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In proofs, how much of a logical leap is too much?
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> How much can be left for the reader to understand and how much is necessary to write in the final proof?

Depends on who your reader is. Things that are obvious to an undergrad math student will not be obvious to a high-school student.

For example, the statement *"for a natural number n, if n^(2) is even, then n has to be even too"* does not need any clarification if you assume your audience intimately understands the fundamental theorem of arithmetic.

On the other hand, if you are showing people a proof that √2 is irrational, and that's one of the first formal proofs they are seeing, you should clarify how do you get from *n^(2) is even* to *n is even*.

>Is there some kind of standard that I should follow or do you gain intuition for it through experience?

Experience of knowing your audience. Also, ask for, and listen to, feedback from more experienced mathematicians.

> are the only tools for proofs the ones that are mentioned in this book or are there other, more advanced proving techniques?

No idea what's mentioned in the book, but the answer is yes, there are more advanced proving techniques.
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Textbooks and published papers leave out a lot of algebra, saying things like “it is easy to show this simplifies to [thing that takes an hour and three pages to convince yourself]”

But logical steps? I’d be very leery of omitting logic unless it’s really obvious, like “the reasoning is exactly the same for x < 0 using -x instead of x.”

I’d also be inclined to put in more details for homework than your textbook authors put in their books.
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Your professor knows/follows your proofs easily and laymen have no clue.

--> your fellow students should be able to follow your proofs easily.  Make that your standard.
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You can generally assume anything that should be known by the reader.  For any given logical leap, you need to ask yourself the question: will my intended readers be able to understand why this follows?  There's another question too, though: will my intended readers be able to *follow* the proof if I don't explain this part explicitly?

What's the goal of a proof?  The goal of a proof is to present your understanding of why a result is true.  So everything in the proof is directed at this.  The point isn't to list all of the logical steps, generally.  Sometimes it is!  Sometimes every step needs a citation, even simple arithmetic.  But unless you're writing a book on logic, you don't need to go to that particular trouble, most likely.
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All the other commenters have given lots of good guidance, but I would be remiss if I did not relate the story that is told about G. H. Hardy, who was lecturing his way through a proof, and remarked that the step he had just made was obvious. Then he paused, reconsidered, and vanished into the preparation-room for twenty minutes, before emerging and announcing with satisfaction, "I was right; it *is* obvious.".
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"logical leap" means you are jumping to a conclusion without proving it (e.g. because you misunderstood the statement of a result that you used). any logical leap whatsoever means your proof is invalid. it is not the same as leaving out technical details that the reader can fill in.
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A lot of the simpler rules/theorems/laws have a shorthand for a reason lol.
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If you're practicing proof in a class (where your reader is the one who grade it), then the rule-of-thumb is that you should imagine your reader to be one of your peer: someone who read through the same books and sat through the same lectures up to the current point and understood the materials, and now do not know how to solve the current problem. The second rule-of-thumb is this: write at the level of the question. If question is asking you to prove something elementary and obvious, write down to basic axioms, but if it's something hard and not obvious, you can skip more details.

That are basically rule-of-thumb for the context of writing proof in a class. Most people who learn math only write proof in this manner anyway, so it's actually more applicable than other rules. That's something I tell my students.

If you're a writing proof for a honest audience (people who are actually interested in knowing how the arguments work, or at least want to check if it's true or not), then you should write at the level of the audience. But of course, this is not a strict rule and there are huge range of preference, even among mathematicians. Some mathematicians write extremely tersely, and some includes a lot of details. So it's not like mathematicians have a common standard.

(in fact, there was a number of controversies in math world where one mathematician write too little details, others have to go and fill them in, and then argument occur whether the others should really get credits for the work)

No, these tools are not only one. The number of tools will increase as your knowledge increases. It's just hard to explain all the tools in a book for general audiences, because they would not have known the necessary knowledge to even use such tool. In fact, for the most part, the point of mathematical theorems and theories is to give you more tools (but not all tools come from theorems).
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Uhhh non. A logical leap is where you go from one place to another with zero valid path in between steps to bring you to your new place. So if you have to make a leap then your proof isn't valid. If you mean not explicitly writing out each individual step you do but it is valid there are shorthands for many of those processes. If there isn't one I tend to write it out. I want my process to be as clear as possible to as many people as possible.
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Well I think it's simple. Every single step in the proof has to be justified somewhere, but the justification can be a lemma somewhere else, and as long as you know about the lemma and everyone reading your proof knows about the lemma, it's fine.

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