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    (Original post by atsruser)
    Is this "long loop" terminology standard? I've never heard of it before.
    First time I've seen "loop" used in the context of proofs is in this thread . So I thought "long loop" just had to be invented. We just need to invent a notion of homotopy of proof loops and we're done for Easter.

    The first time I saw it was in one of those god-awful linear algebra proofs about the equivalence of "Rank A = n" and "unique solutions of Ax=b" and about 1/2 a dozen other things. It struck me as too clever-clever at the time, and I'm not sure that I really believed the proof - it looked like stuff had been missed out.
    Yup, I've never liked it. If there's an equivalence between two facts, then I like to see it demonstrated directly - for the simple reason that you see better why the equivalence is true. Sometimes it may be difficult to prove a direct equivalence, and going through an intermediate lemma is required; but this should be done only when there's little alternative.
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    (Original post by Gregorius)
    First time I've seen "loop" used in the context of proofs is in this thread . So I thought "long loop" just had to be invented.
    Yes, I think it's a pretty nice term - it ought to be made more popular.

    We just need to invent a notion of homotopy of proof loops and we're done for Easter.
    That would be pretty cool - you could prove the equivalence of two sets of proofs. A "proof loop" isn't a very continuous thing though - don't homotopies only work when stuff is continuous? Still, I await your upcoming paper (and subsequent Fields Medal, of which I claim, ooh, 5%, for suggesting a use of the concept) with interest.

    Yup, I've never liked it. If there's an equivalence between two facts, then I like to see it demonstrated directly - for the simple reason that you see better why the equivalence is true. Sometimes it may be difficult to prove a direct equivalence, and going through an intermediate lemma is required; but this should be done only when there's little alternative.
    I agree - however, it's probably the only sensible approach in those "long list of N equivalent properties" proofs - you'd have N(N-1)/2 proofs to do otherwise.
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    (Original post by shamika)
    If any STEP question required that it would be much too difficult I think. It's a very neat trick though, useful for algebra courses at uni!
    I agree, but Lucasium was asking about a loop method that was beyond FM, and I couldn't really think of one.


    (Original post by Zacken)
    Wait, this is boggling me - how does this work? You get logical equivalence between a number of statements just by proving implications from one to another?
    As Shamika said - once it's explained to you it seems quite obvious.

    (Original post by physicsmaths)
    This is very strong. Damn
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    (Original post by Student403)
    That's pretty cool
    I'm glad you liked it, but as others have said, it's not often used except in theorems requiring proof that a long sequence of statements are equivalent.
    I've not seen many questions that this is applicable to, and certainly none in STEP, but it's something to look out for
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    (Original post by joostan)
    I agree, but Insight314 was asking about a loop method that was beyond FM, and I couldn't really think of one.
    It was Lucasium, not me.
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    (Original post by Insight314)
    It was Lucasium, not me.
    That is my bad - being dumb
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    (Original post by joostan)
    I agree, but Lucasium was asking about a loop method that was beyond FM, and I couldn't really think of one.
    Don't get me wrong, I think it's a cool trick (my mind was blown when I first saw it in lectures, but I think at the time I was just overly keen ), just wanted to remind people that they shouldn't worry if they haven't seen it.

    Hope Cambridge is going well
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    joostan

    In which Cambridge college are you studying?
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    (Original post by Insight314)
    joostan
    In which Cambridge college are you studying?
    Spoiler:
    Show
    I'm at Sidney Sussex - the one best known for being opposite Sainsbury's.
    (Original post by shamika)
    Don't get me wrong, I think it's a cool trick (my mind was blown when I first saw it in lectures, but I think at the time I was just overly keen ), just wanted to remind people that they shouldn't worry if they haven't seen it.

    Hope Cambridge is going well
    Thanks.
    Oh definitely don't want folks worrying - for me it was a tip from a (then) second year regarding a problem we were given.
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    This idea shows up here:
    http://msekce.karlin.mff.cuni.cz/~kr.../mendelson.pdf
    page 18, 1.8
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    (Original post by Zacken)
    Oh, and somebody remind me to do a post about multiplying by one creatively.
    Did you ever get round to making this post? I'd be very interested
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    (Original post by Tau)
    Did you ever get round to making this post? I'd be very interested
    Nope, never did. Maybe I'll do it one of these days.
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    (Original post by Zacken)
    Nope, never did. Maybe I'll do it one of these days.
    Thank you very much sir


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    (Original post by Zacken)
    Nope, never did. Maybe I'll do it one of these days.
    I'd like to see it too. I can only think of two tricks that fit this description:

    1. Finding equivalent fractions e.g.

    \frac{2}{3} = \frac{2}{3} \times 1 = \frac{2}{3} \times \frac{n}{n} = \frac{2n}{3n}

    and variants thereof.

    2. Changing units e.g.

    1 \text{ km} = 1000 \text{ m} \Rightarrow 1 = \frac{1000 \text{ m}}{1 \text{ km}}

    so

    5.2 \text{ km} = 5.2 \text{ km} \times 1 = 5.2 \text{ km} \times \frac{1000 \text{ m}}{1 \text{ km}} = 5200 \text{ m}
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    (Original post by atsruser)
    I'd like to see it too. I can only think of two tricks that fit this description:
    Was mainly thinking about it as an integration trick, like:

    \displaystyle \int \frac{x^2}{(x \sin x + \cos x)^2} \, \mathrm{d}x = \int \frac{x}{\cos x} \times \frac{x \cos x}{(x\sin x + \cos x)^2} \, \mathrm{d}x which immediately renders it vulnerable to IBP since z = x \sin x  + \cos x \Rightarrorw z' = x\cos x .

    Stuff along those lines.
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    (Original post by atsruser)
    ...
    Main use I can think of is rationalising denominators/dividing by complex numbers. This probably falls under finding equivalent fractions but is actually a useful technique.
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    (Original post by Duke Glacia)
    1. Adding zero creatively (i) (Zacken)
    2. Trigonometrical symmetry and substitutions for integration (Zacken)
    3. Trigonometrical functions as shifts of others (Zacken)
    4. Implicit differentiation and logarithmic differentiation (Zacken)
    5. Adding zero creatively (ii) (Zacken)
    6. Algebra shenanigans with 'it suffices' (IrrationalRoot)
    7. Discriminants as a useful tool for inequalities and in mechanics (Duke Glacia)
    8. Weierstrass subs (Shamika)
    9. Inequalities and injectivity (Zacken)
    10. Summing arc-tangents (16Characters...)
    11. L'Hopitals rule for limits (Joostan)
    12. Factorising special polynomials and techniques to do so (Aymanzayenmannan)
    13. Strong induction (Zacken)

    Hopefully we can grow this into a useful and extensive collection of tricks and techniques - feel free to share your own!

    So to start with Zacken posted a amazing techniques which could have saved me a lot of time and effort.



    OP written by Zacken
    Thank you so much for this
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    (Original post by Zacken)
    I'd just like to showcase an astounding example of the adding zero creatively trick (even if it's been quoted already by Duke in the OP), it featured heavily in STEP II, 2005, Q8 for example and the examiners report and official solutions didn't even mention it, they decided to mention ridiculous things like hyperbolic trigonometrical substitutions instead of a simple trick.

    So what we have is:

    \displaystyle

\begin{equation*} \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^3y^2}{(1+x^2)^{5/2}}\end{equation*}

    The seperable stuff is easy and you all know how to integrate y^{-2}, but the RHS is slightly more tricky, how would you integrate x^3/(1+x^2)^{5/2} - what I'd do:

    \displaystyle

\begin{equation*} \frac{x^3 +x - x}{(1+x^2)^{5/2}} = \frac{x(1+x^2) - x}{(1+x^2)^{5/2}} = \frac{x}{(1+x^2)^{3/2}} - \frac{x}{(1+x^2)^{5/2}} \end{equation*}

    Whereby integrating is very standard now, since (1+x^2)' = 2x. Oh, and somebody remind me to do a post about multiplying by one creatively.
    Hi there, I was looking through the tricks and hacks and was wondering what the multiplying by one creatively was. If possible could you show me please. Thank you.
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    (Original post by Major-fury)
    Hi there, I was looking through the tricks and hacks and was wondering what the multiplying by one creatively was. If possible could you show me please. Thank you.
    You've probably done it before but just haven't noticed it was 'multiplying by one creatively': consider rationalising the denominator of a fraction: \dfrac{1}{\sqrt{3}}= \dfrac{1}{\sqrt{3}} \times 1=\dfrac{1}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}}= \dfrac{\sqrt{3}}{3}.

    Another example is proving that differentiability implies continuity; this involves both(!) creatively adding zero and creatively multiplying by 1:
     \displaystyle \lim_{h \to 0} f(a+h) = \displaystyle \lim_{h \to 0} (f(a+h)-f(a)+f(a))= 



\displaystyle \lim_{h \to 0} \left(\dfrac{f(a+h)-f(a)}{h} \times h+f(a)\right)=f'(a) \times 0 +f(a)=f(a)
    as required.

    In most cases the general idea is to force a desired expression into the picture.
 
 
 
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