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I'm an advanced soon to be 7th grade student and I do a lot of self-learning. I have done Pre-Algebra, Algebra, and am about half way through Algebra 2. I am wondering what I should do next- Trig, Pre-Calculus, etc.? What course does anyone recommend?

Blake
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    Trig necessarily before you get to anything calculus related. Then during pre-calc you can get to more advanced trig. If you have good trig and algebra skills you will have a tremendously easier time in calculus. – graydad Aug 20 '15 at 03:23
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    If you can find any group theory stuff, it's pretty cool. It has to do a lot with the symmetries of things, and how we can look at that from a mathematical perspective. It won't rely on calculus or trigonometry, and it will be pretty different from what you have seen before. In my opinion, group theory contains some of the most beautiful and interesting mathematics. It's never too early to start it! – Ben Sheller Aug 20 '15 at 03:28
  • What exactly is group theory @BenS. – Blake Aug 20 '15 at 03:33
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    @Blake Don't worry about understanding group theory just yet, though I agree with Ben S. that it's cool. As graydad suggested, trigonometry should be your next step, followed by pre-calculus, then calculus 1, 2, and 3. – layman Aug 20 '15 at 03:34
  • Ok, what about stuff such as basic set theory? Should I just kind of sprinkle some of that in? @user46944 I already know the real, real basics such as complements, intersections, unions of sets, subsets, supersets, etc – Blake Aug 20 '15 at 03:38
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    @Blake Sure, really basic set theory such as what the symbols $\cup$ (union), $\cap$ (intersection), $\setminus$ (minus), etc. that appear in set theory could be useful. You will learn about the first two when you study pre-calculus because you use them when talking about the domain of a function. – layman Aug 20 '15 at 03:41
  • Ok thanks a lot for the help! @user46944 – Blake Aug 20 '15 at 03:42
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    @Blake No problem! Good luck with your studies! And please use this website if you ever have any more math questions. – layman Aug 20 '15 at 03:42
  • Another reason I think it would be good to work through calc 2-3 before doing set theory or group theory: Set theory and group theory are hallmark examples of pure mathematics (not to say they can't be used for applied purposes.) Everything you've done so far in math has been applied. For example, you get explicit numbers and must solve specific equations. $2x = 4$ etc. In pure math things are very general and a lot more abstract. Results are obtained through rigorous proofs. Going through calculus gets you some exposure to rigor and proofs. It makes for a nice transition. – graydad Aug 20 '15 at 03:49
  • Ok, I wasn't planning on going too deep into set theory, I'll just stick with what I know about it, I just know the basics, I won't go deeper yet. @graydad – Blake Aug 20 '15 at 03:55
  • You'll get a bit of set theory intuition from calculus too! Whatever you decide though, I wish you the best. Math is an incredible field and I'm envious of how much self-motivation you have for your age. I wasn't that gung-ho until my 20s – graydad Aug 20 '15 at 03:58
  • Haha thanks you helped a lot I have a better view on what to do now @graydad – Blake Aug 20 '15 at 04:00
  • @Blake By the way, as long as we're talking about set theory, here's something interesting: in axiomatic set theory, you (usually) only consider "pure sets." A pure set is something like {} or {{},{{}},{{},{{}}}}, or {{},{{}},{{{}}},…}; it's something that is either empty or contains nothing but other pure sets. So, if you want numbers, you're going to want to represent them in terms of sets. Usually, 0 is defined to be the empty set {}, so that 0={}. 1 is defined to be {{}}. Question is, what's the best way to continue this? How would you represent an integer? a rational? a real number? – Akiva Weinberger Aug 20 '15 at 04:42
  • Would {{{}}} represent 2 and so on adding a "pure set" as it goes on for integers? Only thing is I have no idea how you would represent negative integers... Or fractions @columbus8myhw – Blake Aug 20 '15 at 04:50
  • That's a good idea. Unfortunately, that turns out not to be the most useful way to do it. (The usual way to do it satisfies the property that there are $n$ elements in the set $n$. For nonnegative integers, at least.) As for the rest, I'll give no hint, but I will ask you another useful question: How can you represent the ordered pair $(a,b)$ as a set, if you have two sets $a$ and $b$? (This turns out to be important for the other ones.) If you wish to discuss this matter more, you can move this discussion to chat; if you want the answers, read any good book on axiomatic set theory. – Akiva Weinberger Aug 20 '15 at 05:03
  • How do I move a discussion to chat? @columbus8myhw – Blake Aug 20 '15 at 05:04
  • I made a chat room, but I think you have to have visited it before I can add you. – Akiva Weinberger Aug 20 '15 at 05:10

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I have read your original question and your interaction with Comments.

There are many possible directions for you to go. If you want to take the traditional path towards calculus and a math major in college, then trigonometry would certainly be a good choice. Independently of that (before, alongside, or after) you might do some basic set theory at the level of a 'discrete mathematics' course, or explore some fundamentals of logic. I think you are wise to explore just the basics of these topics for now. If you are headed towards a career in mathematics you will get the full version of these topics in due course.

More generally, the mathematical sciences include such topics as computer science, probability theory, and statistics. Disciplines that make heavy use of mathematics also include economics, physics, and engineering. Maybe you can discover some of the connections.

You should try to explore some of those fields a bit before you finish high school and go on to college. This might give you increased confidence in choosing the right major when you get to college.

Finally, you should listen to advice from family members, advisers, faculty members, and so on, to hear what they have to recommend. They may know a little about your strengths, weaknesses, and interests. Some academic subjects have stronger links than others to jobs that you can build into a career.

I think the most wonderful choice is from among those that would allow you to be paid what you consider a sufficient salary to do something you really love to do and will remain enthusiastic about. (I was lucky enough to make such a choice--well, the 2nd time, anyway--and I highly recommend it.) After listening to everyone's advice, $you$ should make the choice to head in a direction that allows you to do that.

BruceET
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