Best Way to Learn Pure Math

In summary: It's a GCD, right?In summary, the book "A Concise Introduction to Pure Mathematics" by Martin Liebeck is a difficult read for someone with no prior experience in mathematics. The first chapter or so introduces definitions and basic ideas, but quickly moves onto problems that are difficult for the reader to solve. Chapter 1 of the book is only a taste of what is to come, and the rest of the book is devoted to proofs. This is a difficult subject to learn for someone who is not already familiar with mathematics. I would recommend starting with a book on geometry or number theory before diving into pure mathematics.
  • #1

NoahsArk

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I've wanted to learn pure math for a while. I bought the book "A Concise Introduction to Pure Mathematics by Martin Liebeck." The first chapter or so misled me into thinking that finishing the rest of the book would be doable. Chapter 1 gets into definitions of sets and related vocabulary and basic ideas like if then statements. It very quickly gets into problems like proving ## \sqrt 3 ## is irrational. Even the chapter on decimals is difficult for me.

Is there a better way to learn pure math- like through another book or online course? Other than pure math, the furthest I've gotten to in math is some calculus, and it was mostly from self-study. I'm rusty in most of the math that I do know since I haven't studied it in a while, but my understanding is that pure math is kind of a standalone subject and more related to logic. If there is anything I do need to brush up on or learn before studying pure math please let me know. Thanks
 
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Pure math is a huge range of subjects. All it means is you develop theorems from axioms, and prove things.

I took a look at the book and it moves pretty fast. For your first introduction to proofs, it starts off with some hijinky stuff. Proving every real numberhas a decimal expansion is mind bending because it's so obvious, why would you even need to prove it? Which makes it hard to see what the point is.

You might be better served by doing a book on geometry or just number theory first, to get the pattern down of what a proof looks like and how to know when you've actually solved a problem, and then revisit this book. Or ask lots of questions here.
 
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I appreciate your response! I will look into number theory and brush up more on geometry first. Thanks
 
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NoahsArk said:
I appreciate your response! I will look into number theory and brush up more on geometry first. Thanks
Don't hesitate to use our - in this case - precalculus homework forum and ask what you need to understand:
https://www.physicsforums.com/forums/precalculus-mathematics-homework-help.155/
... and show some of your efforts or thoughts. This is required there.

Number theory can become quickly tricky, depending on where you enter and what you consider. I would suggest trying to prove Bézout's lemma. It says that given any two integers ##a## and ##b## you can find other integers ##n,m## such that ##a\cdot n +b \cdot m= \operatorname{gcd}(a,b)## the greatest common divisor of ##a## and ##b.## All you need is the Euclidean algorithm for that. The Euclidean algorithm is the sophisticated word for division: Given two integers ##N## and ##D## there is are integers ##q## and ##r## such that ##N=q\cdot D +r.## It means: ##D## fits ##q## times into ##N## leaving a remainder ##r## that is smaller than ##D.## (Of course. Otherwise, we could set ##q## one higher and take then the remainder.) Formally we require ##0\leq r< D.##

Hint for Bézout's lemma. Set ##N=a## and ##D=b##. Then write ##a=q_1\cdot b +r_1## with ##0\leq r_1<b.## If ##r_1=0## then stop. Otherwise, write ##b=q_2\cdot r_1+r_2## and so on, until ##r_n=0## becomes zero. Why has it to end up at zero? What if you roll up this chain from the bottom back to the top?
 
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