I am going through Lang's Basic Mathematics. Reading from

https://en.wikipedia.org/wiki/Mathematical_proof

check out methods and go dig deeper from there.

learn proof by induction.

ex: 1+2+...+n = n(n+1)/2

proof: case n = 1 , left side says 1, right side says 1(1+1)/2 = 2/2 = 1 , so the base case is true.

fix a natural number k, assume the proposition is true for k, then we show it's true for k+1:

1+2+...+k+(k+1) = k(k+1)/2 + (k+1)
= [k(k+1)+2(k+1)]/2 = (k+1)(k+2)/2
so the prop. is true for k+1.

Therefore by the principle of induction, the proposition is true for any natural number n (1,2,3....) .

QED

proofs by induction are a good way to practice basic proofs. Another common way is proof by contradiction (ex: show that there is no rational number r that satisfies r^2 =2)

>>8229036
There is only one way to get better at proving and that is doing it yourself.

It is something every person studying mathematics has to learn and the intuition you gain from it will be what makes you good at maths.

For starting out I would suggest doing very easy things like proving something like >>8229060 or some field or set properties.

A proof is nothing more than an argument showing why a particular statement is true. This means the following: Proofs should have words. A very bad habit perpetuated by bad mathematics teaching is doing stuff without words or explaining what's going on. Your work in math in general should be in readable, grammatically correct English. Your proofs should look like an argument. It needs to be air tight and convinving, but the format is something one can read and be convinced the given statement is true, as well as why specifically that is the case.

As others have said, the only way to learn proofs is to write them yourself. I will also add that when you read mathematics (such as the main body of the book you're currently reading), pay attention to how the author is writing their proof. Don't just read it go become convinced of a statement, make sure you understand everything aspect of the argument. When you really understand, you should be able to explain to someone else why the given statement is true.

>>8229036

Hi OP. /lit/ here.

/sci/ is somewhat humorously mystified by proofs.

Proofs are confusing to most people because they usually don't understand why the proofs have to be written in one order and not in another. Much of the confusion comes from the fact that western proof writing (and largely mathematics education up until quite recently) largely came from the proofs of Euclid's Elements, in which the proofs were simply instructions to be followed that would lead to an identical geometric construction. If you follow steps A, B, and C, you will arrive at the final geometric figure D. Constructing geometric figures with a compass and straight-edge are limited by the specific physical restraints of compasses and straight-edges, something not needed in algebraic constructions, which are only limited by written rules, thus the order doesn't appear to 'matter' as much to the uninitiated. How do I know their rules are "logical" and mine aren't? An so on. Looking at proofs as "purely abstract, algebraic entities" is a kind of charming anachronism that many modern mathematics professors like to promote as the Truth(tm).

Anyway, as an exercise, do not begin with a book on modern proof writing, as is often suggested to beginners here. Begin with Proposition I, Book I of Euclid.

As you're constructing the figures, take note of the order of the steps you are taking, and see what happens if you change the order. What happens if I begin by drawing a circle instead of a straight line? Does it become impossible to construct the intended figure? Does it take more steps? Less steps? The important part of the exercise is being able to translate your "moves" (drawing a circle) into "statements". This will give you a good, tangible foundation for abstract proof writing, which you can then draw analogies, rather than sit and stare blankly at the page until lightning strikes.

You can find the Elements online by typing into google "Euclid's Elements Heiberg .pdf".

>>8230518

To note, before the rabid undergraduates begin attacking this with "b-b-b-but m-m-m-muh non-euclidean geometries!!!!" It is important to point out that OP is

1. A beginner
2. The purpose of the exercise is to get into the habit of demonstrating why the order of operations matters, but beginning with tangible, physical constructions, NOT riemannian manifolds

>>8230518
>/lit/here.
Opinion discarded, go back to reading some hentai or mangas weeabo

>>8230585
Lol what? I'm not that anon, but /lit/ isn't for manga or hentai readings.. You're looking for /a/

>>8230591
>samefagging this hard
HAAHHHAHAHAH /lit/ always acting like pretentious faggots but they are really /b/ tier.

Please let the real science men talk.

>>8230599
THANKS FOR THANKING MY LADY

>>8230600
wewlad :^) this thred is amazing guys im all sciency now this tripguy is amazing
OH EAIT IM THE TRIP GUY hhehehauehua

>>8230518
Thanks for the information, why do you mention "humorously mystified by proofs", isn't proving the work of mathematics and logic combined? Thus by proving you are exercising both?

Are proofs just a meme?

File: 1440628203252.jpg (193KB, 629x960px) Image search: [Google]

193KB, 629x960px

I just finished Lang's Basic Mathematics.

Most of the proofs just involve fairly basic algebraic manipulation. But there are definitely a few exercises scattered throughout the book which require some creativity.

In general though Lang does challenge you quite a bit, and forces you to prove things that seem intuitively obvious. Once you get used to his style, it should make doing the proofs quite a bit easier. Don't feel back for referring to the answer section for the first few chapters until you get a feel for it.

I highly recommend that you try to do all of the proofs "left as an exercise for the reader". If you can do those then you are well on your way.