Exercises 2.3 and 2.4 & Did I buy the right book?

[SparseSpeculator]

I liked your book and I worked along the chapter about polynomials as recommended: Making up examples along the way and making sure I understood the formulas. Now I have gotten to the exercises for this chapter.

I don't know how to even try and solve exercises 2.3 and 2.4

Either I am overlooking something or the road to these exercises did not in the least equip me with any of the tools I would need to tackle them. I was motivated enough by the beginning of "A programmers introduction to mathematics" to dedicate an hour each day learning math. The exercises make me reevaluate this decision. Who is this book for? Someone already well versed in mathematical proofs? If not, how am I supposed to develop these skills with the exercises given?

For the live of me I don't know why mathematicians do this... The chapters are challenging but understandable, the first couple of exercises are really easy and then BAM: "YOU THOUGHT YOU WERE ABLE TO DO MATH? YOU THOUGHT WROOOOONG!!!"

I noticed this pattern in other math books: It is like a beginners guide to programming going over basic control structures in the first chapter and then asking about the difference between abstract base classes and interfaces in the exercises.

So, after the venting, as someone who really liked everything up to the exercises: What should I do here?

[j2kun]

I'm sorry to hear you're struggling with the exercises. I know it can be frustrating. For what it's worth, I don't think you need to solve every exercise, and certainly not in strict order, to get through the book. Some exercises are meant to pique your interest in topics I didn't have space to write about, some are meant to be a bit more challenging than others, and some are meant to be like small projects that take you out of the confines of the book.

All that said, I'm happy to help you work through these two exercises. For 2.3 ("Verify the following theorem using the examples..."), note I'm not asking you to prove the formula is true, just to try a few examples to confirm the theorem works, and to hopefully recognize it if you read about RSA in the future. If you try, e.g., a = 3 and n = 8, you get phi(8) = 4 and 3^4 = 81 = 8 * 10 + 1 has remainder 1 when dividing by 8. Feel free to try some more to see that they work.

For exercise 2.4 (in the first edition this is "A number x is called algebraic...", I'll assume you're working with that one), the goal is to determine whether φ = (1+√5) / 2 is algebraic. To prove it is, we need to find a polynomial p(x) for which p(φ) = 0. This is the sort of "follow your nose" exercise where you fiddle with φ and come up with a working polynomial.

But if you try that and it fails, what can we do? We can try to simplify things. If φ were something simple like √5, we wouldn't have much trouble because x = √5 is a root of p(x) = x^2 - 5. But there are these pesky other numbers and fractions in the way for φ. Can we tinker with φ to get it to look more like √5?

Well, what if we temporarily let y = 1+√5, so that φ = y / 2. If y is algebraic for polynomial p(y) = 0, can we modify p to get a new polynomial q where q(φ) = 0? Then we have reduced the problem to proving that y is algebraic. And y looks closer to the simpler case of √5...

I can certainly give more hints, or explain it outright, but please take a moment to try again, for no more than ten minutes. Let me know if you still don't find a solution. √2+√3 is a bit harder, but if you see a general rule that comes from solving the case of φ, I think you will see how to prove it is algebraic.

[SparseSpeculator]

Thanks for your swift reply and the time you took to answer in depth!

I actually had done exercise 2.3, so gave the wrong number there. It was from 2.4 onward, that the exercises looked menacing. Your tips help and I will tackle at least 2.4 with those in mind.

I still think the hand-holding you provide here with your answer is the require level for someone like me to have a challenging but not impossible exercise. Even exercise 2.1 was weird for me. On the face of it, it looked obvious that degree(f*g) = degree_g + degree_f. But then I looked up the proofs online and had to contend with "Fields" and "multiplicative inverses". Which made me unsure whether I was on the right track. I get that those are probably edge cases or overarching concepts that we usually ignore when just playing with formulas less rigorously, but they make getting into proofs really difficult for the layman.

Books like yours that do a good job teaching the concepts both from the ground up and without much formalism are great but they often fall apart in the exercises. Here (at least from my perspective) they usually have one or two warm up questions and then want you to proof stuff that originally took even people like Euler more than one evening.

On the other hand of the spectrum there are books that just teach application. They have plenty of incremental exercises but they teach calculation, not math. I would love to train my underdeveloped proving-muscles but I just cant find the right weights.

Maybe I am just a whiner or my goal of learning this stuff on my time budget is unrealistic. I don't want to be an ungrateful: Your book and advice are great. I just wish to make the most of it.

Anyways, thanks for the reply! I'll stick with your book. But I will either have to skip most exercises or badger you here for most of them.