growth.

Yes, you can use an intermediate value in 2^x, but if you do that, you're saying "This path is actually growing continuously by ln(2) = 69.3% and wow, it just happens to line up with an exact doubling on integer intervals." Which probably isn't what the person writing meant ("1, 2, 4, 8, 16..." as a sequence of growth).

Let me try to give an alternative explanation, since while /u/Darkphibre did provide one, I think it is lacking in some ways, and doesn't really give any explanation about what big O notation really means. Why is O(n) dominated by O(n² )? Why is O(100n) the same as O(0.001n)? Why is O(n log(n)) less than O(n² )? I'll try to make some of these things clear as well.

I'll still talk about things from a programming perspective, since I think it is more approachable than purely abstract usage.

Let's say we have a function, and it takes a list of size n as input. Our goal is to get a measure of how long it takes for that function to run, based on the length of the input list (i.e n).

Instead of measuring in terms of actual time taken, we're going to measure it in terms of the number of operations the computer has to make. An operation is generally some simple instruction; things like adding two numbers together, reading a value from memory, printing a line.. stuff like that.

Now we don't just care about how long it takes for small lists. We want n to scale arbitrarily high. It could be hundreds, thousands, millions of elements long.

Let me give an example of why this can be hard, and how big O helps. Let's say you have one function that prints a hundred lines regardless of what the input list is, while you might have another function that prints one line for each element of the input list.

The first function has to do 100 operations. It doesn't matter what n is, it's always 100 operations. So this function is O(100).

The second function has to do 1 operation for every item in the list. So it has to do n operations. So this function is O(n).

It's obvious in this case that if your list is size 10, the first function is slower. And the same for a list of size 50. But once you get to a list of size 100, they're the same speed. And with a list of size 1000, the second function is 10x faster.

What's important to note here is that not only is the second function faster than the first eventually, but past a certain point it is always faster, and if we pick a certain multiplicative factor, like 10x, or 100x, or 999999x, then by choosing an n large enough we can guarantee that the second one is that much faster for all n beyond a certain point. This is what we mean when we say that O(100) is dominated by O(n).

Now, why do we say that O(n) is the same as O(2n)? If you think of the explanation above, it's obvious. We can't just choose any multiplicative factor and say a function that is O(2n) takes that much longer than a function that is O(n). An O(2n) function will never be 10x faster than an O(n) function, it will always be just twice as fast. This is what we mean when we say that O(n) is the same as O(2n); neither one dominates the other.

And likewise, O(1) is the same as O(100), since 100 is always exactly 100x faster than 1, but it will never be 1000x or 99999x faster. This is why we say O(1) is the same as O(100), and use them interchangeably.

Since O(n), O(2n), O(3n), O(9.5n), O(0.0001n) are all considered the same, most people would just refer to all of them as O(n). Similarly O(1), O(2), O(100), etc. would all commonly just be called O(1).

Big O explained from a phone:

You could calculate complexity by adding up the time it takes for each line of code.

Then you notice that some lines completely drown out the other lines. Those lines are calling functions that do stuff, or are in loops that repeat over and over again.

The simple lines are O complexity. Linear... If you have 10 lines in one function and 100 in another, you have 10O and 100O Not super accurate, but we are simplifying complexity to do performance comparisons at a glance (finding the largest/slowest part of the code). 100O is the slower of the two.

Then some methods, say a sort, have a loop (with 10O of helper code wrapping it), so we have it taking 10O+O² or or a few loops next to each other with 10O+3O^2,, or several loops inside of loops with 10O+2O^5. It's clear that second one is slower (grows faster the more it has to do) than the first, and the last is going to get slow very fast.

In programing the 10O is just noise as soon as larger terms show up (and they do), so we only keep the largest term from inside the method.

Aaaaand, I just realized there's a Big-O in math (which is technically the same/similar, so I'll leave this in the hopes it helps).

...no? As somebody who studies higher math (theoretical physicist), rote memorization is a technique used less often in higher mathematics than spot welding.

Or, to put it another way, is to need to use it. Even simple things like why we multiply instead of adding large numbers of things. These are things I teach my 6 yr old son. I'm also enforcing an understanding of units because for almost all simple math if you know the units you can derive any number of equations that accurately describe a relationship.

This mindset could actually be detrimental to a budding student. Using mathematics is nice, but even a basic understanding of the processes behind is better as it allows you to tackle a wider range of problems. This solid foundation also allows you to easily move up to more advanced topics which would've been harder if you've been taught the shortcuts/approximations/generalizations.

This is one of the main issues addressed by the site, let us go beyond "using" and try to have a deeper understanding of often hand-waived concepts.

Check out the linear algebra article:

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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.

Let me know if I can help. My main intuition is that just like a smoothie can be reverse-engineered into a recipe, a function can be reverse-engineered into circular components.

The math gets tricky because we build circles using Euler's Formula, so there's a lot of background prereqs from that end. But happy to answer any questions.

The context is that it works well compared to trigonometric operations, you can't compare it to "hey kids lets just define sqrt(-1) to i and here are some of its properties".

Great insight. When starting the site, I wrote for a younger version of myself, assuming similar backgrounds/contexts. Explanations need to be tailored to the audience and there are several ways to approach the same concept.

I agree on giving things physical meanings, we often get so abstract that we don't have an everyday example people can get their heads around.

I'd really like to cover them down the line. I'm hoping to do more on that topic as well as the LaPlace Transform.