Should be easy math

[Hawkwings]

It's only confusing because of Chewie's cousin's stupid math teacher. |x| cannot be negative. End of story. Graph it if she doesn't believe it.

Other than that, what Kuroneko and Surlethe are talking about is probably beyond the grasp of most people on the board, not to mention in real life.

To get the best foundation though, college classes really are the best way to go.

I'd really say that High School classes are the foundation of math. You know, good old basic manipulation of numbers, all the way up to basic calculus.

[Havok]

Does anyone know if there is like a Math for Dummies (i.e. ME) or something that might help me not feel so much like, well, a dummy when I read these threads?

The best thing you can do is take some classes. The classes I personally have are Calculus I, II, and III and also Matrix Algebra, and they give me enough of a foundation so I can generally, but still not always follow along with what the smart people are saying. Kuroneko and Surlethe are both very smart people and still often beyond me, but at least, with the classes I do have, I have enough of a foundation so I can spend the time and continue learning on my own to catch up. It does take an investment of time though to understand it, classes or no classes.

If something comes up, specifically, that I want to know about, Mathworld is an excellent resource for learning more, but it does require you already have a decent foundation in math so you can make the most of it.

Also, if you have a specific question, you can ask it in SLAM or Testing (depending on how significant the question is) and people here will try to help you.

To get the best foundation though, college classes really are the best way to go.

Uh... I took math skills in high school. I'm fucked aren't I?

[SpacedTeddyBear]

Damn I haven't heard of the Triangle Inequality in a while. Like Surlethe, I was first introduced to it back in 3rd semester calculus. But I personally didn't really get into the derivation until I took a vector analysis course. According to my old notes the Cauchy-Schwarz inequality is derived by:

Letting a=y·y & b=-x·y. If 'a' is zero, both sides of the inequality reduce to zero. Supposing that 'a' is not zero, and since x·x≥0

0≤(ax+by)·(ax+by)= a^2*x·x+2ab*x·y+b^2*y·y
0≤(y·y)^2*x·x-(y·y)(x·y)^2
Dividing by (y·y) yields
0≤(y·y)*x·x-(x·y)^2

=> (x·y)^2≤ (y·y)*(x·x)
Then take the √ plus a little bit of magic yields.
x·y ≤ |x||y|

[SpacedTeddyBear]

Ghetto edit:
That was the method I was taught anyways. If there are other methods of proving it, please share.

[Master of Ossus]

It's only confusing because of Chewie's cousin's stupid math teacher. |x| cannot be negative. End of story. Graph it if she doesn't believe it.

Agreed.

Other than that, what Kuroneko and Surlethe are talking about is probably beyond the grasp of most people on the board, not to mention in real life.

The Triangle Inequality is not that complicated. Draw two points, A and B. Now draw a line segment connecting those two. Now try to find another way to connect them that is shorter. Unless you suck at art (and drew the line wrong in the first place), you can't do it in Euclidean space.

Triangle inequality just says that if you try to draw the line through some other point, it will be longer unless that point is already on the line segment AB, in which case the length is the same as it was, before. Nothing is surprising about this (at least in two and three dimensions), but it shows up in a lot of mathematical proofs and so it's sort of interesting to know.

[Gaidin]

The sad thing is |x| not being negative shouldn't ever be hard to explain until you get into the wordy definitions that are favored for some reason(usually around Modern Algebra...every class I've had before now never bothered with explaining what is a simple concept to the class). And even then its just the wording of the definition that says that if x < 0 then |x| = -x. And by that time its awkward for a split second and then you read it again, shrug, and move on.

[Faqa]

*wonders if it was good or bad that only half of Surlethe and Kurenko went over my head, as opposed to the usual three quarters*

Teacher's an idiot(and I do mean completely), but your cousin's wrong, for reasons expressed here. Of course, for her to understand it, she needs to learn the correct definition of numbers, not just the intuitive one.

I really don't get why that's not done already. Half my uni work thus far seems to be re-defining the concepts I already knew to make logical sense. Kind of a waste, even if it is fun.

[Fleet Admiral JD]

This may be a semi-relevent question:

What math level is your cousin in, Chewie?

[Kuroneko]

That was the method I was taught anyways. If there are other methods of proving it, please share.

Well, it works, but it's more complicated that it needs to be. The cases when either |x| or |y| are 0 are trivial, so let's assume they are not. In the reals, (a-b)² ≥ 0 implies ab ≤ (1/2)[a²+b²], so that in implicit summation over k, (x/|x|)·(y/|y|) = [x_k/|x|] [y_k/|y|] ≤ (1/2)[ x_k²/|x|² + y_k²/|y|² ] = 1. QED.

[Surlethe]

That's strange. You probably encountered it in the form of determining whether or not there exists a triangle with sides of three given lengths, even if the name itself was not covered.

I don't remember that specific problem, but I do remember a rather simplified form of it from first-year Calculus: x ≤ |x|, which the teacher introduced, and we all went, "huh? oh" when he explained it.

Speaking of vectors, did your class prove it? If so, I'm curious of the method of showing the Cauchy-Schwarz inequality x·y ≤ |x||y| (the triangle inequality for R^n is a direct consequence).

I'm afraid that my notes are at home. When I head home next (probably this weekend), I'll grab them and look through them for the proof.

I think it was along the lines of the method SpacedTeddyBear gave: break it out into components, and then muck through the algebra. Speaking of proofs, in your proof in the post immediately preceding mine, do you mean to prove it in R^n, instad of just the reals?

[Surlethe]

This is the most confusing thing I have read on here to date. Does anyone know if there is like a Math for Dummies (i.e. ME) or something that might help me not feel so much like, well, a dummy when I read these threads?

The best thing you can do is take classes, where you'll have to learn the concepts to get your money's worth out of it. The next best thing is to study on your own so you get an idea of what math is all about. The point here is that you only learn math well through lots of study; the math/physics people on here (Kuroneko, Wyrm, drachefly, and probably several others whose names escape me at the moment) whom I suspect of having Ph.ds have put literally years into the study of mathematics. Those of us who are going through an undergraduate education in mathematics are taking several hours of classes per week, and will do so for several more years.

I'm not trying to scare you; I'm only trying to instill in you a sense that this knowledge isn't free. It's not like Kuroneko and I were born with the ability to discuss Cauchy sequences, e.g.; we've had to study them and think about them and turn them around in our heads until we understood them. The same goes for all the mathematical topics we might discuss. So, if you want to understand, be sure that you understand you're going to have to work for it like the rest of us.

That said, go on! Get cracking!

[SpacedTeddyBear]

It's not like Kuroneko and I were born with the ability to discuss Cauchy sequences, e.g.; we've had to study them and think about them and turn them around in our heads until we understood them. The same goes for all the mathematical topics we might discuss. So, if you want to understand, be sure that you understand you're going to have to work for it like the rest of us.

That said, go on! Get cracking!

Sometimes though, the terminology used to discuss math is what confuses people the most. I'm in my 6th year of physics now, and I can follow you, Kuroneko, and a few others most of the time, but I occasionally have to look certain terms up on wiki simply because I don't use math lingo very much.

[drachefly]

the math/physics people on here (Kuroneko, Wyrm, drachefly, and probably several others whose names escape me at the moment) whom I suspect of having Ph.ds

not yet, but getting there...

[Wyrm]

the math/physics people on here (Kuroneko, Wyrm, drachefly, and probably several others whose names escape me at the moment) whom I suspect of having Ph.ds

I'm flattered that you think I have a PhD, but I have but a Masters in Mathematics. I am currently seeking my way into a PhD program, so this will probably be true in a few years.

[CaptainChewbacca]

This may be a semi-relevent question:

What math level is your cousin in, Chewie?

She's in Algebra 2, which comes after geometry and serves as a pre-calculus class.

[Kuroneko]

I don't remember that specific problem, but I do remember a rather simplified form of it from first-year Calculus: x . |x|, which the teacher introduced, and we all went, "huh? oh" when he explained it.

You never had to deal with problems like: "Does there exist a triangle with sides {3,10,15}?" I wad under the impression that this was semistandard in grade school geometry classes; perhaps it's only in the schools around here. In any case, the common geometry result that the sum of any two sides of a triangle is greater than the third is but another form of the triangle inequality.

I think it was along the lines of the method SpacedTeddyBear gave: break it out into components, and then muck through the algebra. Speaking of proofs, in your proof in the post immediately preceding mine, do you mean to prove it in R^n, instad of just the reals?

In R^n, of course. Imagine a summation over k forb all terms in which k appears, which makes the dot product look like x·y = x_k y_k and the norm-squared as |x|² = x_k². I've left the summation implicit because it doesn't carry over well in text.

If you prefer: [x·y]/[|x||y|] = Sum_k[ (x_k/|x|)(y/|y|) ] ≤ [1/2]Sum_k[ x_k²/|x|² + y_k²/|y|²] = 1, where again the inequality ab ≤ (1/2)(a²+b²) was used on the components and |x|,|y| are assumed to be nonzero, as having either of them zero makes x·y ≤ |x||y| trivial.