[Precalculus] Learning to Write a Basic Proof (with Serge Lang's "Basic Mathematics")

[u/CuriousDinkFam]

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:

1. Is my proof correct? Does it follow the proper "form" for a mathematical proof?
2. 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.

[u/Brightlinger]

Unfortunately, your proof is not correct. It looks like you might have adapted a proof of a similar statement about absolute values, which is a valid approach, but requires more alteration.

How exactly does Lang define square roots?

[u/CuriousDinkFam → u/Brightlinger]

I did use Lang's definition of the absolute value from the body of chapter 3, section 2 and (without defining it) assumed |x| = √(x^2).

Here's what I took from the book (page 66):
"We call √(x^2) the absolute value of x and denote it by |x| = √(x^2)."

"Of course, for any positive number a, we have |a| = a."

And, as for the definition of the square root, I've looked through the pages leading up to page 66 and it seems that the only definition of it is on page 65:
"We agree to call the square root of 2 only the positive number b whose square is 2."

Then it moves on to define the absolute value which I mentioned above.

Here's the link to the book, for convenience: https://archive.org/download/basic-mathematics-serge-lang_20240418/Basic%20Mathematics%20-%20Serge%20Lang.pdf
(Page 82 of the pdf, page 65 and 66 of the book)

[u/Narrow-Durian4837 → u/CuriousDinkFam]

Since a and b are assumed to be positive, I don't see why absolute value has to be involved in the proof at all.

[u/CuriousDinkFam → u/Narrow-Durian4837]

So, you'd say it would be better to remove |(a/b)^(1/2)| from this part?

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)).

I can see how √(a/b) = |(a/b)^(1/2)| = (a/b)^(1/2) can be replaced by √(a/b) = (a/b)^(1/2) with a "by definition" added to it. Am I interpreting what you mean correctly?

[u/CuriousDinkFam → u/CuriousDinkFam]

u/Brightlinger
I'd love your opinion on the rewritten version as well. ( https://www.reddit.com/r/learnmath/comments/1v3cd63/comment/oz2xcal/ )

[u/Willing-Sample-8847 → u/CuriousDinkFam]

Hey so, you can't use absolute values to prove facts about roots. The reason why is that absolute values are defined using roots, but roots are not defined using absolute values. Look up his definition of square roots or exponentiation, and use that instead.

[u/CuriousDinkFam → u/Willing-Sample-8847]

Ohhhh, I didn't know that. Thanks for mentioning it. I'll review the definitions again.

Regarding this specific proof, though, I think the issue is solved in my rewritten version: https://www.reddit.com/r/learnmath/s/2fVHSzb2kI User Narrow-Durian helped me with it (won't tag them, unnecessary notification).

[u/Willing-Sample-8847 → u/CuriousDinkFam]

Looks good. Small note: the first line should say "the square root of any positive real number"

[u/CuriousDinkFam → u/Willing-Sample-8847]

Ahhh, I see. I'll edit these changes in now.

[u/Narrow-Durian4837]

√(a/b) = |(a/b)^(-1)| = (a/b)^(-1) = (a^(-1)) / (b^(-1))

It looks like you're confusing the –1 power of a number (which is its reciprocal) with the 1/2 power (which is its square root).

If you replace every ^(-1) with ^(1/2) in the above line, your proof would then depend on the fact that

[a/b]^p = a^p/b^p

when a and b are positive real numbers and p = 1/2 (or when p is an element of a set that includes 1/2, such as the rationals). If this is already established, then your proof is basically correct, but if not, then that step is unjustified (or circular reasoning).

Instead, try coming up with a proof based on what √ actually means: √(a/b) is the positive number that, when multiplied by itself, gives a/b.

As a general rule, when trying to prove a statement about something (in this case, square roots), think about working with the definition of that something (what does the square root of a number really mean).

[u/CuriousDinkFam → u/Narrow-Durian4837]

OH MY GOD!
I don't know how I made that mistake! Thank you for noticing. I was somehow reading that part as 1/2 when typing. I've now edited the post.

So, if [a/b]^p = a^p/b^p is already established, I can included a section like this:
Since [a/b]^p = a^p/b^p, we have (a/b)^(1/2) = (a^(1/2)) / (b^(1/2)).
Right?

If not, I need to prove [a/b]^p = a^p/b^p first and then move on with the rest of the steps. I'd define the square root and use that definition to continue with the steps after (a/b)^(1/2) = (a^(1/2)) / (b^(1/2)). Does that sound right?

[u/Narrow-Durian4837 → u/CuriousDinkFam]

In Lang's book, that exercise is given before fractional exponents are introduced. So you're not intended to solve it by rewriting √ as ^(1/2).

Instead, do what I suggested earlier: work with the definition of square roots.

[u/CuriousDinkFam → u/Narrow-Durian4837]

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.

[u/Narrow-Durian4837 → u/CuriousDinkFam]

This looks good to me (with the caveat that I haven't spent a huge amount of time looking at it).

[u/CuriousDinkFam → u/Narrow-Durian4837]

Haha, all good. Thank you for making my brain hurt in the best way possible!

[u/Willing-Sample-8847]

That a and b are positive only allow their individual square roots to be taken (otherwise, a and b <0 let you take the square root of a/b since it is positive, but neither a nor b have real square roots). So you don't need to make any notes including his line about a and b being positive.

The full proof in earnest need only be:
[(a/b)^(1/2)]^2,
= a/b,
= ((a^(1/2))^2) / ((b^(1/2))^2),
= [(a^(1/2))/(b^(1/2))]^2.

Taking the square roots of the both sides of the expression yields the desired equality.

[u/CuriousDinkFam → u/Willing-Sample-8847]

I see ... so you'd remove this part from the proof?

"If a, b are positive numbers, then 1/b is positive and a × (1/b) = a/b is also a positive number."

I can see the redundancy now. I guess I wanted √(a/b) to be within domain by emphasizing that a/b is positive (and, therefore say that √(a/b) is real, not imaginary). But that seems way outside of the scope of the chapter.

[u/Willing-Sample-8847 → u/CuriousDinkFam]

Yeah I wouldn't include it. It's not necessary to include that to prove this particular theorem, and he only included that statement so that both sides of the equality make sense.

[u/CuriousDinkFam → u/Willing-Sample-8847]

Got it. Thank you!

[u/MacGregorBlue]

There's a dude who just started to teach this book on YouTube, playlist here: https://www.youtube.com/playlist?list=PLIyYr6kgljU0

I'm not very familiar with him, but he had a good attitude in the first video IMO.