# Best online sources for learning maths to a 16 year old?

Last year I tried Single Variable Calculus (very good course) and look some videos of Multivariable Calculus on MITOCW and I cover almost all the material.

So, when I took that course I realised I want to learn about mathematics. I want to be a PhD physicist (and I know that it cover a lot of Advanced Mathematics), but I don't want to learn mathematics just to do physics, but also for DOING mathematics. I like very much to learn few concepts in new subjects than learning deeper and boring things about the same theme again and again.

So, what are the best sources for learning mathematics to a 16 year old by my own?
Remind I know calculus.

Figure out some goals first. What do you really want to know eventually? Then I can suggest some material to get there.

• Student100
Figure out some goals first. What do you really want to know eventually? Then I can suggest some material to get there.

My goals are to learn advanced mathematics to, for example, understand advanced discussions about mathematics like in Nature web or even in Physics Forums, because I don't understand anything about what their talking about. I want to learn mathematics for doing mathematics professionaly.

My goals are to learn advanced mathematics to, for example, understand advanced discussions about mathematics like in Nature web or even in Physics Forums, because I don't understand anything about what their talking about. I want to learn mathematics for doing mathematics professionaly.

OK, but after calculus, mathematics really isn't linear anymore. There are a lot of different directions you could go in and all are worth studying.

So, is there something really specific you want to understand? Or is some general mathematic ok?

OK, but after calculus, mathematics really isn't linear anymore. There are a lot of different directions you could go in and all are worth studying.

So, is there something really specific you want to understand? Or is some general mathematic ok?
Yes micromass, as I know very few about that directions (differential equations, linear algebra, differential geometry...),

fresh_42
Mentor
Have you tried what you can learn in wiki? On the one hand there's really a lot that can be learned and on the other it might help you to figure out where you want to deepen your understanding. There's a long way to go from 1-dimensional real calculus to complex Riemannian manifolds. Books in those fields are expensive and you might get recommendations that will not fit your requirements. I have been tempted by myself to name you a book "Real and Abstract Analysis". But the chances are high you might get frustrated. Therefore my recommendation to go to wiki.

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If you know calculus already, you could try dipping into some analysis through Spivak or Apostol. Though I must add, I've already had a full year and a half of university calculus and I'm finding the transition difficult.

Can you tell me about Spivak and Apóstol?
Are there free legal online sources for that?

Ok, first of all thanks for you answers!

Learning by only reading a book is very difficult for me.
I'm familiar for learning in online couses like MITOCW and EDX and at the same time I can complement it with a book related to that.

So do you know about some good Mathematics online courses?

Ok, first of all thanks for you answers!

Learning by only reading a book is very difficult for me.
I'm familiar for learning in online couses like MITOCW and EDX and at the same time I can complement it with a book related to that.

So do you know about some good Mathematics online courses?
Sorry, no. If you want to understand higher mathematics and physics, you will need to be able to learn from a book. You're not going to get an adequate understanding of e.g. complex analysis by just watching some videos. That might have worked for easy stuff like calculus, but it won't work for much longer.
If you do have trouble self-studying, you could always find somebody who is willing to tutor you.

• Student100
Ok, good to know that.

But, can an you give other links for books about other fields of mathematics?
Because I want to know about as many fields as I can to have a better knowledge about general advanced mathematics better than knowing deeper a field and having no idea about another fields.

Student100
Gold Member
Last year I tried Single Variable Calculus (very good course) and look some videos of Multivariable Calculus on MITOCW and I cover almost all the material.

So, what are the best sources for learning mathematics to a 16 year old by my own?
Remind I know calculus.
How well do you actually know calculus? If I gave you problems, such as, $$\int_{0}^{1} \frac{\sqrt{arctan(\theta)}}{\theta^2+1}d\theta$$ $$Prove \lim_{x\to 10}6=6$$ $$y'= \frac{dy}{dx} \:\text{for}\: y=x^2y^3 + x^3y^2$$ $$\text {Find the Maclaurin series of the function}\: f(x)=e^x\: \text{and its radius of convergence.}$$ $$\text{Show: For fixed a and fixed}\: f : R^n → R, \: \text{the greatest value of}\: D_uf(a) \text{ over all unit vectors u is |∇f(a)|}.$$ $$\text{evaluate}\:\iiint_E (x^2+y^2)dV, \text{where E is the region bounded by the cylinder}\: x^2+y^2=4\: \text{and the planes z=-1 and z=-2}$$ $$I =\oint_C \frac{3z+2}{z(z+1)^3} \,dz \: \text{Where C is the circle |z| =3}$$

Are you comfortable doing these? Obviously, some are more difficult than others, but the idea is to show you that the statement "I know calculus" demands more understanding than a course in single variable and watching multivariate videos online. There's plenty more to study in subjects you're already introduced to! If I were you I would move to linear algebra, then go through a multivariate book such as Anton. Listen to Micromass, he won't steer you wrong.

How well do you actually know calculus? If I gave you problems, such as, $$\int_{0}^{1} \frac{\sqrt{arctan(\theta)}}{\theta^2+1}d\theta$$ $$Prove \lim_{x\to 10}6=6$$ $$y'= \frac{dy}{dx} \:\text{for}\: y=x^2y^3 + x^3y^2$$ $$\text {Find the Maclaurin series of the function}\: f(x)=e^x\: \text{and its radius of convergence.}$$ $$\text{Show: For fixed a and fixed}\: f : R^n → R, \: \text{the greatest value of}\: D_uf(a) \text{ over all unit vectors u is |∇f(a)|}.$$ $$\text{evaluate}\:\iiint_E (x^2+y^2)dV, \text{where E is the region bounded by the cylinder}\: x^2+y^2=4\: \text{and the planes z=-1 and z=-2}$$ $$I =\oint_C \frac{3z+2}{z(z+1)^3} \,dz \: \text{Where C is the circle |z| =3}$$

Are you comfortable doing these? Obviously, some are more difficult than others, but the idea is to show you that the statement "I know calculus" demands more understanding than a course in single variable and watching multivariate videos online. There's plenty more to study in subjects you're already introduced to! If I were you I would move to linear algebra, then go through a multivariate book such as Anton. Listen to Micromass, he won't steer you wrong.
OK, I can solve the 3 firsts and the six one.
Thanks for all your replies. I will be reading the linear algebra book.
Thank specially to micromass. I read on your profile you are available to do some mentoring about mathematics. Could you make me some mentoring?

Student100
Gold Member
here is an online lecture series in linear algebra that is very clear:

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/
I like Strang's book (for the applications), but his video lectures are weak on proofs and follows kind of a weird course format to me. Determinants being all the way towards the middle of the course seems strange to me, since you can use them to argue so many of the things he's already covered. Anton's book is probably better for a first course.

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Thank specially to micromass. I read on your profile you are available to do some mentoring about mathematics. Could you make me some mentoring?
Sure, feel free to send me a Private Message anytime.

fresh_42
Mentor
The skills to solve the problems above are very much needed in studying physics. However, advanced mathematics is sometimes of a more abstract kind. I have one from that category:

If you draw half a circle above [0,1], i.e. with radius 1/2. Then your line is of length 1 and the circumference is π.
Next you draw 2 halfcircles within the above of radius 1/4 and the centers 1/4 and 3/4. Then the baseline is still 1 and the sum of all halfcircle lines add still up to π. This way you go on drawing halfcircles each time with half the radius than before.
The more steps you take the closer will the smaller halfcircles approach the baseline. But still the length of the baseline is 1 and the sum of all halfcircle lengths π. So as your circles converge to the baseline, why doesn't π equal 1?