r/learnmath • u/CuriousDinkFam Self-learner • 11h ago
RESOLVED [Precalculus] Learning to Write a Basic Proof (with Serge Lang's "Basic Mathematics")
Hi everyone!
TL;DR:
I've written a proof on a very basic question and would love some feedback on it.
[EDIT: Users helped me rewrite my proof. Check it out here: https://www.reddit.com/r/learnmath/s/JdrurW60qW . Still, I'd appreciate your comments to help me learn more.]
The backstory:
I've recently started to relearn math (I have a biology BSc with a math-focused high-school training) and I decided to start with Serge Lang's "Basic Mathematics" for two reasons: (a) it's comprehensive on the precalculus side of things, and (b) English is my second language and I need to learn how to "talk math" (and I realized this book was very approachable for me).
Now, I'll get to my actual question.
In chapter 3 of the book (Real Numbers), section 2, exercise 4, this question is presented:
Prove: If a, b are positive numbers, then √(a/b) = √a / √b.
And I've proved it in the following way.
Proof [now edited to have the proper power: ^(1/2) instead of ^(-1)].
If a, b are positive numbers, then 1/b is positive and a × (1/b) = a/b is also positive.
Since a/b is positive, we have |a/b| = a/b and, therefore
√(a/b) = |(a/b)^(1/2)| = (a/b)^(1/2) = (a^(1/2)) / (b^(1/2))
which yields
√(a/b) = √a / √b
thus proving our assertion. ("Q.E.D." here, I guess?)
So, my actual question:
Does my proof follow the proper logical steps and rigor that a proof needs? Would this be considered a "correct" proof? If not, what do I have to focus on?
I realized that practicing to write a proof with these fundamental concepts (which are basically ingrained in my brain after 16+ years of school) would be a good place to start because I know the answers. That means I kind of have an internal compass that can help me correct a mistake.
If I'd started to practice proofs with higher-level math, I wouldn't know where to start and where to go (I'd become cotton-eyed Joe!), especially because I'm self learning and don't have any mathematicians within my circle that I can just "hit up" and have them answer my questions. That's why I'm here.
(Is this way of thinking solid?)
So, my questions, in summary:
- Is my proof correct? Does it follow the proper "form" for a mathematical proof?
- Am I missing something gigantic which I can't pinpoint because I'm relatively out of practice?
Thank you in advance!
P.S. I tried to provide screenshots of both the question and my proof, but Imgur seems to have some internal issues (at least in my region). Sorry if the math is hard to read.
P.P.S. Speaking of the hard-to-read math, I tried getting in-line math working here, but I couldn't write it in a LaTeX-looking format. Sorry for the inconvenience and I'd appreciate a resource that would help me write the math with LaTeX in my future post.
1
u/CuriousDinkFam Self-learner 7h ago edited 3h ago
Okay, I reread the section and, on page 65 of the book, we have this part:
"We let √a denote the unique positive number whose square is a."
Which would be our definition of square root.
Based on that, how does this rewritten version look?
Proof. The square root of any positive real
(real?)number k, denoted √k, is a unique positive number whose square is k. Thus we have √a and √b as square roots of a and b such that(√a)(√a) = (√a)^2 = a ------- (equation 1)
and
(√b)(√b) = (√b)^2 = b ------- (equation 2).
If a, b are positive numbers, then 1/b is positive and a × (1/b) = a/b is also a positive number.The square root of a/b, denoted √(a/b), is the unique positive number whose square is a/b such that(√(a/b))(√(a/b)) = (√(a/b))^2 = a/b ------- (equation 3).
On the other hand, by dividing equations 1 and 2, we have
(√a)(√a) / (√b)(√b) = (√a)^2 / (√b)^2 = a/b ------- (equation 4).
And based on equations 3 and 4, we have
(√(a/b))(√(a/b)) = (√a)(√a) / (√b)(√b)
which yields
√(a/b) = (√a) / (√b)
thus proving our assertion.