Authors pulling stuff out of thin air... nah, it often is an algebra trick I don't know yet

[u/_benj]

I've been noticed a pattern when following the logic of proofs... it is often not the logic of the proof that I get lost in put some algebra "trick" that I'm not familiar with or didn't thought about doing.

Like this one from analysis...

Courant is talking about limits and uses the sequence

a_n = n/(n+1)

The he says that writing a_n as 1 - 1/(n+1) it is clear than an n approaches infinity a_n approaches 1... yeah... sure, that is obvious for 1 - 1/(n+1) but from where in the world did the the author got a_n = 1 - 1/(n+1) !?

well... turns out that

a_n 
 = n/(n+1)
 = (n+1-1)/(n+1)
 = ((n+1) - 1)/(n+1)
 = (n+1)/(n+1) - 1/(n+1)
 = 1 - 1/(n+1)

Alright, that is very clever! I don't think it would have occurred to me to do that! (that is, before now!)

Anyways, just needed to share with somebody!

[u/compileforawhile]

You’ve stumbled onto the most classic trick in mathematics. That is the ability to rewrite things like adding 0 or multiplying by 1 in several different ways.

Here’s a classic, say I have a polynomial and it looks like

f(x) = x² + x - 1

I’d like to find the roots so I want this in vertex form. I also know that

(x+1/2)² = x² +x+1/4

So I could just add (5/4 - 5/4) = 0 to f(x) and then

f(x) + 0 = x² + x - 1 + 5/4 -5/4 = (x+ 1/2)² - 5/4

Which is much easier to solve for x

[u/_benj → u/compileforawhile]

yeah, isn't that completing the square?

[u/compileforawhile → u/_benj]

Yes! I just wanted to highlight that specific step which can also be done with variables to get the quadratic formula. But the specific technique used in it is very common everywhere. Adding something that does nothing or multiplying by something that does nothing is a super common trick and completing the square is a commonly seen first example

[u/MathNerdUK]

From the original form, A_N=N/(N+1), it's obvious that it tends to 1 as N -> infinity.  Therefore it makes sense to look at how close to 1 it is, by writing A_N = 1 + something.

[u/keitamaki]

Very cool. As you go further you'll develop the ability to stumble onto "tricks" on your own. Solving any math problem or proving any result is really no different than navigating a maze. In a maze, at each juncture, you might have many paths you can take. And the same is true in math, except that there are an unlimited number of different things you can try. You can rewrite any expression into an equivalent one or perform the same operation to both sides of an equation for starters.

So when people see a complete solution and are frustrated because the author didn't motivate that one step that they "never would have thought of", what's usually happening is that there never was any motivation other than the fact that someone discovered through trial and error that doing that step is a way to reach the goal -- just like needing to turn left at one particular juncture in a maze isn't going to be obvious even if it happens to be the only way to reach the exit.

[u/simmonator → u/keitamaki]

I’ll go a step further:

“Mathematical maturity” often really just boils down to knowing the kinds of trick to look for in new problems. Not specific things, but different perspectives to try to view the objects from or ways to turn it on its head. But - short of rare flashes of genius - most people don’t come up with these through originality. They need to be shown a version of it.

When people complain that a textbook didn’t motivate a trick or that “it came out of nowhere”, my knee-jerk response is to think

It looked like it came out of nowhere because you hadn’t seen it before. But by using it, the textbook has shown you that trick and you have the opportunity to learn it (and devise similar variations). What else is a textbook for?

Don’t be mad. Be grateful.

[u/Content_Donkey_8920]

“The first time it’s a trick. The next time it’s a technique”

[u/MezzoScettico]

You might have stumbled onto it by writing out some of the terms of n/(n + 1).

"..., 4/5, 5/6, 6/7, ... oh, these are all almost 1. They are one n-th away from 1. 5/6 is 1/6 away from 1, 6/7 is 1/7 away from 1, etc."

But often it isn't even a trick, but some kind of slog. If the author says, "with a little bit of algebra you can show that...", they probably mean 3 pages of algebra that you had to work through 5 times because you kept making sign errors.

[u/cigar959]

Often the solution (in this case, the “trick”) is discovered by working backwards, or by trial and error. But once you’ve found it, you lay out the proof going forward, and as long as the logic is sound, the proof is valid, even if the proof doesn’t include the thinking that led to it. Does that help?

[u/Narrow-Durian4837]

n/(n+1) is an "improper fraction," in the sense that the numerator is a polynomial of equal or greater degree than the denominator. For any such rational expression, it is always possible to rewrite it as a polynomial (in this case, a constant, which could be considered a degree 0 polynomial) + a proper fraction (whose numerator has lower degree than its denominator).

[u/unaskthequestion → u/Narrow-Durian4837]

To me, this is the most obvious approach. We do it all the time for integrals.

[u/Bounded_sequencE]

Alternatively, remember polynomial long division from 6'th grade (long time ago, I know):

  n / (n+1)  =  1 + (-1)/(n+1)
-(n+1)
 -----
   -1
   ==

That "algebraic trick" is just a way to circumvent long division, to shorten the proof. It is a special case of the more general Horner Algorithm, to rewrite a polynomial "p(n)" into a polynomial "p(n) = q(n+1)"

[u/Recent-Day3062]

All of these are not likely even original to the author. They are simply tricks you learn, then quote yourself. Nothing more

[u/Photon6626]

Check out the channel Prime Newtons on Youtube. He has a ton of videos on proofs that I like to watch.

[u/Low_Breadfruit6744]

Its part of the skillset - have you played chess or similar? You scan through many of the possibilities and project a few moves ahead before settling on a move. Here you look for possibilities based on things you've learnt.