To all those who took high school or college Algebra courses...

[AllanD]

I've failed Algebra twice. The semester always starts out with using i(fake numbers), and literally no teacher could explain it to me to make sense.

 

[colonel kurtz]

math classes, IME, always come down to the quality of the teacher

like i said in the post before, i had a horrible teacher for trig/precalc. then i went into calculus with a good teacher, and aced that shit no problem.

same thing with stats...i bombed statistics SO hard the 1st time i took it. had one of those teachers who was a super-smart guy...used to do statistical analysis for pharmaceutical companies, but he was a terrible teacher. he had a monotone voice, never graded/checked homework, and would give quizzes at the end of class on shit he just lectured about - before we got a chance to go over any of our notes or do the work. he also graded harshly on tests, so between our entire grade being tests and quizzes, i was eating tons of shit. by the time thanksgiving came around, i figured out that even if i aced the rest of the class, a D was the best i could do.

i just stopped going and got an incomplete...then i took it the next semester with a different teacher, and got like 103% in the class. shit was SO easy with a teacher who knew how to explain the concepts, and who gave regular feedback on what you were doing, so that you knew what was going on when heading into an exam.

 

[LeSedna]

I've failed Algebra twice. The semester always starts out with using i(fake numbers), and literally no teacher could explain it to me to make sense.

If ever I can help in layman's terms, that's how I saw it in the beginning and how i made it easier for me to get, and how I created sense for it and made it so easier to understand in the end, just keep in mind I haven't done any maths in like 4 years :

- you can think of i, for now, only by i*i=-1 or i^2=-1
- as a result, obviously, sqrt(x)=-1 is solved by x=i , instead of x=Ø in the IR realm only
- you're now working in C which you can consider IR + i IR instead of just IR, every number has now two terms, x=a+bi
- it's easy to represent because it has 2 terms so you can make it have coordinates and create a representation of any number in the plan. x=a+bi would have the coordinates (a,b) in a plan where the horizontals are the reals and the verticals are the imaginaries (seems like in english you call it the fakes?)
- don't think it has something useless, or not real, because it is called "fake". It is actually a confusing name, but it's just another maths tools that is extremely powerful and creates links between previously unrelated things, and opened so many doors to mathematicians 2 hundred years ago by not creating stops in their calculations, it was the "dark matter" of mathematics of their time maybe . It is probably called fake because it's a made-up way to solve sqrt(x)=-1 and so i=sqrt(-1)for example, but it turns out it is used in many tangible ways
- you can have another look at it with this : multiplying a number by 2 or 3 would make the representation of it increase/decrease its scale by 2 or 3, or homotethy (not sure the term is the same in english sorry). Basically, multiplying by a real term a number (real only or not) is like looking at it while changing the scale, (not technically true but you see my point, it's like zooming in or out the shape of it in a representation)

but multiplying by i a whole number makes it rotate by 90° or Pi/2 in the trigonometric (opposed to a clock) direction. Look :

1 (or 1+0*i) becomes i (or 0+1*i) , so its coordinates move from (1,0) to (0,1)

i becomes -1

-1 becomes -i

-i becomes 1

2+3i becomes -3+2i, its coordinates move from (2,3) to (-3,2).. again a 90° rotation.

Multiplying by 2i is a 90° rotation with a *2 homotethy

Multiplying by -1 is obviously a 180° rotation, every positive becomes a negative and negative becomes a positive... But see, multiplying by -1 is a 180° rotation... how awesome is that, because we just said i^2 = -1... which makes sense : a 180° rotation is two 90° rotations. So you can write it i*i, or -1...

and a 270° rotation, or -90° rotation, is -i. If you follow me, it's also i^3 which is 270°. And -90° is a 90° rotation in the other direction, so instead of multiplying by i... you can also divide by i ! Dividing by i is rotating backwards, and it's only logical, because 1/i = i^3 = -i !

Multiplying by a whole number like a+ib is nothing else than rotating by a certain angle + a certain homotethy, based on this numbers coordinates

I don't know how far you are in this, but if they talked about the euler formula, and linked it to trigonometry, it makes even more sense. exp(i*Phi) is the point on the circle with a radius of 1 which is placed at the angle Phi from the origin. It's also written exp (i Phy) = cos Phi + i sin Phi, and there you find its coordinates again (cos Phi, sin Phi) and now a whole world of rotations, cosinuses, imaginary cosinuses called "hyperbolic" and written cosh, sinh, tanh, which are cousins of the reals cosh, sinh, tanh which are just the projection of them on the real line of our plan, creates tremendous possibilities. For example, in physics, if I recall correctly, if you take a rope by its extremities, and consider its weight, it will hang in a cosh fashion, and that's also one of the efficients way to build a bridge, because the forces calculation can end in such a way depending on how you see it (if you divide it in different parts, consider the angle of the forces applied to it, and make the number of those parts tend to infinite, you'll tend to a cosh-ish formula... don't take my words on that cause I really don't remember the details, but you see what I mean)

If you see what I mean, 1+i=sqrt(2)/2*exp(i*Pi/4), you just passed from (x,y) coordinates to (rad, angle) polar coordinates

I found that thinking in terms of coordinates just made some sense to me, and made the term i less "fake". Other than that, you can just treat it with its rule i^2=-1 to just solve any problem, it really changes nothing to algebra as you knew it. It just opens the possibilities. It's used in topology, can be used in fluid mechanics, quantic mechanics, almost everywhere except maybe "basic" electricity. But you could use it in electromagnetics too, because in a wave's formula you find cosinuses terms, you see where Im going ! Also, the complex (we call it complex in france, i don't know how you call them in english) make any transformation writable in a single operation, because of its properties.

Even if you just consider the reals, the "out of the box" effect of discovering i creates more sense to them. sqrt(x)=1 has two solutions : 1 and -1, and it makes sense because if you think of multiplying = positive rotation, then rotating twice by multiplying a number by itself has 2 possibilities : 1 cause it multiplies by itself which is rotating by 0°, so it ends to 1, and -1 because it starts at 180° and multiplying by itself is multiplying by -1 which is 180° rotation and 180+180=360° so it ends to be 1.

Well don't just think of it in terms of rotations, but I know for me it's what just made it clearer and down to earth !

I was passionate about maths, so I followed the maths "prepa" course in france which is like an accelerated (2 or 3 times) uni course which leads to the national engineer/physicist/financial/prestige schools (and the national pilot course), and then i discovered this was easy and college maths aren't real "maths" and are only the 1% tip of the iceberg

 

[Plankis]

If you're having trouble with mathematics I can really recommend http://www.khanacademy.org/

 

[TesticleMilkshake]

Help a JBroll out.

I've been working on preparation for the classes I'll be teaching, and since I'm doing a developmental math class I need a little help from people who aren't math nuts...

Jeff

Hello fellow member of academia! Although I'm Humanities and Liberal Arts, I fucking loved algebra and statistics. What level of algebra are you teaching? Also, when you say high school students, what kind of group are we talking? Required for HS diploma, remedial math, APs or honors? Maybe some potential math, science majors in the college realm? I think the best thing that you can do is to provide a reference list of the formulas and equations (highlight what chapters they appear in) you'll be covering this semester; hand it out with the syllabus. And then maybe make a master list of terms, sample formulas? Also, if you are preparing for a state-wide exam (in NY we have the regents) the best thing to do would probably be to teach the test. Barron's made test books for NYS that I found extremely handy. Also, reference where algebra would come be handy in real life situations, regarding college, future jobs, etc. If you need help organizing and outlining, please feel free to message me or send me an email (ahr2128@columbia.edu)

Barron's books (new editions produced yearly) :

[ame]http://www.amazon.com/Integrated-Algebra-Lawrence-Leff-M-S/dp/0764138707/ref=sr_1_fkmr0_1?ie=UTF8&qid=1340507131&sr=8-1-fkmr0&keywords=barrons+algebra+NY+regents+review+2012[/ame]

[ame]http://www.amazon.com/Lets-Review-Lawrence-Leff-M-S/dp/0764135910/ref=sr_1_2?ie=UTF8&qid=1340507131&sr=8-2&keywords=barrons+algebra+NY+regents+review+2012[/ame]

[ame]http://www.amazon.com/Review-Algebra-Trigonometry-Barrons-Course/dp/0764141864/ref=sr_1_1?ie=UTF8&qid=1340507131&sr=8-1&keywords=barrons+algebra+NY+regents+review+2012[/ame]

 

[JBroll]

Well, this was bumped pretty hard. Wow...

Jef

 

[dale_a_smith]

thanks to every one who replied..

I was kinda skipping a step and making an arse of it..

I'm on simultaneous equations with 3 unknowns now, so I may be back soon for help.. haha