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tberton

Math Thread of Fancy Counting

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Alright, because we've been talking about it in the Random Thought thread, I figured a thread about math was worthwhile. Feel free to share intersting facts, proofs, puzzles and questions!

 

I'll get us started with two YouTube series - Vi Hart's math videos and Numberphile from Brady Haran. Both are really good at explaining neat stuff about numbers.

 

I'd particularly like to mention

 from Vi Hart, which is where this thread gets its name. It's ostensibly about logarithms, but really about the fundamentals of algebra. Vi's style is a bit overwrought at times, but I highly recommend that video, especially if you think you just don't "get math," because it does a really good job of explaining how things like multiplication and roots are just different ways of showing how numbers relate to each other.

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Your link to Vi Hart's video goes to the Numberphile channel FYI.

 

Cool stuff about Infinity:

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Whoops, thanks for that.

 

Infinity is one of my favourite math things. I read a book when I was 17 called The Infinite Book and it was so interesting. Cantor's Diagonal Proof was the first time my mind was blown by math.

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 Cantor's Diagonal Proof was the first time my mind was blown by math.

 

Ha! I was already preparing to respond saying that was one of my favourite things too.

I think this is the relevant video:

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I wish I still remembered how to do differential equations in any meaningful capacity from college. Math, like anything else, atrophies if not used.

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0.99999... = 1

 

People always flip out over this for some reason. It is one of the simplest things to prove and yet people go through crazy mental gymnastics to continue to convince themselves that it isn't true.

 

I love math, math is great. I was never especially good at it or cared for it all that much until I had a decent teacher in high school that was able to give me just the right perspective for all the right things to click. It was a fairly small high school too without any kind of advanced math class so my senior year, me and another girl who had taken all of the math classes the school had to offer were put in the back corner of the room for one of the lower level classes and spent the entire year going through all of the chapters and exercises in a calculus book. As great as that math teacher was, he was clueless when it came to calculus so we basically had to teach ourselves by going through the book and helping each other out with concepts that the other was struggling with. It was surprisingly effective and we were able to get through all of the standard calc 1 content and started delving a little bit into calc 2 and integration by parts.

 

Hearkening back to the random thought thread, I also find it pretty frustrating to constantly hear people claim they are bad at math. I've found it to be one of the easier subjects to learn just by going through a math book and reading the concepts, then going through some sample problems with solutions to see the various ways to tackle the problem and learn any underlying fundamental patterns. If people like puzzles, they should love math because it's the best puzzle.

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0.99999... = 1

 

This is a great one, because it gets at some of the stuff mentioned upthread - ideas about infinity and what "=" actually means. I think that's a big misunderstanding in math. People think that "equals" in math means "becomes" or "transformed".  But what it actually means - and this is what Vi illustrates so well in that video - is that the two things on either side of "=" are the same thing. Just different ways of representing the same underlying idea. Anything you do to one side has the exact same effect on the other. So .99999... = 1 because anything you do to .99999... does the exact same thing to 1.

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This is a great one, because it gets at some of the stuff mentioned upthread - ideas about infinity and what "=" actually means. I think that's a big misunderstanding in math. People think that "equals" in math means "becomes" or "transformed".  But what it actually means - and this is what Vi illustrates so well in that video - is that the two things on either side of "=" are the same thing. Just different ways of representing the same underlying idea. Anything you do to one side has the exact same effect on the other. So .99999... = 1 because anything you do to .99999... does the exact same thing to 1.

 

It's just crazy though when you try to show this to someone. Show them 1/3 = 0.333... and they have no problem. But as soon as you take it one step further and tell them that if those two are equal then 3*(1/3) = 3*(0.333...) becomes 1 = 0.999...  they just can't handle it. I had one super stubborn person sit there and try to explain to me that even though 0.999... goes out to infinity, if you took 1 - 0.999... you would get a super tiny number like 0.0000....1. I guess infinity is just one of those concepts that is hard to accept for a lot of people and they try to redefine what '=' means to make things work the way they want them to.

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Hearkening back to the random thought thread, I also find it pretty frustrating to constantly hear people claim they are bad at math. I've found it to be one of the easier subjects to learn just by going through a math book and reading the concepts, then going through some sample problems with solutions to see the various ways to tackle the problem and learn any underlying fundamental patterns. If people like puzzles, they should love math because it's the best puzzle.

 

Some people have a knack for reading math books, but they can be really hard. The main reason, I think, for this is that the person who wrote the book by necessity needed to know more math than the book is teaching. So when you read a high school calculus book, you're learning something for the first time whereas a team of people with PhDs and years of experience teaching wrote this book to be mathematically rigorous in a way that obscure meanings to the lay person. The authors knew how to write a mathematically rigorous proof, they knew set notation, etc.

 

I think it's easy to see written math and want to skim and skip over equations. It's something I have to actively fight against whenever I go back to look over a text book.

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Yeah, that's why I find videos for this kind of thing to be much easier.

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Some people have a knack for reading math books, but they can be really hard. The main reason, I think, for this is that the person who wrote the book by necessity needed to know more math than the book is teaching. So when you read a high school calculus book, you're learning something for the first time whereas a team of people with PhDs and years of experience teaching wrote this book to be mathematically rigorous in a way that obscure meanings to the lay person. The authors knew how to write a mathematically rigorous proof, they knew set notation, etc.

 

I think it's easy to see written math and want to skim and skip over equations. It's something I have to actively fight against whenever I go back to look over a text book.

People often say that you should read math books with a pen. It annoys me a bit ("proof is left as an exercise for the reader"), but it's true. You're not meant to take things at face value, you need to read it half a dozen times or more, and work through the implied details on paper yourself. I took some fairly advanced math classes last fall and dreaded them for that reason. Ended up putting in the work (which turned out to be just as much as I dreaded) and surprised myself with some really good grades. Now I sort of regret not taking more. I know if I did/had I'd hate it in the moment, but appreciate it once I got through it. Can't stay in school forever though.

The rest of this post belongs in the drunk thread

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Yeah, I mean reading mathematical texts requires "active reading" styles that most other subjects don't require. You can't sit back and read a math book, you have to actively participate. I always got way more out of my math lectures than text books, as a result. I actively engaged with listening to a lecture in a way that is draining with a textbook.

 

If I could just take one advanced math class a semester, just for funsies I would. If I lived near my alma mater, I would definitely think about taking them up on their 1 free audit per semester where alums are allowed to attend lectures.

I get wistful when I look at my abstract algebra or numerical analysis books.

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Yeah, I mean reading mathematical texts requires "active reading" styles that most other subjects don't require. You can't sit back and read a math book, you have to actively participate. I always got way more out of my math lectures than text books, as a result. I actively engaged with listening to a lecture in a way that is draining with a textbook.

 

If I could just take one advanced math class a semester, just for funsies I would. If I lived near my alma mater, I would definitely think about taking them up on their 1 free audit per semester where alums are allowed to attend lectures.

I get wistful when I look at my abstract algebra or numerical analysis books.

 

I'm the exact opposite. I couldn't help but fall asleep during lectures as they were usually really dry and mostly presenting the same material that was in the book. I always felt like I internalized everything better by reading through the material, sometimes multiple times, then reading through several sample problems and their solutions to get the general pattern for how those types of problems can be solved, and then going through the rest of the problems in the assignment on my own. I would usually save the problems I got stuck on until the end, at which point I had done enough problems to figure out what I was missing, or just go back through the material and read it a little more closely to get the hint I needed.

 

A lot of it usually came down to figuring out what trigonometric proof you needed to apply to transform whatever weird sine/cosine statement into something more digestible. That was always the part of math that frustrated me because a lot of it came down to how well you had memorized all of these proofs so you could recognize where an equation could be simplified and apply them appropriately. 

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Trig equality proofs in pre-calc were my jam and my favorite part of my math class that year.

Like I said, some people have a knack for reading math textbooks, I am not one of those people.

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I remember learning once in a discrete math class the proof for why you can determine if a number is divisible by three by adding up the digits and seeing if that sum is divisible by three.  That proof is probably still one of my favorites.

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I remember learning once in a discrete math class the proof for why you can determine if a number is divisible by three by adding up the digits and seeing if that sum is divisible by three.  That proof is probably still one of my favorites.

 

Math is fucking weird. 

 

And that's about the extent of how much I can contribute to this thread.  God bless all of you who are good at math.  I actually aced every math class I ever took, but if you asked me even the most basic question weeks later, I couldn't answer.  No retention at all. 

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I remember learning once in a discrete math class the proof for why you can determine if a number is divisible by three by adding up the digits and seeing if that sum is divisible by three.  That proof is probably still one of my favorites.

 

I'm not actually familiar this one. Mind sharing?

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I'm not actually familiar this one. Mind sharing?

Say you have a number. You can express that number as the sum of each digit times a power of ten. For example,

1234 = 1(1000) + 2(100) + 3(10) + 4(1)

You can also express this as

1234 = 1(999 + 1) + 2(99 + 1) + 3(9 + 1) + 4(1)

Which can be rewritten as

1234 = 1(999) + 2(99) + 3(9) + 1(1) + 2(1) + 3(1) + 4(1)

If you add a number divisible by 3 to another number divisible by 3 then the sum is clearly divisible by 3. All the terms with 9s in them are obviously divisible by 3. This just leaves the individual digits of the number. And obviously this expands out to any number. I think the formal proof uses the modulus of the terms but the principle is the same.

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This thing continues to be enchantingly absurd to me:

It's the epitome of bonkers maths stuff that (apparently) does actually work.

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That, truly, is some fancy counting.

 

I really forgot too much of this stuff since the one semester of advanced math I took at college. No more proof by induction for this guy.

 

Although I recently ran into a problem in my amateur game dev stuff where it was nice to be able to call back some of this stuff. I wanted to spawn a ball on a 2D plane that would move in a random direction, so I assigned a random value between 1 and -1 to x and y of its spawn vector. But then it spawns at different speeds, of course, the resulting vector would be much shorter if the values end up being close to 0. And I just had to normalize it to prevent that.

 

The eventual process was: add together simple math functions to perform a complex thing -> realize there is a prebuilt function for the complex thing -> use that instead.

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My days of Complex Analysis and Stochastic Calculus are mostly behind me, I think. Now I basically use relatively uncomplicated maths and some slightly more complicated stats. There's definitely beauty in having to come up with proofs for results you've never seen before based on things you have seen before and I was good enough doing that, but applied maths if definitely much more my thing.

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I got the same answer James! That is indeed a cool puzzle.

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That -1/12th thing smells like BS, but then I don't know if such averaging of infinite series is really used in physics. I would think its not used in pure math at least?

Also what if you don't shift the second S in 2S?

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