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PostPosted: Mon Jul 10, 2017 3:12 am 
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I was good at math, getting top grades all through uni. It's definitely come in handy tons of times, particularly when optimizing things. In the 3d world it's a lot more common than in 2d, but still useful. Synthesizing curves like rainwarrior said, finding faster ways to do something, etc.

I think it's similar to CS education vs a self-taught programmer. With a good math background, you can jump straight to the best way, and not have to guess or research (kinda like tokumaru's logarithm surprise recently. Sorry for calling you out specifically, but that came to mind).

A self-taught programmer will know how to get something done, but he won't know the area of solutions, O notation, what is best used where, the tradeoffs. When getting my software engineering degree, I recognized multiple times things that I previously had no idea about, even though I could solve the issue. I had no idea some kinds of solutions even existed, let alone when I should use what. I only tried to use my hammers even when a screw appeared in front.


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PostPosted: Mon Jul 10, 2017 4:55 am 
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Gilbert wrote:
even for maths I sucked at arithmetic badly

I suck at arithmetic very hard. I cannot stand when people ask "how many is 7 times 56" and I have to use a calculator and they say "I thought you were skilled at math ?". This is NOT math. Math is about solving equations, geometry problems, etc... This is arithmetic, and it is *NOT* math.


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When I look at stuff like this (which flies completely over my head) I realize I suck at math. I barely managed to pass calculus in college.

Same here, however the calculus level required was very high so barely passing is already an acomplishement. The uni I was in is reputed to be extremely severe with math, basically no matter what you study you have to be very good at math or they'll make you fail. The grading system even have a "math" average and a "non-math" average, and you needed to pass both. My math average was always very close to the minimum for passing.

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I graduated my university degree with a "not very bad" grade, and even continued to study for a master degree in maths afterwards.

I was an expert in the art of barely getting the grade required to pass. That until the Master's degree, included.

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Wikipedia is also generally a very bad place to learn anything to do with math.

Yes. Wikipedia is a good source of info for many things but I noticed it sucks HARD in several area, and definitely suck when it comes to either math or religion, or health for that matter. The content is both extremely incomplete and inacessible due to using specialist terminology which is not available to random person looking things up.

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Really simple eh? :lol: like...why does it work? I've kind of gathered it's an infinite sum of derivatives of a function (with other terms in the formula I don't yet understand) which approximates the shape of the actual function, so if you can keep computing the derivative of a function you can get as much precision as you want for the function you're going for, right?

Seems like you understood it fully. The concept is extremely simple, you're re-building a function which for a said point have the same derivate, 2nd derivate, 3rd derivate, etc... as another function (in this case the sin function). It thus embraces the shape of the sin function arround the point (but not elsewhere !), and it will be polynomial, hence computable.
Now actually computing a taylor series is not always simple, but the concept is extremely simple I think.

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In the 3d world

Am I the only one who thought you said "In the 3rd world" for awhile ?


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PostPosted: Mon Jul 10, 2017 5:03 am 
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Bregalad wrote:
Am I the only one who thought you said "In the 3rd world" for awhile ?

I did too.


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PostPosted: Mon Jul 10, 2017 5:31 am 
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I'd argue wikipedia isn't particularily good overall ("approximate knowledge of many things", to quote Adventure Time, but hey, it's free). For math, it basically looks to me like a reference guide for those who already know how to "read" math. It's probably incomplete, too, even if i can't judge that.

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PostPosted: Mon Jul 10, 2017 5:48 am 
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Bregalad wrote:
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In the 3d world

Am I the only one who thought you said "In the 3rd world" for awhile ?
No you aren't :wink:


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PostPosted: Mon Jul 10, 2017 6:09 am 
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rainwarrior wrote:
GradualGames wrote:
Oh, cool! That makes sense (seeing the pattern there...still don't have any idea how somebody figured this out to begin with). Well, I still am confused because...I understand you can approximate the sin function with the derivative(s) of sin at just one point. What I don't get is how does it work that you can approximate a whole function just from the derivatives at one point?

Yes, it works not just for sine, but for any continuous function.

e.g. if you pick a point on a curve f(a), you can approximate a point f(a+b) nearby by taking the slope and following a straight line to that nearby point.

The slope is just the first derivative, though, you can make a better approximation if you take into account whether it is curving up or down vs. that straight line slope... so you can take the "slope" of that derivative slope, or second derivative, to improve the estimate. Instead of following the straight line, you follow the straight line, plus a continual curve up or down to adjust from it...

This process can be repeated until you have the accuracy you want. If the function is a simple polynomial eventually you get a derivative that is just 0, and at that point you're calculating it exactly.

If you want more accuracy around a specific region of the function, you can pick your starting point there. Considering a sine, you might realize that an approximation of sine(0.3) near 0 might be more accurate than sine(50π+0.3), even though the target value should be the same. Each successive layer of approximation will maybe get you one more "curve" in your approximated function. Of course with sine you know it's periodic so you can just modulo 2π to keep everything in the "close" range, but there are lots of non-periodic functions out there you may want to approximate.


Anyhow, that's just the Taylor Series idea, there's lots of other approximation methods. Many approximation methods have undesirable instability that you should be careful to avoid. Again, hard to know where they can apply until you understand them and have experience with them, but I recommend taking it as far as you're interested. It's OK to use stuff you don't understand if it solves your problem adequately, just it's hard to know what you're missing until you do.


Wikipedia is also generally a very bad place to learn anything to do with math. I find most of the articles are written by experts who expect a lot of pre-requisite knowledge. Wikipedia isn't supposed to be a tutorial, either, but it's not even structured in a way that is suitable for learning these concepts anyway. If you're lucky there's good learning material in the "external links" but that's a crap shoot. Much better to learn math from the traditional sources: textbooks, school programs, good teachers, etc.


That bit about the slope really helped me understand how that is working. Since a "sampled" derivative is the slope so if you keep taking (additional derivatives), it gets more and more accurate. Makes sense. It's still kind of amazing to me though...I mean looking at the Taylor series for the sin function on wikipedia, there isn't any trigonometry to be found, yet it works.

This further helps me realize how much of math is breaking things down into smaller steps, much like programming. I mean, I've been doing sinusoidal motion in my games since Nomolos, by accelerating an object in the opposite direction relative to a center point. It was just somehow easier to understand because I was down at the step by step, "add the next acceleration value" level, whereas math on paper has all the steps kind of expressed all at once. (though come to think of it, so is code, but maybe since I was writing it myself with a notation I was familiar with, a programming language, it was easier to think about...)

It's still damn impressive it works and somebody figured it out to begin with. Like, I really have trouble calling these things "simple." Maybe they're simple to use and perhaps to understand in some cases, but...actually being the guy who invents something like this...that amazes me.

Proofs are another thing I really never got a good grip on. The furthest I got is a few lights maybe turned on in my discrete math course in college and I did a little bit of induction, but that's it. After that whenever a proof came up in a class I was usually quite lost.


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PostPosted: Mon Jul 10, 2017 10:59 am 
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I was also always scared off of complex numbers and the concept of the "imaginary unit." I think the word "imaginary" created a lot of unwelcome cognitive dissonance. It's real math, right? It solves real problems, right? Then there's nothing remotely imaginary about it.

I realized recently that even negative numbers can be thought of as "imaginary", in so far as the "minus" is more of an "action" or a "direction," it is a tool. It's not a "real" thing in so far as, you can't count a negative amount of something in the real world. Negative numbers only exist as an abstraction. 5 less of something. Etc. It's the action of taking away. It's not an actual observable THING, the way "5 things" is something you can see with your eyes. So one could say negative numbers are "imaginary" too.

Naming things is hard. Math is full of really really scary sounding or bad names. That's probably the biggest problem a lot of folks have with it, right there.


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PostPosted: Mon Jul 10, 2017 11:38 am 
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I discovered fractals at a very early age, and despite not "understanding" what i was, I thought the resultant pictures cool enough to abstract around not really understanding complex numbers.

Access to FRACTINT and James Gleick's Chaos: The Software helped a lot, too.


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PostPosted: Mon Jul 10, 2017 11:55 am 
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Just wait until you get to imaginary space & imaginary time, very real physics concepts ;)


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PostPosted: Mon Jul 10, 2017 12:02 pm 
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calima wrote:
Just wait until you get to imaginary space & imaginary time, very real physics concepts ;)


Are they real in so far as someone has observed imaginary space and imaginary time in a lab, or just that there are equations that use these concepts? (I know very little about physics besides the basics that are useful for game development)


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PostPosted: Tue Jul 11, 2017 12:03 am 
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Imaginary space has been observed (the virtual particles that affect light speed in a vacuum), but imaginary time is just a theory so far.


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PostPosted: Tue Jul 11, 2017 3:29 am 
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GradualGames wrote:
I realized recently that even negative numbers can be thought of as "imaginary", in so far as the "minus" is more of an "action" or a "direction," it is a tool. It's not a "real" thing in so far as, you can't count a negative amount of something in the real world. Negative numbers only exist as an abstraction. 5 less of something. Etc. It's the action of taking away. It's not an actual observable THING, the way "5 things" is something you can see with your eyes. So one could say negative numbers are "imaginary" too.

Be careful when using the words "imaginary" and "real" in math. Those refers to complex numbers which obviously you didn't want to take in consideration here.

There is a lot of applications in math where negative numbers have to be excluded, but even more situations where they are an extremely practical tool to describe things. They are a natural invention following the invention of substaction, which very much exists.


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PostPosted: Tue Jul 11, 2017 6:03 am 
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Bregalad wrote:
GradualGames wrote:
I realized recently that even negative numbers can be thought of as "imaginary", in so far as the "minus" is more of an "action" or a "direction," it is a tool. It's not a "real" thing in so far as, you can't count a negative amount of something in the real world. Negative numbers only exist as an abstraction. 5 less of something. Etc. It's the action of taking away. It's not an actual observable THING, the way "5 things" is something you can see with your eyes. So one could say negative numbers are "imaginary" too.

Be careful when using the words "imaginary" and "real" in math. Those refers to complex numbers which obviously you didn't want to take in consideration here.

There is a lot of applications in math where negative numbers have to be excluded, but even more situations where they are an extremely practical tool to describe things. They are a natural invention following the invention of substaction, which very much exists.


The very paragraph above that one stated that I felt the word "imaginary" in complex numbers was a bad name. Then, I simply proceeded to describe how the concept of "negative" does not describe any real, observable thing in the real world---it is an abstraction, which helps us ARRIVE at real observable results in the real world. It's very real on paper, yes, but it is "imaginary" in so far as it is an abstraction we are using to solve problems. The "imaginary unit" in complex numbers, is another abstraction---the ONLY point I am making is that traditional names in mathematics sometimes scare people off, but not for any good reason---they're just bad names. I have less of a problem with the name "complex number" than I do with the name "imaginary." Because it is clearly not imaginary, it solves real problems. I have no ideas for a better name for it. But math is full of things with scary sounding names, which, when you break it down, turn out to not be all that difficult to understand after alll---that's my only point. It's a dumb reason to be scared off of it, but lots of people are.

In other words, I'm very aware that "imaginary" has a long-standing and traditional use in mathematics. I'm using "imaginary" in the english sense of the word, something that exists in your mind that doesn't exist in the real world, to point out that math is full of abstractions that are not representations of real physical objects but of operations to arrive at real results that may represent real objects. Does that make sense? I'm just pointing out that "imaginary" is simply a NAME for an abstraction, much like we arbitrarily name routines and functions. It's a bad name, because it isn't imaginary at all, it solves real problems.


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PostPosted: Tue Jul 11, 2017 6:30 am 
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Let me put it another way:

The simplest abstraction, positive natural numbers, are the easiest to "see" in the real world. e.g. COUNT things. They are the most "real" of anything in mathematics. (note: using a crude definition of real, meaning, your average person trying to understand what aspects of math actually exist not on paper at all. I'm not saying math isn't real or true, of course it is)

When you add negative numbers, now all numbers have a "-" next to them. Is there anything in the real world you can see and touch that has this property? No. Minus is an operation. An abstraction, which is real only on paper and in our minds. When applied to a real problem, we might wind up with "5 apples minus 3 apples is 2 apples" 2 apples, which is real and we can see it.

The point I'm really trying to make (and probably failing to explain what I'm saying) is that an operation such as - is no more real than the square root of -1. They are abstractions which allow us to solve increasingly difficult problems. They aren't something we observed somewhere in the real world. That's why I have a problem with calling the square root of -1 imaginary. In terms of the usual definitions and rules for multiplying negative numbers, the square root of -1 doesn't make sense (thus the term 'imaginary'), but -1 was an abstraction to begin with, e.g. "imagined." Then, mathematicians discovered you could solve more interesting problems by defining the square root of -1 as part of a term to use in a complex number. It's an abstraction. All abstractions exist in our minds and thus could be SAID to be "imaginary," re-iterating that I'm well aware of the traditional use of the word and do not mean to conflate negative numbers with complex numbers.

Now that I've established that---nobody wants to think of something in mathematics as being "imaginary." It's just a bad name. Much like if I named update_column "pink_elephant." I have no idea what it means and it doesn't adequately describe what it does. And worse, the name "imaginary" is lying to me and makes me think something impossible is magically working anyway. Which obviously isn't true.


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PostPosted: Tue Jul 11, 2017 7:42 am 
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Another example. Division by zero. 1 / 0. What is that? We say its NaN because it isn't defined, unless you are talking about *approaching* infinity, then it makes sense. They're all exceptions and edge cases and rules and names, just like in programming. Thus you could call any of these "odd" things "imaginary" if you wanted to. I just think it's a dumb word to use because these rules, definitions and abstractions all hold universally and all can be used to describe real world phenomenon accurately in many cases, so I just think it's dumb to use the word "imaginary," though I understand why it was used to begin with. When it was first discovered it was probably surprising that it worked, and caused the same cognitive dissonance that is bothering me and which probably bothers a lot of people when they learn math more advanced than highschool math.

Another funny thing about abstractions in math is that when you boil them all down, you have digits which originate in the abstraction of counting discrete objects. Yet we've found ways to describe continuous phenomena with the "programming" of arranging our rules and abstractions to continue to work with what ultimately are symbols for counting discrete objects. It's really quite amazing when you think about it. We went from counting things as cavemen, to creating this vast network of symbols and abstractions to describe the real world IN TERMS, ultimately, of the symbols we use to count discrete objects. It's quite impressive! In summary: Math is programming, but unfortunately was invented long before software engineers came along and said: "Let's give things good names rather than greek letters and brainfucky names like 'imaginary'!"


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