Hello I am thirteen and self studying real analysis

[bit188]

Hello! I am thirteen and self studying real analysis... I'm wondering where to go after this; kinda looking for a general outline of what to do from here on out. Any help is greatly appreciated. Thank you.

 

[JohnDuck]

Just find some subject that looks interesting, and if it turns out to be above you find something simpler on the same subject. That's what I've been doing lately--I picked up a book a while back on chaos theory (because of Jurassic Park ), but I found it was a bit too much and traded it in for an introduction to nonlinear differential equations.

 

[malawi_glenn]

Hello! I am thirteen and self studying real analysis... I'm wondering where to go after this; kinda looking for a general outline of what to do from here on out. Any help is greatly appreciated. Thank you.

well it depends on what you are intressted in. maths, physics, cemistry?

 

[math_owen]

If you're 13 then your progress is truly amazing. And it would be awesome to see how you do in later years. Excellent work young friend.

What you want is Calculus by Spivak. This is "the" calculus book. But it is really an analysis book sprinkled with some basic abstract algebra. It's expensive, so I'd get an used copy. But trust me, it's what you want.

When I was in analysis class (after calc 1 - 4) I read Spivak's book. My analysis book for class was Real Analysis by Bartle. It is good too, but, I would suggest Spivak's first. After those two comes a classic, Rudin's Principles of Mathematical Analysis. It is affectionately known as Little Rudin, because there is an even more advanced book by him.

Good luck young mathematician.

 

[math_owen]

bit188 I don't think I answered your post the way you wanted. But I would still get that book by Spivak. If you can move through that then I'd say you really know analysis well enough to move on.

The oversimplification of pure math is these 3 areas; analysis, algebra, and topology/geometry.

If you honestly can read Spivak or Bartle, then I would say next is Topology by Munkres, and at the same time Abstract Algebra by Dummit and Foote.

From there, the math world is yours.

Best of luck!

 

[bit188]

Thanks for the advice, math_owen; sadly, I can't even afford Spivak used (I checked Amazon); however, I can afford an older edition of Baby Rudin.

Mostly, though, I have stick with Dover's books. They're fairly comprehensive and easily understood, at least from my experience. Currently, I'm using Kolmogorov and Fomin's Introductory Real Analysis.

Sadly, though, I'm worried about how far inexpensive math books will take me before they "run out". Eventually they're going to get expensive -- hopefully I'll have a job by then!

Thanks again for the great advice. Analysis is actually going pretty well, and it's a lot easier than differential equations and linear algebra -- for me, conceptual stuff is far easier than computational, and so writing proofs seems to be right up my alley. There's just so much room for error with computational mathematics; get your algebra wrong, do a bad substitution when integrating -- it can be infuriating at times. Glad that's over!

So... introductory topology and algebra, and that's it? Then I can do whatever, essentially?

...then I want to be an analytic number theorist. Algebra was fun and geometry was interesting enough, but once I hit calculus -- it was incredible. And I continue to be amazed at analysis every single step of the way.

One particular experience that I remember is when doing Laplace transforms in ODE's; that's when I first met the gamma function. The whole "one half factorial" thing was incredible to me, and I wanted to do more with that. However, my limited mathematical background is holding me back from doing so!

...And now I've just begun real analysis, and I've got a long way to go still! I know that math's going to keep me on my toes my whole life long.

And -- I've begun to develop a passion for physics as well. I'm far, far behind in physics -- I'm only doing introductory mechanics with calculus. Even though I enjoy physics very much, sometimes it's like banging my head against a brick wall until I get the problem right. It's far more difficult than mathematics for me. Maybe I lack the physical intuition needed to be successful in that area.

The other area of "hard" science that I enjoy is computer science. Math and comp sci are easiest for me, while physics is exceptionally difficult; comp sci just seems to come naturally. Maybe I think like a computer? I'm learning MIPS (assembly language) and Java, as well as reading Introduction To Algorithms and Operating Systems: Design And Implementation right now. I use Mac OS X for two reasons: it's UNIX based and (as a possible illustrator) I like the interface.

Visual art is another passion of mine. I like to draw, and I would love to be a children's book illustrator some day; my style is extremely simplistic -- it's something that would suit a show on Cartoon Network. You can see my Photobucket page http://s185.photobucket.com/albums/x75/bit188/".

Anyhow, I will continue to do mathematics, computer science, and (attempt) to do physics my entire life long. They are my passions (in addition to literature and visual art).