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Blast into Math! Pure mathematics: the proof of the pudding is in the eating

9. Continue learning theorems and proofs. Think about the techniques employed in the proofs.
How are they similar? How are they different? What are the key ideas? Focus on what makes
the proofs work so that you can prove the theorems yourself in your own words without
needing to look at any books, notes or references.

The first correct proof we write is usually long and complicated. If we look at the proof again a day or
two later, we can often simplify it. In mathematics truth and simplicity are beautiful. After you have a

correct (beautiful) proof it is helpful to set it aside for at least a day and then re-read it later. You may
be surprised to see that you can simplify your proof and make it even more mathematically beautiful.

2.5 Exercises

Mathematics is experiential: it is only possible to learn through experience. What is “mathematical
experience?” Working on math problems! In pure mathematics research we solve problems which have
never been solved, and we prove theorems which have never been proven. It is not easy. We spend
days, months, and often years working on one problem. Each day we try various ideas, solve equations,
compute examples, and on most days our work can be summarized as writing on pieces of paper and

throwing them away. Although it seems futile, it is through this seemingly pointless struggle that we
reach a moment of mathematical glory, when an idea works, and we have overcome our mathematical
challenge. It is a totally amazing feeling of satisfaction! The only way to reach that moment is through
the hours of work which seemed to lead nowhere. So when you the reader are working on problems
but seem to be getting nowhere – that’s okay! Mathematicians around the world are making the same
struggle. Keep a positive attitude, and know that you are not working alone, you are working within a
global community of mathematicians.

Please make an earnest attempt

to solve every exercise.

Please carefully and conscientiously

write out all your solutions.

Take pride in and enjoy your work!

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