How should I get into 'higher' mathematics?

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Mooseeeeey
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How should I get into 'higher' mathematics?

Postby Mooseeeeey » Wed Feb 02, 2011 11:25 pm UTC

Hello everyone. I'm 13 and have been wanting to look at online courses/lessons that can help me get introduced into things like Geometry(Taking next year) and Calculus(3 years from now) and pretty much any other math type that would help me. Anyone have suggestions?

aneeshm
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Re: How should I get into 'higher' mathematics?

Postby aneeshm » Thu Feb 03, 2011 3:58 pm UTC

For an introduction to the world of mathematical thinking, there is nothing better than Euclidean geometry, specially at your age.

I recommend this book - it's old, and very good.

afk2011
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Re: How should I get into 'higher' mathematics?

Postby afk2011 » Thu Feb 03, 2011 7:26 pm UTC

Personally, if you are just interested and want to get into higher mathematics, at your age I would maybe consider reading some non-intensive but extremely interesting books that highlight some interesting mysteries/discoveries without boring you with rigorous math. For example, Godel, Escher, Bach, or also I really like "Prime Obsession" by John Derbyshire. It differs from person to person, but for me number theory was extremely interesting, and there are a number of small theorems in number theory that someone just starting out with higher-level mathematics could prove.

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Yakk
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Re: How should I get into 'higher' mathematics?

Postby Yakk » Thu Feb 03, 2011 8:53 pm UTC

Here is something I played with at your age:

Take a polynomial, like f(x) := x^+3x+2. Evaluate it at x=0, x=1, x=2, x=3, x=4.

These are your unit distance samples.

Look at f(1)-f(0), and f(2)-f(1), etc. These are the difference between your samples.

Now look at [f(2) - f(1)] - [f(1)-f(0)]. These are the differences between the differences in your samples.

Keep going. Anything interesting happen?

How about the polynomial f(x) = x+2? Or the polynomial f(x) = x^3 + 1?

Figure out some rules. Try the rules "backwards".
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Mooseeeeey
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Re: How should I get into 'higher' mathematics?

Postby Mooseeeeey » Thu Feb 03, 2011 11:30 pm UTC

Hey Yakk, I've been trying the problem you gave, and either I'm a complete idiot or you made a typing error.
Should the original f(x):=x^(3x+2) or x^(|3x+2|) or x^(missing number) +3x+2?



Oh and, aneeshm, that book link said lowest price was $322. Cheaper elsewhere or is that actual price?

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Kurushimi
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Re: How should I get into 'higher' mathematics?

Postby Kurushimi » Fri Feb 04, 2011 1:27 am UTC

I think you and Yakk are talking about different things, because the phenomenon I think Yakk is talking about should work with any polynomial.

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Re: How should I get into 'higher' mathematics?

Postby Dopefish » Fri Feb 04, 2011 1:34 am UTC

I'm fairly sure they meant f(x) := x^2+3x+2 (aka [imath]x^2+3x+2[/imath]), which would actually be a polynomial, which have interesting relationships with their n-th differences. (The other non-polynomial possibilities don't have any particularly special properties pertaining to their nth differences so far as I'm aware.)

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z4lis
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Re: How should I get into 'higher' mathematics?

Postby z4lis » Fri Feb 04, 2011 2:06 am UTC

I don't see why you can't go pick up a book on basic set theory and logic. The first experience I had with "real math" was a book on formal logic. It was a hard-bound, really old, purple thing that I found at a local bookstore. It was difficult, but really interesting to work through. You might also try doing some discrete math, like graph theory. Also, don't let your age discourage you. I'm not really sure what it matters. I know people who are 30 who can't understand things that a 13 year old can. If you enjoy mathematics enough to want to learn "higher" math, then it's probably because the concrete math is too simple for you, and you're ready to explore more abstract areas. Just don't get discouraged if you pick up a book and have no idea what's going on. Find a new book and read that, instead. There are tons of books that I "just couldn't possibly understand" earlier on in my readings that I won't even think twice about now. Just ask questions on here if you get stuck.

Also, you may find some really famous problems that nobody has solved, and try and solve them. You'll quickly find that they're unsolved for a very good reason. :P Don't worry about those.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

Mooseeeeey
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Re: How should I get into 'higher' mathematics?

Postby Mooseeeeey » Fri Feb 04, 2011 5:09 am UTC

All right well I'm back, and I believe I have found the pattern for f(x)=x^3+1
I solved f(x) up to x=6, and believe I found the pattern. As I got to line 3 on my piece of paper I noticed they were all multiples of 6 counting up. I did a Google search for something like "f(1)-f(0)" and looked at a webpage and I'm guessing(based on title of the page) this property is called the Zero of polynomial function? Am I close to the expected results?

Also, I asked my dad to take me to the book store tomorrow to see if we can get any books on these subjects. :D

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Re: How should I get into 'higher' mathematics?

Postby Robert'); DROP TABLE *; » Fri Feb 04, 2011 1:44 pm UTC

You've found the right result, but the proper term for what you've found is the "differential" of the function. The zero of a function is the x value for when [imath]f(x)=0[/imath]. The differential is another function, based on the first, that tells you how [imath]f(x)[/imath] changes as you change the x value. Taking the differences of [imath]f(0), f(1)[/imath], etc, gives you the values for the differential function.
...And that is how we know the Earth to be banana-shaped.

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Yakk
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Re: How should I get into 'higher' mathematics?

Postby Yakk » Fri Feb 04, 2011 2:43 pm UTC

Oops, sorry about dropping that 2! Yes, it was supposed to be x^2. Keep on trying with different polynomials. Vary the coefficients and the degree.

See what rules you can invent. Check to see if you can break them. See if you can do any predictions, and how much detail you can put into those predictions.

As noted, going backwards is interesting. From the differences, can you work out what the polynomial was?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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the tree
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Re: How should I get into 'higher' mathematics?

Postby the tree » Fri Feb 04, 2011 4:30 pm UTC

Presumably you're doing some sort of precalculus or algebra or whatever you're calling it at the moment and you're only just being properly introduced to polynomials. As it happens, polynomials are kind of really really important when it comes to nearly the rest of mathematics. Learn to factor them, learn to multiply them with ease, learn to divide them with one another, try not to get too bored because all of that is important when it comes to getting a feel for really using them.

Already in this thread you've had a glimpse of differentiating polynomials which is basically what the first stage of calculus is, but with more general rules (what you're finding is a derivative, not a differential, they're related but not the same thing). If you do the same exercise for [imath]x^2[/imath], [imath]x^3[/imath], and [imath]x^4[/imath] then you can guess the derivative of [imath]x^n[/imath] and then wahey you've gone dun some calculus.

Spoilered for not that important right now:
Spoiler:
The derivative of [imath]f(x)[/imath] (which should properly be called "the derivative of [imath]f(x)[/imath] with respect to [imath]x[/imath]") is often denoted [imath]f \, '(x)[/imath] (pronounced f prime of x) or [imath]\frac{\mbox{d}}{\mbox{d}x}f(x)[/imath].

There are various rules for finding things like the derivative of f(x)+g(x), f(x)g(x), f(x)/g(x) and f(g(x)) if you're really, really, really keen on learning these then you may be way ahead of your class by the time that you get to calculus.


The 'zero' of a polynomial (or of any type of function) is also known as the 'root' and that should be what you're doing right now. Finding roots of and factoring quadratic equations takes up a large part of the precalc experience, even though you'll just learn a few rote methods you'll be a lot closer than you realise to one of the most fascinating parts of modern algebra.

Spoilered for way beyond your level:
Spoiler:
To find solutions for [imath]ax^2 + bx + c=0[/imath] you'll often use the equation [imath]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/imath] which is all well and good, there's a similar (but much bigger) equation for cubic equations (polynomials of degree 3) and one for quartics (polynomials of degree 4). Now something called the Fundamental Theorem Of Algebra states that an [imath]n[/imath] degree polynomial must have up to [imath]n[/imath] roots which might suggest that there is a similar equation for solving a polynomial of degree five, but it turns out there isn't, at all. That's not to say we can't find it, we actually know that it can't possibly exist.


If you just do the work you're being set in school and then once you're done try to investigate a bit further, we can help you out and point you in the right direction. If what you learn in school is a generic guided tour of a city given in the same way to every tourist group that passes through then it's up to you to actually go into the buildings, try the food, talk to the locals and ask more challenging questions of your tour guide.


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