

Ask HN: What is the right way to learn Mathematics? - roansh

Please note that I am not asking for good resources, I am asking for the &quot;right way&quot;.<p>The reason I learned to code in the first place was because of my fascination with the idea of AIs in childhood, but when I realized later on that to learn it you need Math, I was in despair. I have decided to (RE)learn everything from the very basics. I have been watching videos and solving examples on KhanAcademy since last two weeks now. (Thanks to Sal). But once in a while I stop and ask myself -- whether it&#x27;s the right way. Whether I will have true insights into Math when I am done with this. I really need to know the right path before I spend years going in the wrong direction, only to find nothing.<p>The areas I am interested in are -- Artificial Intelligence, Image Processing, and Computer Graphics.
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ChaitanyaSai
It might also help to know what parts of math are intuitive and what parts
aren't. Our brains just can't visualize tensors, but we can visualize
quantities. Trying to relate tensors to the spatial geometry of the world
around us -- the world we can perceive -- may not lead to much.

I found this book helpful

[http://www.amazon.com/Number-Sense-Creates-Mathematics-
Revis...](http://www.amazon.com/Number-Sense-Creates-Mathematics-
Revised/dp/0199753873/ref=sr_1_1?s=books&ie=UTF8&qid=1423298459&sr=1-1&keywords=the+number+sense)

~~~
roansh
Thanks! I'll look into it! :)

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brudgers
Learn the mathematics that's applicable to the area in which you are
interested. That means it's recursive:

    
    
        function learn x
            if x.prerequisties.know()
            then study x
            else learn x.prerequisites
    

Not iterative. So start with AI if that's what you want to know.

That's not to say that building up iteratively from foundations is bad. But if
it's not directed from the top level, then it may not lead there.

~~~
roansh
Thanks for writing. I like your approach. I actually have started from the
very top level. Let me explain further as why I asked this question in the
first place with examples.

Two days ago I (re)learned that (-1) x (-1) = (1). I read many resources
explaining why that is true, but it just doesn't strike as 1 + 1 = 2.

To give one more example, just to make clear what I am asking about -- Today I
learned about slope of the line, which is = change in y / change in x. Even
that doesn't strike. I have already learned these things believing that they
are true in my school years. I don't want to learn in the same way.

~~~
cottonseed
It is important to distinguish between axioms (things you assume), definitions
and theorem (things you prove from axioms, stated in terms of definitions).
So, for basic arithmetic, your assumptions might be the defining properties of
numbers: associativity, distributivity, etc. In this setting, 1 + 1 = 2 is
basically a definition, while (-1)* (-1) = 1 is something to be proved:

    
    
        0 = 0* (-1) (0* x = 0 for all x)
          = (1 + (-1))* (-1) (x + -x = 0)
          = 1* (-1) + (-1)* (-1) (distributivity)
          = (-1) + (-1)* (-1) (1* x = x for all x)
    

implies:

    
    
        1 = (-1)* (-1) (equality; add 1 to both sides)
    

Finally, slope = rise/run is a definition. It is not something you "see" to be
true.

So assumption vs. definition vs. theorem can explain part of it. Another part
is familiarity. 1 + 1 = 2 is ubiquitous in daily life. Depending on what you
think about, (-1)* (-1) = 1 may not be. You probably have lots of
interpretations of 1 + 1 = 2 (counting, number line, shopping, money, etc.)
How many such interpretations of (-1)* (-1) = 1 do you have?

I highly recommend this book, written by two mathematicians:

[http://www.amazon.com/5-Elements-Effective-Thinking-
ebook/dp...](http://www.amazon.com/5-Elements-Effective-Thinking-
ebook/dp/B008JUVDUE/ref=sr_1_1?s=books&ie=UTF8&qid=1423244565&sr=1-1&keywords=5+elements+effective+thinking)

It's now about mathematics per se, but about learning how to learn and think
better. Good luck.

~~~
roansh
Awesome! Really liked the way you proved (-1) _(-1)=1.

What I am doing is trying to relate every single new thing I am learning with
real life examples, only to make sure that I have _learned* that thing. Most
of the times I fail at that. But I guess it's not always possible.

~~~
ralmeida
That may not always be possible (or even necessary) to relate to a real-life
example, but as long as you develop some sort of intuition, of "feeling" about
what you learned, you did not fail at learning that.

For example (taken from the article I mentioned on the other topic), when you
think of a derivative, what is the first thing that comes to your mind?

The equation that defines it? ([http://calnewport.com/blog/wp-
content/uploads/2008/11/deriva...](http://calnewport.com/blog/wp-
content/uploads/2008/11/derivative.png))

Or this intuitive image? [http://calnewport.com/blog/wp-
content/uploads/2008/11/tangen...](http://calnewport.com/blog/wp-
content/uploads/2008/11/tangent_to_a_curve.png)

~~~
roansh
I am yet to tackle Calculus, so nothing comes to mind. But I am agree with you
that, relating all of math concepts to real life won't be necessary as long as
I truly understand it.

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wodenokoto
This article hit the front page a few days ago, I think you will find it very
informative.

[https://medium.com/@warrenhenning/a-software-engineers-
adven...](https://medium.com/@warrenhenning/a-software-engineers-adventures-
in-learning-mathematics-62140c59e5c)

~~~
roansh
Thanks for writing! I actually read that article that day, and it was really
on time. It's a good motivation, but it doesn't talk about _how_ one should
learn Math. I find the commends below very informative and I've concluded that
I need to spend some time on finding better ways to learn, and what suits me
well.

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brogrammer90
Check out the book "The Calculus Direct" by John Weiss.

~~~
roansh
Thanks for the suggestion, will check that book.

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Mz
1) Make sure you understand the concepts, not just how to crunch the numbers.

2) Make sure you understand how the math relates to the actual real world and
is a means to model actual reality.

I know you did not ask for resources, but really good ones are hard to find.
So I will note that a good source for good math articles (as well as good math
answers) on HN is Colin Wright:
[https://news.ycombinator.com/submitted?id=ColinWright](https://news.ycombinator.com/submitted?id=ColinWright)

