Number Theory Proof: If a|b and b|a then a = b or a = -b

Let a and b be nonzero integers. We prove that if a divides b and b divides a then a = b or a = -b. I hope this helps someone who is learning to write proofs.

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Thank you:)

[PhillipRhodes]

Hey Math Sorcerer, here's a question about proofs like this (and any others I suppose):

I have no problem following what you did there, but had you asked me to construct his proof from scratch I can tell you one of the big places I would get hung up. And that is know exactly what "things" I can take as given (or "axiomatic" if you prefer) and thus don't need to be proven as part of the proof being constructed.

Example: "for two integers to be multiplied and equal 1, they have to be either 1 and 1 or -1 and -1". I can see intuitively that that's true, but I can't see intuitively whether it's *merely* intuition, or whether it's something that *must* be accepted as true.

Also earlier when you said something about "the definition of division"... I don't think I've ever actually seen it written out explicitly anywhere what "the definition of division" is in a formal sense. It's just kinda presented as "here, here's how you divide these two things", but without necessarily saying "and thus you can assume the following definition(s) in future proofs".

So the question is, how do you internalize all those foundational rules and/or know what can simply be assumed to be true without any additional explanation required. I mean, obviously one could start everything from, say, the axioms of Zermelo-Fraenkel set theory, but that doesn't seem to be required for everything. For example, you never had to reference ZFS here.

Any help in this regard is greatly appreciated.

[sheatzu]

Hey there, thanks for all the great content. I was wondering if you could please do more basic proof writing (direct proofs/proof by

[Maths_.]

Proof writing is my favourite part of mathematics
Please upload more videos on proofs :)

[mrpalma]

Greetings from Brazil.
Your videos always help me understand a little bit more math every day.
Thank you so much

[johnchristian]

That was a really nice little problem, im still on my math journey and proofs seem so ethereal to me

[bigredwhale]

Hey math sorcerer where do you teach? Love your calculus lecture series btw.

[moon-scqt]

This was very useful. Thank you very much😊

[naas_the_serpent]

Thank you so much! I wasn’t even doing maths but theoretical philosophy :D in logic we unite!

[lambdami]

I think it would be simpler to notice that a|b => |a|≤|b| and b|a => |a|≥|b| so |a|=|b| and the theorem follows

[hussainfawzer]

What’s the software you’re using do these videos..
I mean the writing software

[singhravi]

Sir, To prove c*k=1 we can take a non zero integers c=x;x not equal 0 and k=1/x;x not equal 0, Instead of taking c=k=1. Can we do like this and why you take c=k=1.

[ahmedkeraani]

hey please excuse my stupidity if im wrong(I have a very basic understanding of math) but if ck=1 doesnt that also mean that c=1/k and k=1/c instead of c = k = 1 ? im propably getting something wrong but thanks for your patience

[anuragmalik]

Would you believe me if I told you that I received the notification for this video at the same time I was attempting the exact same problem from Daniel Solow's 'How to read and do proofs?'

[georgesinaihack]

What proof type is this?
What class would you typically see this proof in?

[sais]

I thought I would never need to use Algebra again in my life I was Wrong 🤣🤣

[ShaolinMonkster]

bruh, i read this proof today, what are the chances

[ChefFarisMom]

can you please elaborate "Let 𝑎
and 𝑏
be integers that are not both zero. Show that for all integers 𝑘
, gcd(𝑎−𝑘𝑏, 𝑏)=gcd(𝑎, 𝑏)
."

[gregorymorse]

This is simply wrong. a^2=1. Therefore a=+/-1. Therefore b=+/-1. Therefore a=b and a=-b. Rather than "or" which is ambiguous as it could be exclusive or. Gg.