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Question: Why do people prove/solve questions in such a complex way when there is an obvious, simpler way to do it?

For example, this question on proving divisibility entailed an answer using advanced notation (in the sense that pre-college people like me would never understand it) and methods to express something fairly easy. Instead of saying $\Pi$ why didn't he/she just write it as a product to allow for more people to understand?

To me, solving a problem as simply as possible make your solution all the more elegant and rigorous. But then again I could be just frustrated that seemingly easy problem have answers that somehow manage to incorporate Calculus 54 (a joke btw) and I can't understand the answer/proof, therefore, I must derive my own (which fails most of the time).

Any answer is appreciated.

EDIT: If you downvoted please explain why It'll help me improve asking questions on this site.

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  • $\begingroup$ If this question already has an answer or is not on-topic please tell me. $\endgroup$ – Aops Vol. 2 Oct 7 at 4:18
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    $\begingroup$ A related older post: Is it OK to answer a question with a higher level of mathematics than I expect the OP to know? (And maybe also some of the other posts linked there.) $\endgroup$ – Martin Sleziak Oct 7 at 4:42
  • $\begingroup$ @MartinSleziak: Thanks, should i delete the question? $\endgroup$ – Aops Vol. 2 Oct 7 at 4:45
  • $\begingroup$ easier for who ? $\endgroup$ – Roddy MacPhee Oct 7 at 11:39
  • $\begingroup$ @RoddyMacPhee: The easiest possible solution $\endgroup$ – Aops Vol. 2 Oct 7 at 17:30
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    $\begingroup$ I don't think the notion of "the easiest possible solution" is as well defined as the simple phrase suggests. In some cases advanced studies allow a shorter proof, so there may be a trade-off between using the most elementary principles and giving a solution that is not only shorter but easier to verify. I'm not a downvoter BTW. $\endgroup$ – hardmath Oct 7 at 20:16
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    $\begingroup$ They didn't write out a product with an ellipsis for the same reason instead of using the $\prod$ notation for the same reason you might use $\sum$ instead of writing out a sum with an ellipsis---it's more compact, it's less ambiguous, and the time it takes to learn the notation is very short (supposing you're already used to sigma notation for sums). $\endgroup$ – Travis Willse Oct 7 at 23:36
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    $\begingroup$ In Meta, upvotes/downvotes mean yes/no or agree/disagree. Unlike in the site proper, here they do not mean good/bad question. $\endgroup$ – GEdgar Oct 8 at 12:38
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    $\begingroup$ Readers pondering similar questions might find of interest the comments about induction vs. intuition, generalizations, etc in this answer $\endgroup$ – Bill Dubuque Oct 8 at 13:40
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First, to address your edit about the downvote: up- and down-votes on Meta are often construed as being in favour or against the proposal of the question. Here I expect that the downvotes are from people being against your initial proposition that proofs on the site should be only solved in "obvious" ways. Since for some proofs, using only things are obvious[$1$] would run to half a book, it should be clear why people disagree with a blanket statement.

Second, to answer your question: answers at different levels can give different insights into a problem and how to solve it. There's a lovely book (I think it's Michael Steele's The Cauchy-Schwarz Masterclass) that looks at the Cauchy-Schwarz inequality and provides about $20$ different ways to prove it, and studies the insights that that leads to. Seeing things in new ways can make the difference between "that's all there is here" and "here's an interesting new direction to study". Many of the proofs are not obvious, but they're no less valuable for that.

Your implicit question (and I may be putting words in your mouth) seems to be actually a complaint that you didn't understand the previous answers. If that's true, then a better question from you would be what can I do to improve so I can understand this? You've made a good start -- writing out your own solution is a fantastic idea, and seems to have led to you learning more as well (judging by the comments on your answer). If your next question were why is this notation used in this answer? that would also be an excellent step; it's essentially your question here but rephrased without judgement.

Lastly: proof-writing and -reading are non-trivial skills. Don't expect to understand all but the simplest proofs on a read-through, and do expect to put some work in to follow the ideas in there. They will reward you in the long run.


[$1$] There's also the issue that the longer a proof becomes the harder it is to keep the ideas in mind, so using only elementary ("obvious") language would make the proof harder, not easier to understand.

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    $\begingroup$ Thanks for the reply, I guess my question was worded weirdly. Is there any book/class where I cant learn the notation found in the prementioned proof. $\endgroup$ – Aops Vol. 2 Oct 7 at 21:06
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    $\begingroup$ @AopsVol.2 I think the only notation that bothered you was $\Pi$; that's the short-hand for multiplication. So $\Pi_{i=1}^n x_i = x_1 \cdot x_2 \cdots x_n$ and that's all there is to that notation. More generally, since you say you're pre-college, you might look at a book on Discrete mathematics as an introduction to notation and topics that you might find interesting: discrete.openmathbooks.org/dmoi3/frontmatter.html seems like it might be at the right level of challenge for you, it's interactive(!) and you can look up more on induction in it too :) $\endgroup$ – postmortes Oct 8 at 5:45
  • $\begingroup$ en.m.wikipedia.org/wiki/List_of_mathematical_symbols $\endgroup$ – Roddy MacPhee Oct 8 at 23:56
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While I understand your frustrations toward difficult notations and more advanced explanations than those which you are used too, it's all part of learning math. Mathematicians develop definitions, notations, and ways of writing that may seem alien and unintuitive at first, but you'll see that once you get into it those notations and definitions make sense and in fact are necessary for the communication of mathematics.

If you find notation that you don't understand, try googling it, or search it on http://detexify.kirelabs.org/classify.html and then google the name. Googling things is also an important part of learning mathematics. I often come here to find answers (answers to my own questions) that I don't immediately understand, but that's part of the "fun" of math: you get to think deeply about a near endless stream of new ideas. However, that's the difficulty too. When reading math (from textbooks, homework, or this site), expect to spend lots and lots of time doing so. It's not like reading normal English -- sometimes it's "natural" to spend hours on one page of a book.

Also a note on "pre-college": don't let this stop you. College is not some magic place where you learn the great secrets of math -- it's more like a place where you learn to discipline yourself to learn mathematics on your own, and that is something you can start on now. Best of luck on your mathematical journey!

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They didn't write it out as a product, because it can be arbitrary length. That's the proper use of the notation. There are some other products, that you might not be able to write in full regardless because they have infinitely many terms. Sometimes a simple answer isn't obvious ( Goldbach's, Collatz, Twin prime, abc, Beal's, Grimm's, Legendre's conjectures e.g.) . Your answer, is just a different notation for the same concept mostly.

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