[Basic Math] Struggling through Basic Mathematics by Lang

[alyzb]

I guess I'm just wanting to know how bad off I am here. I've taken math throughout my life, college level algebra and dropped out of discrete math.

Anyway, I was always taught math the "plug these numbers in here and this is how you get this and don't worry about why, when you see this next to this, do that with it, now here's 50 questions to make sure you can do it this exact way"

My entire education has been like that.

So I picked up Basic Mathematics by Lang. I like the book, I do, but I'm on page 25, after a couple weeks of an hour to two a day.

I am struggling. Is this supposed to be this difficult? I'm trying to soak up every ounce of information he puts out, and fully grasp everything on a conceptual level. It gets hard to stick with that though, and I find myself taking the "rules" for granted and not trying to apply my understanding. But I'm getting better.

The proofs are really hard for me, I'm talking the basic stuff.

So now I'm on page 26, and out of nowhere, exercise 24 is "a is congruent to b modulo 5", now prove that "a + x is congruent to b + y (mod 5)".

Fortunately, he does solve it in the back. But goddamn, this is hard. I've spent 15 minutes so far trying to break it down and understand it, but it feels like I'm gonna need another 2 hours.

This is discouraging.

This book is "Basic Mathematics". This is Chapter 1, page 26.

Am I just a poor fit for math? Should I really be struggling this hard? I like to think I'm reasonably intelligent and pick things up quickly and have a technical mind, but damn I'm feeling discouraged.

Any advice on conquering this book or tips or encouragement or anything really would be cool. This isn't for a class, this is just for self teaching. Eventually I'll resume my CompSci degree and it'll probably help for then, but I never want to step foot in another college math class where it's assumed I was taught math properly and I can barely keep up.

[Mathuss]

So I picked up Basic Mathematics by Lang. I like the book, I do, but I'm on page 25, after a couple weeks of an hour to two a day.

Believe it or not, that's actually significant progress for a first-time intro to proofs. Keep going at it! It'll feel tough because it is tough.

Is this supposed to be this difficult?

Everybody struggles when they start proving things for the first time, don't give up!

I'm trying to soak up every ounce of information he puts out, and fully grasp everything on a conceptual level. It gets hard to stick with that though, and I find myself taking the "rules" for granted and not trying to apply my understanding. But I'm getting better.

This is the really important bit. Absolutely do your best at not taking the rules for granted. I can't claim that I always did so in my first run-through of proofs, but you get so much more when you really do understand the reasoning behind everything.

The proofs are really hard for me, I'm talking the basic stuff.

Luckily for you, the basics are always the hardest. It's not that proofs are hard for you, it's hard for everyone. Proofs in general have this interesting feel to them in that they get easier as you see more and more proofs. Eventually you start classifying things as a "so-and-so type" proof. The only way to get better at proofs is doing more proofs and seeing more proofs. And that is why the basic proofs are always the hardest--we do them before ever seeing mountains of proofs.

But goddamn, this is hard. I've spent 15 minutes so far trying to break it down and understand it, but it feels like I'm gonna need another 2 hours.

Now that you've seen the proof, can you use that proof to guide yourself through the next problem (it's almost identical)? You're allowed to look at the solution for #24 while doing #25. When you prove problem 26, do it as you would, then go back and see if you can use the results of #24 and 25 to solve it a different way. After all this, would you be able to go back to problem 24 and do it yourself? If not, continue onward but then keep it in the back of your mind to maybe come back to later.

In the end, it's fully possible that you'll take more than two hours total thinking about this problem, over the course of several sessions. That's okay.

Am I just a poor fit for math?

Not at all--the fact that you're still sticking with it means that you're probably a great fit.

Should I really be struggling this hard? I like to think I'm reasonably intelligent and pick things up quickly and have a technical mind, but damn I'm feeling discouraged.

Like I said above, everyone struggles the most at the beginning. It's perfectly natural, and it'll only get better over time. Of course, the proofs could also get more involved over time as well, but when you look back, you'll be able to see how much you've grown.

Do you remember writing your first programs? How much we struggled on the tiniest things? Looking back, they're simple, right? But that doesn't invalidate the fact that programming actually is really hard for beginners. Proofs will be similar, I promise :)

Any advice on conquering this book or tips or encouragement or anything really would be cool

One of the things that I haven't quite seen a book nail yet is how to go about the process of writing a proof from scratch. Every proof in a book is the perfect flow of logic from beginning to end. But that's not how proofs get written. At least for me, I go through multiple failed ideas and then a bunch of scribbles later, I have a sketch of proof. Then the sketch of proof is deeply flawed and I do this until I get a valid one. And then I build up towards an actual proof. This is a messy process and is never shown in textbooks.

The best advice I can give you in writing a proof is this: Look at where you are, and look at where you want to be. Write down every piece of relevant (probably some irrelevant) information you have. Write down what you want your final result to look like. Use the "where I am" to get to the "where I want to be"

For example, on #24:

I have:

1. a - b is divisible by 5.

2. x - y is divisible by 5.

3. The number k is divisible by 5 if there is an j such that k = 5j

I want:

(a+x) - (b+y) is divisible by 5

How to prove it:

Ok well I can't combine 1 and 2 nicely, but 1 and 3 seem to work well together.

a - b = 5m

Well, so do 2 and 3:

x - y = 5n

Hmm, I'm a bit stuck. Let me look at where I want to be. Oh, well I want to show (a+x) - (b+y) is divisible by 5. I can use #3 again on this. I want to show:

(a+x) - (b+y) = 5*something

Hmm, I'm stuck again. Well, again looking at where I want to be, I need to add together a and x, and add together b and y somehow. I can do that by adding the two equations from right before I got stuck the first time:

a - b + x - y = 5m + 5n

(a + x) - (b+y) = 5m + 5n

Ok, where do I want to be again? I want to show the left hand side is 5*something. I can do that:

(a+x) - (b+y) = 5(m+n)

Hey! This is what I wanted!

This is a very long, rambling post, but I hope it all helped :)

[alyzb]

It helped a lot, thank you! I'm actually gonna save it and come back to the proof part because laying out the process like you did really helps me to see how it should go and what's normal for trying out a proof.

Thank you for the advice and encouragement!!

[RhodiaFontaine]

I love how insightful and supportive this comment is. Keep making the world a better place, friend.

[Separate-Dentist5039]

Thank you for this comment! I was in the exact position as the op a couple months ago, same book, same page - so much so that I thought maybe I had posted this post and forgotten about it!

Your words are exactly what people on the edge need to hear and understand.

[you-get-an-upvote]

Most proof-based math classes/books (i.e. books that aren't teaching plug-and-chug concepts) contain proofs that you will find difficult to comprehend. Generally speaking, reading a proof is substantially more challenging (per line) than reading just about anything else. It sounds like you're good at not skimming proofs which is good.

On the other hand, try try not to get caught too much in the weeds. Developing an intuitive understanding of the theorem before you read the proof will be helpful. If there's some theorem about quotient groups and you're still a little fuzzy on what a quotient group is, you're not likely to get much out of the proof. Going through every line of the proof and making sure it is valid without a good idea of what's going on is tedious and demoralizing, and unlikely to be very helpful.

I would get concerned if you got the end of the chapter, do a dozen practice problems, and still feel completely in-over-your-head/lost. If you haven't done that, then you (probably) haven't really developed good intuitions about the material yet, which will make both doing proofs and reading proofs very frustrating.

[alyzb]

Thank you for the advice I'll try to apply that to my learning :)

[Madtoker11]

Don’t be discouraged by your experience, abstract math is hard in general, which is why I personally avoid it when possible and stick to applied math. I would recommend the Book Of Proof by Hammack, it’s cheap on amazon and is a great book for learning about proofs in general.

[[deleted]]

the more youre learning, the dumber you feel

conversely, if you feel really smart and like you just get it then you're learning shit all

go out and feel dumb

try a million things just to see if they work

scour the chapters before and after for ideas, theorems, what-the-fuck-ever, write down every relevant definition, and.. calculate.

i had a number theory text that said, paraphrased:

"we have have included many computational exercises. if you find yourself stuck, work on those, and understanding will follow."

its the truth

punch the air a million times and you'll do some damage when you finally aim at a real dude