How Do You Differentiate Between Pure and Applied Mathematics and Start a Proof?

[michaelknight]

Hello, all :) I was just wondering a few things:
1) what is the difference between pure mathematics and applied mathematics, and which classes do you need to take in order to get your Ph.D. in either subject?
2) I know this is a really large branch off, but I was wondering how do you start a proof? When I do proofs, the beginning is always the hardest part for me, but when I get going, I can do proofs fairly easily. I guess what I'm asking is how do you pick the subject for the proof, and then proceed from there?

Thank you for anyone who is willing to help :)

 

[daveyinaz]

http://en.wikipedia.org/wiki/Applied_mathematics

http://en.wikipedia.org/wiki/Pure_mathematics

The subject of the proof you are trying to do will depend on the class you are taking and the homework exercise you are doing.

 

[20Tauri]

My understanding is that pure mathematics deals more with proofs and theorems, and applied mathematics deals more with computations and modeling, i.e. real world applications.

I don't know about doctoral curricula, but my grad school research suggests that MS students in applied and pure math largely study similar things, with the applied mathematics students spending more time on differential equations and/or statistics/probability-type courses than the pure math students, who seem to do more algebra and topology.

 

[ArcanaNoir]

Applied mathematics is generally about actual numbers, and in pure mathematics the numbers are largely irrelevant. It's more about the properties.

For example, applied mathematics might have you either computing the tension on an anchor from two strings, or perhaps trying to find such an equation. Pure mathematics would be examining how many ways you can screw with a matrix and still get useful information from it, or perhaps how you can take two separate bits of math (say calculus and matrices) and stick them together and do calculus on matrices. So pure math is more about what kind of math can you do and applied is more about finding the physical answers.

The beginning is always the hardest part of a proof. You just have to do so many that you can kind of tell in the beginning what direction to go in.

I found a lot of interesting properties of groups when I was doing the exercises in my algebra book. Exercises are a great place to start, because they give you something to do with the math. The more you move the math around, the more you see how it moves, and you can start to see little patterns, and then you say "I wonder if this is ALWAYS true?". You can find lots of books of exercises for any branch of math.

 

[Matrix0]

I am not an expert in mathematics, but I can provide some general information about the questions you have asked.

1) The main difference between pure mathematics and applied mathematics is the focus of the subject. Pure mathematics is more theoretical and focuses on abstract concepts and ideas, while applied mathematics is more practical and focuses on using mathematics to solve real-world problems. To obtain a Ph.D. in either subject, you would need to take a combination of advanced mathematics courses and conduct independent research in your chosen field.

2) Starting a proof can be challenging, but it is important to have a clear understanding of the problem and the concepts involved. One approach is to break down the problem into smaller parts and try to prove each part individually before putting them together to form the final proof. It is also helpful to look at examples and previously solved proofs to gain a better understanding of the techniques and strategies used. Ultimately, practice and persistence are key in developing the skills needed for successful proofs.