How to understand math intuitively?

[xavra42]

Hi, I am just starting to really get into math and I was wondering if there were any books for understanding math on a more intuitive level. I remember playing with blocks in like 2nd grade and you would put two lengths together to add. If you put a number of same sizes blocks together you would be a rectangle or a square if the values were equal. You can see this visually in the proof (a+b)^2 = a^2+2ab+b^2 if you actually split up the length of each side by two parts. But this doesn't really work ( or does it? ) when you start applying this idea of multiplication to other things like f=ma it doesn't make any sense. Another one I noticed is that the volume of a sphere is the integral of its surface area. I can't put this idea into words but i get the jist of it.
Anyways I think would be really cool if see mathematics this way rather than a series of deductions and rules. If mathematics doesn't work that way, then oh well, but I am very curious!

 

[chiro]

Hi, I am just starting to really get into math and I was wondering if there were any books for understanding math on a more intuitive level. I remember playing with blocks in like 2nd grade and you would put two lengths together to add. If you put a number of same sizes blocks together you would be a rectangle or a square if the values were equal. You can see this visually in the proof (a+b)^2 = a^2+2ab+b^2 if you actually split up the length of each side by two parts. But this doesn't really work ( or does it? ) when you start applying this idea of multiplication to other things like f=ma it doesn't make any sense. Another one I noticed is that the volume of a sphere is the integral of its surface area. I can't put this idea into words but i get the jist of it.
Anyways I think would be really cool if see mathematics this way rather than a series of deductions and rules. If mathematics doesn't work that way, then oh well, but I am very curious!

Hey xavra42 and welcome to the forums.

Understanding math intuitively in a deep manner usually takes a little while, so don't worry too much about not having the intuition right away.

There are many resources that discuss math visually for all areas of math including calculus, geometry, algebra, and so on. For suggestions it's best that you ask more specific questions about the areas of math you are interested in.

In my own understanding of mathematics, I have found that every single area of mathematics is about three things: representation, transformation, and constraints. I'll explain this in a more intuitive manner.

Representation is how you describe something. Usually in mathematics we describe things built up on numbers of various types (whole numbers, non-whole numbers, complex numbers), but this is not the whole story. You can for example break up 2 into 1 + 1 or 1.5 + 0.5. We do similar things with other things like functions, and other objects.

Transformation is concerned with taking an object and turning it into something else. The object can be a single number that is transformed into another number (function), it can be a geometric object that is squished, stretched, or transformed in some other way (geometry, mechanics in physics and engineering, topology) and it could be anything that you can describe and analyze (analyze means to 'break down').

The constraints are basically restrictions we force on something. They might be the axioms for a particular theory or mathematical construction, or they might be just some rules to make things manageable mathematically.

Without constraints, we can't really do anything because in a nutshell, 'anything can happen' and that is not possible to analyze (remember breaking down) which means we can't do anything with something with no constraints.

When you combine these three things, you have covered pretty much all of mathematics. When you realize what these things relate to in a piece of mathematics (like calculus, topology, algebra, and so on) it will begin to become a lot clearer and more understandable.

In terms of proving things in mathematics, we use all three of the above. We have structures, objects, and related definitions (representation), we have a target result that we are trying to get to (which has its own representation). We also have assumptions (constraints) that we either start with or introduce through the proof. Finally we usually transform things throughout the proof to eventually get to the target result.

The transformations may conserve things (like 2 = 1 + 1) or they may not conserve the original object and create something new like an approximation. Approximations are used a lot when they simplify things without changing them too much for the specific purpose.

You can represent all of these things in many ways including visually, and many teaching resources will do this. If you understand the above three concepts, you will eventually be able to follow all kinds of mathematics.

 

[xavra42]

Thanks for the reply chiro! I will definitely try to think in terms of representation, transformation, and constraints. The problem is that I know what mathematics does, but I want to know more about what it is. For example, formulas in physics can give you an intuitive grasp because you can feel acceleration, you can experiment with torque, and you can see any physical phenomenon obeying these principles. I want to feel this way with math. But mathematics sacrifices any context in order to become broadly applicable. I have read another thread similar to this one but the response is that there is no reason to know, just do it. I hope this isn't true.
Also, i went through a local university library to find books that would hopefully answer my question and it seems most of them are about the history/founding of mathematics, which is cool, but not what I am looking for. Amazon doesn't have good previews of the book so I don't want to buy something and have it be completely out of my level.

 

[chiro]

Thanks for the reply chiro! I will definitely try to think in terms of representation, transformation, and constraints. The problem is that I know what mathematics does, but I want to know more about what it is. For example, formulas in physics can give you an intuitive grasp because you can feel acceleration, you can experiment with torque, and you can see any physical phenomenon obeying these principles. I want to feel this way with math. But mathematics sacrifices any context in order to become broadly applicable. I have read another thread similar to this one but the response is that there is no reason to know, just do it. I hope this isn't true.
Also, i went through a local university library to find books that would hopefully answer my question and it seems most of them are about the history/founding of mathematics, which is cool, but not what I am looking for. Amazon doesn't have good previews of the book so I don't want to buy something and have it be completely out of my level.

Mathematics is just another language. Why it is used has to do with issues like abstractness, consistency, formality, and specificity.

The abstractness simply means that you can use to analyze many different kinds of things. This translates eventually in turning things into numbers and structures.

The consistency means that the language itself is consistent in the way that you can't contradict yourself when you use the various tools of mathematics.

The formality also relates to consistency, but what this means is that the definitions are clear and unambiguous (some call this rigor in part with the consistency). This is good because it means that doing analysis won't give results that don't make sense as long as your initial assumptions and later work make sense.

Finally the specificity means that you can define something as specific as required. This relates to formality, but it's a little bit deeper than that: the best way to think about this is to think about the languages that are spoken and written, and then think about all the times where people spend days arguing about what a simple definition really means. Mathematics helps get around this problem by creating definitions for things that are very specific in terms of what they represent in the clearest kind of terms.

Now the thing with regard to your question, is that mathematics is the study of analyzing things that vary in the deepest way possible.

Numbers are quantities that have variation. Anything that has potentially more than one possible representation is a variable. Anything that can take on different things is in fact a variable. What we do is we give a variable some kind of structure and the general form of something that varies is given by what is called a set and the theory for this is set theory.

So what we do is we take something that varies and we build a calculus around it. The simplest calculus that is intuitive is based on the algebra that you learned in primary school and in high school where we add, subtract, multiply, and divide things.

By taking something and converting them to numbers, we can then do calculus on those numbers which then gives us something else at the end of our analysis. We then interpret our results in the context of what that something is which will tell us something (hopefully useful) about our original something (sorry for bad grammar).

Mathematicians don't worry about what the variation actually refers to: they mostly care about doing the above four things for the language itself, as well as developing techniques to manage and effectively use various forms of variation at the most abstract level.

So this is what mathematics is: a way to deal and make sense of variation in the most general way possible. You then have people that use these results for applied purposes and also people who just study and develop things out of curiosity or for the sole purpose of going higher with the abstraction process.

In pure mathematics, a lot of the work deals with trying to come up with understanding large classes of things using the smallest theories and definitions possible. Think of it like compressing huge amounts of understanding in a single formula. This is what calculus did when it was created, and many many similar things are discovered and created very frequently in the mathematics fields.

 

[xavra42]

I totally understand how mathematics is set up and being a language. But, from my understanding, mathematics is an abstraction of a certain behavior/pattern that allows for calculation. You are definitely going to have to be vague and have rules; I have no problem with that. It just seems that the school textbooks don't focus on what that initial pattern the math models. I vaguely remember sine and cosine modeling the behavior of a circle and how its x / y values vary as you move around it. I want to more stuff like that. The books have proofs but they are usually just an accepted theorem that is manipulated a few times in order to a new useful one. I think if i can understand how these things work I can apply it much more easily because then I can see the behavior and think of functions that exhibit similar behavior. Right now I can only just memorize what a function does and when it applies. Again I am only a beginner so please feel free to correct me if I am wrong in my understanding of mathematics.

 

[algebrat]

what level math are you working on xavra42? it's sort of relevant to what type of idea we should offer

 

[chiro]

I totally understand how mathematics is set up and being a language. But, from my understanding, mathematics is an abstraction of a certain behavior/pattern that allows for calculation. You are definitely going to have to be vague and have rules; I have no problem with that. It just seems that the school textbooks don't focus on what that initial pattern the math models. I vaguely remember sine and cosine modeling the behavior of a circle and how its x / y values vary as you move around it. I want to more stuff like that. The books have proofs but they are usually just an accepted theorem that is manipulated a few times in order to a new useful one. I think if i can understand how these things work I can apply it much more easily because then I can see the behavior and think of functions that exhibit similar behavior. Right now I can only just memorize what a function does and when it applies. Again I am only a beginner so please feel free to correct me if I am wrong in my understanding of mathematics.

From what you have said, it seems that the kind of stuff that Wolfram Alpha is doing is perfect for your needs.

What they do is they have a web application that allows you to have a visualization of some particular mathematical system and then you vary the parameters (sometimes with slider bars) and the model will update itself dynamically and be re-drawn for your to get an idea of how changes in parameters effect the changes visually for the model.

There are also a lot of what are known as Java applications which do the same thing where you get a web application with textboxes or sliders to change values and everything updates instantaneously.

Using this we get a tonne of results which you can check out at your own leisure.

There are lots of these kinds of demos that you can utilize for your own learning and these forms of learning are becoming a lot more popular.

 

[coolul007]

For me it's the fundamentals, the numbers, the number line and the primes, prime factorization, the rules of Algebra. These are the building blocks that lead to the rest.

 

[Studiot]

Mark Levi's book was written for you.

http://www.amazon.com/dp/0691140200/?tag=pfamazon01-20

 

[xavra42]

From what you have said, it seems that the kind of stuff that Wolfram Alpha is doing is perfect for your needs.

What they do is they have a web application that allows you to have a visualization of some particular mathematical system and then you vary the parameters (sometimes with slider bars) and the model will update itself dynamically and be re-drawn for your to get an idea of how changes in parameters effect the changes visually for the model.

There are also a lot of what are known as Java applications which do the same thing where you get a web application with textboxes or sliders to change values and everything updates instantaneously.

Using this we get a tonne of results which you can check out at your own leisure.

There are lots of these kinds of demos that you can utilize for your own learning and these forms of learning are becoming a lot more popular.

Thanks for that site it helped a lot. I also picked up this book https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20 and I am liking it so far, but it will take me a while to get through. I have a lot of stuff I want to learn and especially on the application side, but getting the deepest possible understanding is what I am working on right now. Again thanks for the help!