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PostPosted: Tue Jul 11, 2017 7:52 am 
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Even "brainfuck" is arguably a less brainfucky name than its original name, P Prime Prime.

My first guess about the use of Greek letters and the like in mathematical notation rather than whole-word names is to let more of an expression's numerator or denominator fit within the width of a page.


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PostPosted: Tue Jul 11, 2017 8:00 am 
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tepples wrote:
Even "brainfuck" is arguably a less brainfucky name than its original name, P Prime Prime.

My first guess about the use of Greek letters and the like in mathematical notation rather than whole-word names is to let more of an expression's numerator or denominator fit within the width of a page.


I think the thing is---so many concepts in math boil down to something that's not *that* hard to understand ultimately. But there's something odd in our culture which makes people just sort of throw up their arms and go: "Might as well be chinese to me!" Likely do to the unfamiliarity of the symbols used. Those who are able to get a little bit further might still throw up their arms later when they encounter something which appears to be the definition of an impossibility: i, the square root of -1. But I'm trying to say that ....if we taught math more carefully and properly, we could mitigate the cognitive dissonance by explaining exactly what math and abstractions actually are. Instead we just ask students to take the definitions and abstractions as they were originally formulated a hundred, two hundred or more years ago so they seem archaic and no effort is put into explaining them in simpler terms.

It seems as though some folks are blessed with the ability to intuit abstractions right away. They just "get" that they are abstractions without ever consciously verbalizing this fact. Then there are folks for whom for whatever reason abstract thinking is a bit more of a challenge---it was for me. I think I'm just now reaching some kind of threshold in my mental development which is making it easier to grasp and accept abstractions as they are. My point is I feel if mathematics were taught more carefully, more people might be able to develop that ability more quickly rather than it just being a "you're born with it or you're not" kind of deal.


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PostPosted: Tue Jul 11, 2017 8:13 am 
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Perhaps L. Ron Hubbard was right about people not being able to learn without visual aids, which he called "mass". Hubbard also realized that instruction can't be productive until definitions of words are agreed upon, which later became Layne's law of debate, and that a "steep" learning curve would be most easily understood as having effort on the vertical axis and achievement on the horizontal.

Not that that excuses counterproductive application of his Study Tech though.


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PostPosted: Tue Jul 11, 2017 11:27 am 
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GradualGames wrote:
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.
Brief moment of offtopic pedantry: 1/0 is +∞. (lim x→0 1/x).
0/0 is the only one that's NaN because what it converges to depends on which limit you take.


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PostPosted: Tue Jul 11, 2017 12:11 pm 
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The limit of 1/x differs based on whether x approaches 0 from the left or right side. From the left side, 1/x decreases without bound (-∞); from the right, it increases without bound (+∞).


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PostPosted: Wed Jul 12, 2017 6:12 am 
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GradualGames wrote:
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.

Sin(x)/x is not even close to approaching infinity when x is near zero (hint, it approaches 1), but is still NAN for x=0.
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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.

I disagree. If you're counting potatoes, then you're right. But when counting meters above sea level for example, negative numbers are a very real thing that happens in real life. A wider situation is locating a point on an infinite straight line; you will have to resort to negative values because the line does not end or start anywhere, so you cannot set the zero adequately to use only positive numbers, no matter how you try.

Temperature is often used as an example but it's a horrible example because actually 0K is a true zero and there's no reason to have negative temperatures at all, except we didn't know about absolute zero back when the concept of "temperature" was invented.


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PostPosted: Wed Jul 12, 2017 6:39 am 
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It's difficult to communicate what I'm really trying to say, here. What I'm saying is that the concept of "negative" is an operation that we essentially define to mean "reverse direction" on the number line, and it helps solve problems algebraically. We can't count a negative number of something unless we're talking about taking something away, which is an *action* not a thing in the real world. It's abstract to begin with is what I'm saying. It's real, of course, but not real the way a physical object is real. This is the only point I'm trying to make.

Defining i as the square root of -1 seems like we're defining a falsehood. But if you think about it like I'm trying to suggest, it's simply another abstraction which enables yet more algebraic solutions to work. The normal way we apply "negative" to multiplying negative numbers makes the square root of -1 seem like a falsehood, but since "negative" was a rule or action to begin with, what we're really saying is: "IF -1 had a square root and followed this rule, let's call it i." And the fascinating thing is that it lets us solve more problems.

For some reason it's a lot easier to understand and internalize "negative" than to accept an abstraction where we must suppose a falsehood is true (i is the square root of -1). Get where I'm going with this? They're both very "real" in that they solve real problems, but one is a hell of a lot harder to grasp if you aren't blessed with an innate ability to grasp definitions and abstractions. It seems to me with enough effort math pedagogy could be improved so that concepts such as complex numbers would not be so intimidating.


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PostPosted: Wed Jul 12, 2017 9:07 am 
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Bregalad wrote:
But when counting meters above sea level for example, negative numbers are a very real thing that happens in real life.

What? How?

Edit: To answer the thread title, bad. :(


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PostPosted: Wed Jul 12, 2017 9:32 am 
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The distance between sea level and the sea floor still has positive magnitude; it's just in the downward direction. Negative real numbers are an abstraction including whether the direction is forward or backward. Complex numbers are a generalization of this direction to a plane.


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PostPosted: Wed Jul 12, 2017 10:04 am 
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tepples wrote:
The distance between sea level and the sea floor still has positive magnitude; it's just in the downward direction. Negative real numbers are an abstraction including whether the direction is forward or backward. Complex numbers are a generalization of this direction to a plane.

That's what I thought. Who the hell says "-5 meters above sea level"?


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PostPosted: Wed Jul 12, 2017 10:12 am 
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tepples wrote:
The distance between sea level and the sea floor still has positive magnitude; it's just in the downward direction. Negative real numbers are an abstraction including whether the direction is forward or backward. Complex numbers are a generalization of this direction to a plane.


I haven't studied complex numbers in depth yet, but at a high level, what makes complex numbers different from just having a pair of X and Y coordinates, then?


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PostPosted: Wed Jul 12, 2017 11:03 am 
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Nothing, it's a regular 2d vector.


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PostPosted: Wed Jul 12, 2017 11:08 am 
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GradualGames wrote:
I haven't studied complex numbers in depth yet, but at a high level, what makes complex numbers different from just having a pair of X and Y coordinates, then?

Complex numbers can be used for various 2D geometry problems, though I think the notation is a bit confusing for someone who is really trying to do 2D geometry and would rather just work with a 2D vector.

The original reason for i is simply to solve the quadratic equation where there are no "real" answers. It is not the square root of -1, but rather it is the "imaginary" number that when squared will equal -1. i2 = -1

When trying to solve a quadratic, if that square root term b2-4ac is positive you get 2 solutions, if it is 0 you get 1 solution, if it is negative you get 0 solutions. However, you can think of the 1 solution case as two solutions that just happen to be equal to each other. If you use complex numbers, even the negative/0 solutions case becomes again two solutions, just requiring an "imaginary" component. So with complex numbers this thing that had 3 different types of outcome now only has 1. This kind of uniformity is part of why complex numbers can be very useful.

So... okay maybe it doesn't immediately sound useful for a quadratic, but as soon as you want to solve cubic (x3) or quartic (x4) equations, you'll discover that the "equivalent" to the quadratic formula for them is maddeningly complicated. Complex analysis becomes a light leading out of the tunnel for this.

There's all sorts of useful things that come out of complex numbers, particularly to do with exponents, and periodic functions like sine and cosine fit in here as well in ways that might surprise you. It becomes a very good way at looking at a lot of problems that aren't obvious at all when you first hear about i.

calima wrote:
Nothing, it's a regular 2d vector.

The thing that makes a complex number different than a 2D vector is that you can multiply two complex numbers (and all the consequences that carry on from that). 2D vectors can be multiplied (scaled) by a scalar (single value), but not by another vector. Everything that follows from this is what makes complex numbers useful for things that 2D vectors alone are not.


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PostPosted: Wed Jul 12, 2017 11:12 am 
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rainwarrior wrote:
GradualGames wrote:
I haven't studied complex numbers in depth yet, but at a high level, what makes complex numbers different from just having a pair of X and Y coordinates, then?

Complex numbers can be used for various 2D geometry problems, though I think the notation is a bit confusing for someone who is really trying to do 2D geometry and would rather just work with a 2D vector.

The original reason for i is simply to solve the quadratic equation where there are no "real" answers. It is not the square root of -1, but rather it is the "imaginary" number that when squared will equal -1. i2 = -1

When trying to solve a quadratic, if that square root term b2-4ac is positive you get 2 solutions, if it is 0 you get 1 solution, if it is negative you get 0 solutions. However, you can think of the 1 solution case as two solutions that just happen to be equal to each other. If you use complex numbers, even the negative/0 solutions case becomes again two solutions, just requiring an "imaginary" component. So with complex numbers this thing that had 3 different types of outcome now only has 1. This kind of uniformity is part of why complex numbers can be very useful.

So... okay maybe it doesn't immediately sound useful for a quadratic, but as soon as you want to solve cubic (x3) or quartic ([i]x4) equations, you'll discover that the "equivalent" to the quadratic formula for them is maddeningly complicated. Complex analysis becomes a light leading out of the tunnel for this.

There's all sorts of useful things that come out of complex numbers, particularly to do with exponents, and periodic functions like sine and cosine fit in here as well in ways that might surprise you. It becomes a very good way at looking at a lot of problems that aren't obvious at all when you first hear about i.


Aren't complex numbers used in 3D graphics a lot too, via "quaternions?" That's probably the main application I'll wind up reading up about...it's the only application I can think about that would produce interesting results with things I like (game programming). I just have no idea what those results are. Haha

...You get why the concept is intimidating to those who haven't studied math very hard, right? Once one has internalized that no negative number can be a square, the definition makes you think that complex numbers are based on a falsehood. It produces a similar cognitive dissonance to that one proof where you can say the sum of all natural numbers is -1/12 (which I really don't understand to this day. I can see it shows up when you graph it but I really don't get it yet. I'd have to spend some quality time with it...) lol.


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PostPosted: Wed Jul 12, 2017 11:37 am 
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GradualGames wrote:
Aren't complex numbers used in 3D graphics a lot too, via "quaternions?" That's probably the main application I'll wind up reading up about...it's the only application I can think about that would produce interesting results with things I like (game programming). I just have no idea what those results are. Haha

I would say no about quaternions; the primary use for them in 3D graphics is simply as a way to express a rotation and its axis in a compact/efficient way. You don't really need to understand the "complex" consequences of that to work with them for that purpose, which is merely a geometric purpose. (Just as I said above, how I think using complex numbers for 2D geometry problems is probably unnecessarily confusing.)

So... while the word quarternion will come up almost certainly, its actual use is really to just represent a 3D geometric rotation and it need not be understood any more than that to work with quaternions in 3D graphics. (There are 3D graphics applications of the true complex quaternion, but they're obscure.)

However, if you want to write a raytracer, for example, you'll probably very quickly run into cases that require complex numbers to solve effectively. An intersection of a ray and a sphere is a quadratic equation, but a ray and a torus is a quartic. I mentioned how they have some hideous "formula" solutions for such a thing but in practice they do not tend to be numerically stable (the floating point errors compound wildly). You're going to have a very hard time writing a practical solver for this kind of thing without dealing in complex numbers. (Just complex numbers here, not quaternions.) ...though you might manage with just some calculus and an iterative solver, if you can trade the CPU / accuracy for it. ;)

GradualGames wrote:
...You get why the concept is intimidating to those who haven't studied math very hard, right?

I do, but I can't address your complaint that you don't like the terminology. I didn't make up the words, I can only try to help you understand what they mean.

GradualGames wrote:
you can say the sum of all natural numbers is -1/12

That one is an interesting paradox, and it definitely is something that follows from a particular sequence of logical conclusions, but probably the most interesting thing about this is finding out which premise that you've accepted made this thing that should not be true. ;) There are possible paradoxes in all formal systems of logic, unfortunately, but that doesn't necessarily make the less paradoxical parts of it unuseful, and you should guard yourself against sophistry that would rob you of that utility.


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