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every time you learn a new equation they tell you to prove how this formula was derived, but why is this made ?

i mean the derivation of an special formula is applied once and can not be used to prove another formulae

so why in the process of learning you must know how to derive each math formula you learn ?

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    $\begingroup$ The alternative is to just memorize results handed down to you by some alleged authority. $\endgroup$
    – zhw.
    Jul 14, 2017 at 19:21

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I believe there are many different kinds of good answers to your question, but I'll just focus on one:

If you're lucky, you may indeed go through life and successfully apply the formula without being able to derive it.

However, you may find yourself in a situation that is slightly different from the kind of situation that the formula was intended for. In that case, if you knew how the formula was derived in the old kind of situation, you may be able to change that derivation to derive a new formula that is more appropriate for that new kind of situation.

Indeed, if you know the derivation, then you are immediately more likely to know the assumptions being made for that formula, and thus be able to gauge whether it is appropriate for you to use the formula in some kind of situation or for some problem in the first place, by seeing if those assumptions indeed hold true for the situation or problem you are faced with.

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    $\begingroup$ Indeed. Learning does not mean ``being able to solve exercises in a book''. $\endgroup$ Jul 14, 2017 at 18:55
  • $\begingroup$ @AndrésE.Caicedo That would be the short way of putting this, yes :) $\endgroup$
    – Bram28
    Jul 14, 2017 at 18:56
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    $\begingroup$ And, it can be a memory aid. Sometimes its easier to remember the derivation than the formula itself. And sometimes (say when you work with others) it helps to explain the result if you know where it came from, $\endgroup$ Jul 14, 2017 at 18:58
  • $\begingroup$ @kimchilover Agreed (personal example: Bayes' formula. I don;t even try to remember the formula, since I know I can immediately derive it from $P(A \cap B)=P(B \cap A)$ and $P(A \cap B)=P(A|B)*P(B)$) You should post this as a separate answer, as I believe there are many different answers to the question as posed. $\endgroup$
    – Bram28
    Jul 14, 2017 at 19:02
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There's a few reasons, even if you don't intend on deriving similar formulas in the future.

First, it's hard to remember formulas. For many people, it's easier to remember reasons - maybe not the whole proof, but if you remember just the outline of the proof it can help you remember the formula. For example, I can't for the life of me remember the formula for the arc length of a curve - but I do remember the gist of the way it's derived, by taking an infinitesimal fragment of the curve and applying the Pythagorean Theorem. That's enough for me to remember that I'm looking for something like $\int \sqrt{?^2 + ?^2}dt$, and then it's just a matter of filling in the blanks.

Second, it's hard to remember the uses of formulas. Yes, most formulas have a one-to-three-word phrase that describes their use; for example, the quadratic formula is for "solving quadratic equations". But even in these simple cases, there are literally thousands of random special situations where you can use them - for example, the quadratic formula can be used to solve $x^4 + 2x^2 + 3 = 0$, and by analyzing the formula you can see a way to tell whether a quadratic has a real root without having to take any square roots. Knowing the reason behind the formula - in this case, remembering that the proof of the quadratic formula doesn't depend on using $x$ specifically rather than $x^2$ - helps you see these additional uses without having to do huge amounts of memorization.

Third, it's hard to remember the non-uses of formulas - that is, the situations where a formula might look like it applies, even though it doesn't. Taking the quadratic formula example from above, it's a common mistake for students to try to apply the quadratic formula to situations like $x^3 + 2x + 1 = 0$, or to $x^2 + 2x + 1 = 1$ without rearranging the formula first. Understanding the proof behind the quadratic formula makes it obvious that the formula simply doesn't apply in either of these cases; the first because the proof is about squares rather than cubes, and the second because the proof gives a factorization of the quadratic rather than just "the answer".

Fourth (and finally, for now), it helps unify formulas. To take an example from multivariable calculus: Green's Theorem, Stokes' Theorem, and the Divergence Theorem are all basically the same formula - each applies in different situations, but they all arise the same way. If you've never seen a proof of these theorems, or if you didn't understand the proof you were shown, then you wind up just memorizing three separate formulas and having to memorize a laundry list of situations in which one applies and the others don't. If you do understand how these formulas arise, then to you they're more like one theorem than three; and when to use which one becomes obvious, requiring no memorization at all. This sort of thing happens a lot - and even when it doesn't, there are still a lot of situations where several formulas are closely related, and understanding their relationships can help understand how to use them.

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Richard Feynman, speaking on studying physics, gave the following advice (but I think the same advice applies to studying mathematics):

"It will not do to memorize the formulas, and to say to yourself, 'I know all the formulas, all I gotta do is figure out how to put 'em in the problem!'

"Now, you may succeed with this for a while, and the more you work on memorizing the formulas, the longer you'll go on with this method--but it doesn't work in the end.

"You might say, 'I'm not gonna believe him, because I've always been successful: that's the way I've always done it; I'm always gonna do it that way.'

"You are not always going to do it that way: you're going to flunk-- not this year, not next year, but eventually, when you get your job, or something--you're going to lose along the line somewhere, because physics is an enormously extended thing: there are millions of formulas! It's impossible to remember all the formulas--it's impossible!"

--Richard Feynman, Feynman's Tips on Physics

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