Paragraph 1: The 'art' in mathematics comes in connecting ideas to create much more than you started with. The music you listen to probably attracts you because of interesting symmetries and patterns, but even the best of music (or painting, or sculpture, or interpretive dance, or [insert other non-mathematical nonsense here]) will inevitably pale in comparison to even elementary abstract algebra, where symmetry is studied in its purest possible forms. It's helpful to have some understanding of mathematical 'elegance' - a proof that is clear, concise, and shows a deep connection between two ideas (rather than just proving their equivalence) is itself a thing of beauty. The ancient Greek proofs of the infinitude of primes or the irrationality of the square root of 2 are elementary and elegant; for examples of inelegant proofs, check your high school texts.
It's damned near impossible to understand without experience - without the big light-bulb moments and such nonsense - because mathematics is something that has to be *done*, not watched. It'll happen, though.
Paragraph 2: It's a common fault of *everything* before college, as far as I can tell, to put definitions in front of intuition and necessity. Imaginary numbers, for example, popped up in a few places - the biggest of these was solutions to polynomial equations. There are irreducible polynomials of degree 2 in the reals, for example (x^2+1 being the most obvious), but it was discovered that allowing such 'imaginary' nonsense to be regarded led to solutions to very complicated polynomials. The only irreducible (not factorable, in other words) polynomials in the complex numbers are the linear functions, which leads to a great deal of simplification when dealing with polynomials. Since all we needed to 'complete' the real numbers was introduction of some weird symbol 'i' (with addition and multiplication of combinations a+bi defined as you've seen), we quite like it.
Think of it like this... you can already see the graph. You can already see how complex addition works, and hopefully how complex multiplication looks (most helpful: look at the numbers in terms of their magnitude and the angle they make with the positive x-axis; complex multiplication of two numbers is simply adding the angles and multiplying the magnitudes), so the complex numbers behave very interestingly... they can be visualized easily *and* they're 'structured' enough to allow for something that works just as multiplication. Further, the complex plane includes the real line that you're already (hopefully) happy with. The most clear reason to care, really, is that the complex plane is an extension of the real numbers that allows polynomials to be reduced to products of linear functions.
Other useful things they brought about take more advanced knowledge. The Fundamental Theorem of Algebra, for example (discovered by Gauss when he was 20 or 21... the fucker), was unproven until advances in complex analysis made it seem all but trivial - for millenia the brightest people in the world were stumped, but a few new ideas pop up and all of a sudden algebra has a Fundamental Theorem that can be understood by middle-school students and proven by bright high-school students.
Paragraph 3: Same thing here - I no longer tutor regularly (in fact, I have a hard time *not* thinking that contributing to high school bullshit is a crime against all things beautiful in the universe) but I saw that happen myself. Definitions are put before motivation, so the students might as well not listen - it could be robbing them of real understanding later in life, and it certainly isn't bringing anything but rudimentary equation-matching abilities for all but a few lucky bastards. I wouldn't be anywhere right now if I didn't learn that stuff on my own, with a good text and actual motivation, so even thinking about robbing one younger incarnation of myself of real mathematics is just disgusting.
Paragraph 4: No projectile motion or physics in math classes - generally, algebra-based physics is given *before calculus* (yeah, skullfuckingly brilliant idea there) and only equations to memorize are really present. I didn't see imaginary numbers in my 'Algebra II' class, but it may have been elsewhere - in my third and fourth years, I took classes at a proper university, so I don't know what happened in my math classes. At the school I tutored some people had a basic understanding, but rarely anything remotely resembling a clue - they got bogged down by things to memorize and never saw the meaning of anything.
Jeff
It's damned near impossible to understand without experience - without the big light-bulb moments and such nonsense - because mathematics is something that has to be *done*, not watched. It'll happen, though.
Paragraph 2: It's a common fault of *everything* before college, as far as I can tell, to put definitions in front of intuition and necessity. Imaginary numbers, for example, popped up in a few places - the biggest of these was solutions to polynomial equations. There are irreducible polynomials of degree 2 in the reals, for example (x^2+1 being the most obvious), but it was discovered that allowing such 'imaginary' nonsense to be regarded led to solutions to very complicated polynomials. The only irreducible (not factorable, in other words) polynomials in the complex numbers are the linear functions, which leads to a great deal of simplification when dealing with polynomials. Since all we needed to 'complete' the real numbers was introduction of some weird symbol 'i' (with addition and multiplication of combinations a+bi defined as you've seen), we quite like it.
Think of it like this... you can already see the graph. You can already see how complex addition works, and hopefully how complex multiplication looks (most helpful: look at the numbers in terms of their magnitude and the angle they make with the positive x-axis; complex multiplication of two numbers is simply adding the angles and multiplying the magnitudes), so the complex numbers behave very interestingly... they can be visualized easily *and* they're 'structured' enough to allow for something that works just as multiplication. Further, the complex plane includes the real line that you're already (hopefully) happy with. The most clear reason to care, really, is that the complex plane is an extension of the real numbers that allows polynomials to be reduced to products of linear functions.
Other useful things they brought about take more advanced knowledge. The Fundamental Theorem of Algebra, for example (discovered by Gauss when he was 20 or 21... the fucker), was unproven until advances in complex analysis made it seem all but trivial - for millenia the brightest people in the world were stumped, but a few new ideas pop up and all of a sudden algebra has a Fundamental Theorem that can be understood by middle-school students and proven by bright high-school students.
Paragraph 3: Same thing here - I no longer tutor regularly (in fact, I have a hard time *not* thinking that contributing to high school bullshit is a crime against all things beautiful in the universe) but I saw that happen myself. Definitions are put before motivation, so the students might as well not listen - it could be robbing them of real understanding later in life, and it certainly isn't bringing anything but rudimentary equation-matching abilities for all but a few lucky bastards. I wouldn't be anywhere right now if I didn't learn that stuff on my own, with a good text and actual motivation, so even thinking about robbing one younger incarnation of myself of real mathematics is just disgusting.
Paragraph 4: No projectile motion or physics in math classes - generally, algebra-based physics is given *before calculus* (yeah, skullfuckingly brilliant idea there) and only equations to memorize are really present. I didn't see imaginary numbers in my 'Algebra II' class, but it may have been elsewhere - in my third and fourth years, I took classes at a proper university, so I don't know what happened in my math classes. At the school I tutored some people had a basic understanding, but rarely anything remotely resembling a clue - they got bogged down by things to memorize and never saw the meaning of anything.
Jeff
