Atm I’m a comp sci major, but I told my advisor I’m planning on switching to Mathematics because CSC just isn’t for me =\
Both majors are similar though.
Well, here’s a weak example… Obviously there are much more difficult subjects in Discrete, and the topics vary greatly. I didn’t have a problem with Automaton, but for some reason Induction proofs kicked my ass.
There is an example here, which is pretty basic… All I can say is take my word for it that it gets insane. When you are writing 3 page papers for a single problem on tests, you know its a fuckin whopper.
i suck at math. last year i tried tutoring my friend in differential equations, which back in the day i 4.0ed easily, , but its been too long, and i was like, fuck dude, i’m useless in math now.
i want to learn more about fractals, but there is no fractals for dummies book. the math is probably beyond me anyway. PWNED. anyone read mandlebrot’s book?
I’m not entirely sure if you saw the pdf you gave me, but this actually consists of 3 problems whose content is less than a page each. Nevertheless, the induction was giving you problems. For the example you gave me, it is really a matter of understanding and becoming familiar with proving things by induction. I’m assuming your professor didn’t give you a lot of practice problems to do because that’s what ended up helping me with it.
The first step P(n) = true is pretty trivial really. For the P(n+1) step, you must use the fact that P(n) is true to show P(n+1) is true. What helps me is that I plug in n+1 to see what my goal is. Since you have to use P(n) somewhere in that inequality, you can work backwards until you get stuck, then you can work forwards from your assumption that P(n) is true until you get stuck. Then somewhere in the middle is where the ingenuity and some algebra tricks come in to get you to the goal. These aren’t too bad once you’ve seen a bunch of them.
If you’re willing, I’m willing to listen to what else gave you problems. So far it seems the proof technique was the stumbling point for you, but that was just one example…
I took Calc 2 and Physics Mechanics at the same time, and made an A in Calc 2, but never made higher than a 70 on my Physics tests. I still ended up with an A because they curved the hell out of my class, though (the class average was in the 30-40 neighborhood).
I used to be indifferent about math until taking Calculus AB at my school. Now, I find it enjoyable and much more interesting imo. My teacher is also pretty cool and he’s teaching some BC stuff that might help us, like the shell method. I just don’t want to take the AP test, mainly because I always get myself in a nervous wreck, even if I know the material. Did any of you who took the AP test find it hard?
Mind you I’m no expert, but in the 5 minutes I fished around on the internet, here’s what I gather. Creating fractals appears to the the result of a finite number of iterations on a certain equation. The number of iterations seems to be one for every pixel on the screen.
Essentially, if you want a Mandelbrot set, you create members of this set using an certain iterative process on the equation
f(z)=z^2+c
where z is a complex number and c is a constant. Each value of this set gets plotted on a screen and gets assigned a certain color depending on the iterative process.
What’s interesting is that depending on the constant value that you plug in for c, you can get an entirely different set from the one you had before.
I had a couple of links to give you, but then I noticed some pop up ads came up, so just google “complex numbers + fractals + mandelbrot sets.” The essence of creating a Mandelbrot set is really as simple as plugging in a value into an equation say, 25,000 times.
Enjoy
If the class was geared towards math majors this is how it should be taught. Linear Algebra is about maps between vector spaces, which the book mentioned below covers well, not necessarily elementary row operations.
You might be thinking of Axler’s Linear Algebra Done Right. Determinants are introduced way at the end, but he makes a point to avaoid them as much as possible. The funny thing is that another great linear algebra book, Shilov’s Linear Algebra, starts right away with determinants. Shilov’s book might be the best out there and it’s a $15 Dover edition now. I’m currently refreshing my knowledge with Axler and I’ve gone through a good amount of Shilov, but like a lot of other people I haven’t used it much lately and it sucks to forget.
Well, you really should know linear algebra before doing calc 3. The whole point of differential calculus is using linear maps to approximate stuff.
In the sense of set inclusion, R is a proper subset of C, but in terms of cardinality using bijections, R and C are the same size. http://en.wikipedia.org/wiki/Cardinality_of_the_continuum#Sets_with_cardinality
Like I said I’m in the same boat as so many others. I’m currently reviewing my linear algebra and vector calc because I forget so much. I hardly know any topology (whatever I learned from Rudin), and my stats background is pretty weak.
Yea you’re right,that was a bit ambiguous. I should’ve specified that.
Shilov seems interesting (and dirt cheap too!). Our class used Howard Anton’s book. I’m also in the middle of reviewing Linear Algebra as well, mainly so I can start studying this Functional Analysis book I got, but also because I like vectors and matrices and all that stuff.
Ok, I read over the wikipedia article on Mandlebrot set.
http://en.wikipedia.org/wiki/Mandelbrot_set
From what I understood in the article, a complex number c is in the mandlebrot set IF z(n+1) = z(n)^2 + c remains bounded as n hits infinity.
The article gives an example, for c=1 and z(0) = 0, the function goes to infinity, so its not in the set. whereas c=i and z(0) = 0, the function is bounded, so i is in the set.
Then to generate the picture, http://en.wikipedia.org/wiki/File:Mandelset_hires.png, the x axis is the real component of c, and the y axis is the imaginary component of c. A pixel is black if its in the set, and white if not.
What I don’t understand is:
- how do you add color to the picture
- how do you tell mathematically whether or not a function is bounded?
Don’t be man, asides from the MC part, Free Response is quite easy (partial credit ftw), not to mention the curving.
The nationwide average score on the Calc AB/BC AP exam is like a 4ish every year.
Any time you’re able to get credit for a college class, take that opportunity. A 5 will usually give you 8 credits (Calc 1 and 2), and a 4 will give you 4 credits (Calc 1).
Once you get a feel for what the AP test asks you won’t be worried. Look at these: http://www.collegeboard.com/student/testing/ap/calculus_ab/samp.html?calcab
It’s basically the same 5 or 6 questions every year for free response. For AB, guaranteed area between curves, volume of revolution (shell method is an AB topic as well), at least one FTC problem (water flow, velocity/distance traveled, rate of ticket sales), probably a slope field and separable diff. eq. Plus google “Lou Talman” for full solutions. Your test, if you take it, will be exactly like every single test on there.
…which explains why you added two positives and got a negative. You’re really good at this!
I’m taking Electrical Engineering in school right now, and I must say the math can really frustrate me sometimes, especially since things are mostly theoretical.
There are only two things that really bother me in math with regards to my major:
Laplace Transforms & Fouriers Series
This shit can get ridiculously complicated when you are trying to implement them in actual electrical circuits and models.
wikipedia up twos bit complement
I think you misunderstand. I aced the class. I had issues with it 10 years ago, not yesterday. I was trying to give examples of what I had issues with WHEN I took the class. I just googled a document to show what type of problems Discrete Math contains - and I remember Induction gave me problems. I thought I mentioned that the problems I had difficulty with WERE 3 page+ answers - but I didn’t find an example like that to show. Theres a WHOOOOOOLE lot more to Discrete than Proofs… If I remember right the class AVERAGE when I was taking littel discrete was 63 (me, and 3 others had 90s which killed the curve - we made our own study group together.) Big Discrete was easier, and I also aced that in a similar manner. I think that class average was in the 50s.
Discrete is no joke. If you want to learn more, get a book… And I agree, a good teacher and reading your book and working the problems makes it doable - I put the time in to pass it. Im not saying its impossible, im just saying I had to study more for this class than any other math class I ever took, and I went through all the typica CS math courses.
I’ll check it out. That phrase sounds familiar.
I thought I quoted Black Jesus but I guess not. Now I deleted that post, possibly causing confusion lol.
Ok, so I went back to my CV book by Wunsch. So, essentially, the criterion for inclusion into the Mandelbrot set is the following:
- pick a complex number c
- plug it into f(z) = z^2 + c
- define the following sequence:
z_0 = 0
z_1 = f(z_0) = (z_0)^2 + c
z_2 = f(z_1) = (z_1)^2 + c
etc
Now here is the kicker:
if this sequence converges to a finite number, you’re in the Mandelbrot set.
so, for example, c = -2 is in the Mandelbrot set since:
z_0 = 0
z_1 = 4 - 2 = 2
z_2 = 4 - 2 = 2
z_3 = 4 - 2 = 2
so the sequence converges to 2
EDIT If you want to be rigorous about it, pick n = 1, then notice for n > 0 where n is an integer, |z_n - 2| < epsilon for any positive epsilon. It should also be noted that just iterating 4 terms SUGGESTS convergence. It does not PROVE convergence.
Regarding your original question, it can be shown that any convergent sequence is bounded. That was sort of the hidden assumption in the wiki article you cited. However, being bounded isnt enough for convergence. For instance, the sequence sin(n) is bounded at 1 and -1, yet it diverges as n goes to infinity.
Using your example of c = 1 demonstrates an example of something that is not in the Mandelbrot set
z_0 = 0
z_1 = 1
z_2 = 2
z_3 = 5
etc
and so the sequence does not approach a finite number as n grows arbitrarily large. Hence, c = 1 is not in the set.
The nice thing is that it can be shown that if z is in the Mandelbrot set, so is z conjugate. I think this is the reason for its symmetry. What I’m still confused about (in the 20 minutes I spent on this subject) is how come I see one picture symmetrical with respect to the a line on the imaginary plane (say z = i) with the complex plane, and yet in another picture I see the same set rotated 45 degrees to the left.
Now, if you’re in the set, you get assigned a black pixel. What’s interesting is that if you’re not in the set, you get assigned a color depending on how fast the sequence at c diverges. My guess is that the computer uses some O(h) function to compare how fast z_n converges relative to another function, perhaps for instance, 1/h?
Also, it seems as though there is only one Mandelbrot set, but Julia sets do not appear to be unique. A glance at the snippet on Julia sets in this book suggests that one assigns values to both z and c. Furthermore, one need not use z^2 + c. (for example, z^2 + (lambda)z could be used).
Anyways, that should get you started. One other interesting (and computationally efficient) thing: If c is in the Mandelbrot set, so it its conjugate. Hence twice the computational power at about half the cost if you assign this to your algorithm.
Yea, looks like I did :lol: sorry. Glad to see your efforts paid off. To be honest, some of the discrete stuff I’m learning now seems a bit wierd, but its because I’m so used to relying on all the calc/analysis stuff I’m used to. But no, Discrete math is pretty useful once you get used to it. It’s just a different set of axioms that you need to get accustomed to.
After finishing real analysis, I never wanted to see that blue book (Rudin) again. On the other hand, abstract algebra is infinitely interesting. Different strokes for different people I guess.
You intentionally left out the part where I mentioned using Windows calculator.
Damn, Jesus, why you gotta try to go Judas on me like that? That hurts man. I’m taking back the fat bitch I got you for your birthday.