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Not only am I hoping you can answer my question, but perhaps refine my question itself. Unfortunately it is something I do not know how to ask, but I will give it my best attempt. Either I ask it, or it keeps pestering me.

A bit of history: I have little experience in mathematics, but want to harness this tool/language. I'm in early high school. Math seems different when I hear about it than when I'm doing it in my textbook.

My question is, how do people come up with formulas ? I understand when my teacher explains to me how something works, or asks me to test it with numbers e.g. draw a triangle and measure the sides, but how do people first come up with the formula? Do they do the same? Play around with numbers until they see a pattern and then write different ideas down and test them?

When math gets really complicated, how do people come up with it? Do they completely rely on algebra or do they just magically think formulas up (of course not but what is the alternative)?

I want to learn math, not just how to use a formula. If I start by learning my basic operations, and basic theorems, where do I go from there? How do I actually truly become acquainted with the way of thinking so that I can come up with formulas?

Thanks, I'm sorry if it seems naive. But I would appreciate if you tried to answer. I know there are seemingly many questions, but I asked them all in one post because the person who can answer the first set of questions, is the person who could answer the last set too. Please give me ideas for how to become better at mathematics. I know practice, which I have to start taking more seriously, but I feel like I am just practicing functions - orders of operations, algebra. I can't actually progress and do something myself. How do I get there?

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closed as too broad by daw, user91500, apnorton, quid, user642796 May 12 '15 at 17:26

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ It is a skill like any other. It takes time, patience and experience that's all. You obviously write well, but what would you tell somebody who asked you how to learn to write? What would you tell somebody who wanted to know where the ideas come from? We all get a lot of practice now with the internet, but there was a time about 20 years ago when it was practically a lost art. But you couldn't ask somebody how they learned to write, they'll tell you just practice and get better at it bit by bit over time. Ideas just bubble up from our subconscious, I don't think anybody really knows how. $\endgroup$ – Gregory Grant May 12 '15 at 10:16
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    $\begingroup$ You are thinking about running before you can walk. I wouldn't worry about these things yet; you have to focus first on acquiring the knowledge before you can understand how it came to be. It will take a long time but I guarantee that it is worth it. $\endgroup$ – Rammus May 12 '15 at 10:18
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    $\begingroup$ Learning the different methods of proof and how proof works will likely help you to get some intuition for how formulas came to be. It is also helpful to find out what (if any) problem the creator of said formula was trying to solve when they created it. This will get you thinking like that person and hopefully you can then understand the formulas origins. $\endgroup$ – Rammus May 12 '15 at 10:25
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    $\begingroup$ Mathematical thinking needs mathematical language; from the beginning in ancient times, the mathematical language has changed a lot but was and is a highly "specialized" language, made of "formulae". In order to "think mathematically" you have to "speak mathematically" i.e. to learn the language. Simple "algebraic" thinking needs a simple language : school algebra. "Very complicated" math thinking (research level) needs a much more complicated language. The only way is to study and practice with math language. $\endgroup$ – Mauro ALLEGRANZA May 12 '15 at 10:25
  • $\begingroup$ On a more specific level, the way to derive a formula depends on the formula. Some (ex: area of a triangle) are easier than others (ex: area of a circle). And I think there are over four hundred known proofs of the Pythagorean Theorem. $\endgroup$ – Akiva Weinberger May 12 '15 at 12:38