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+----- Thread: Theory of Relative Squares (/thread-454.html)
RE: Theory of Relative Squares - Guy1234567890 - 06-29-2013
Just as a note, you only have to show that the proposition is true for one case (usually the case where n is the smallest) and then you can immediately show that it is true for the k+1th case.
RE: Theory of Relative Squares - fl3tching101 - 06-29-2013
(06-29-2013, 01:40 AM)Guy1234567890 Wrote: Just as a note, you only have to show that the proposition is true for one case (usually the case where n is the smallest) and then you can immediately show that it is true for the k+1th case.
Yea, I saw that on the wiki, but we were taught to show true for the three cases. Also we were taught to make a middle step that was just replacing n with k lol... I quite hate Mathematical Induction so
Lol when I posted this I really didn't expect such discussion
I know there are like a thousand different shortcuts and ways of finding a square, I was just posting a quick, fun(ish) way to find a square...not trying to rewrite a textbook here lol. The origin of this was actually that one day in math, we were doing some group work and one of my friends was in the group, and I just said one of the steps out loud, a square of like 17 (I talk to myself a lot through math problems, really just to help others around me or have someone smarter correct a problem 
) and he immediately spit out the answer. The like 10 people around him including me all just looked at him...turns out he had memorized the squares of the numbers through 25. So wishing I could do that lol, I randomly remembered that moment and was thinking about it, trying to figure out a shortcut to find squares quickly. These were the shortcuts I figured out in the car on the way home from a short trip lol.
RE: Theory of Relative Squares - Thor23 - 06-29-2013
(06-29-2013, 01:40 AM)Guy1234567890 Wrote: Just as a note, you only have to show that the proposition is true for one case (usually the case where n is the smallest) and then you can immediately show that it is true for the k+1th case.
No, if you you only had to show it was true for one case, then you wouldn't be proving anything. If I had the equation 2+k=4, and showed that it was true when k=2, that doesn't mean it would be true for 3. You'd have to show that the k+1'th term was equivalent to simply substituting k+1 in for k in the original equation. If it works for both k and k+1, since k can be anything, then you've proven that it works for all k's.
(06-28-2013, 10:06 AM)xdot Wrote: The plugin/TeX server seems to ignore multi-line scripts.
Ah, so that's what it was. Fix'd.
Finding products of 2 and/or 3 digit numbers is pretty easy if you just foil them in your head. No memorization required.
RE: Theory of Relative Squares - Guy1234567890 - 06-29-2013
Yes, you must show the proposition to be true for both k and k+1 (that wasn't the part I was talking about). However, the assumption is that it does hold for k, and so that step is unnecessary. However, I was just referencing the 3 specific cases, two of which were unnecessary.
RE: Theory of Relative Squares - Thor23 - 06-30-2013
Ah, ok then. It seemed like you were saying that if you can show it works for one case, then it would work for all cases by extension.
RE: Theory of Relative Squares - Guy1234567890 - 06-30-2013
Yeah, the step where you assume it works for all n=k therefore it must work for all k+1 is called the inductive step XD
Upon re-reading what i wrote here it does not make much sense*
RE: Theory of Relative Squares - Thor23 - 06-30-2013
But you can't assume it works for all n = k because you don't know if that's true. You can only assume it works for one k, then show that if - based off that assumption - k+1 also happens to be true, then it will work for whatever k happens to be. If k+1 doesn't work, then your assumption must have been wrong.
RE: Theory of Relative Squares - Guy1234567890 - 07-01-2013
(06-30-2013, 09:01 PM)Thor23 Wrote: But you can't assume it works for all n = k because you don't know if that's true. You can only assume it works for one k, then show that if - based off that assumption - k+1 also happens to be true, then it will work for whatever k happens to be. If k+1 doesn't work, then your assumption must have been wrong.
Actually, the assumption is not that it works for all n. Rather, it is that it works for all k+1. Then after showing this to be true, and because you know it works for a specific k, you must therefore know that it works for all whole number values greater than the "seed" value.
Effectively, you show it to work for n=0, and hence n=1, and hence n=2 etc. because each time you show it works for a specific n, it therefore also works for that n+1, and hence for that n+1 and so on.
RE: Theory of Relative Squares - Thor23 - 07-01-2013
Alright yes, I think we're finally on the same page. Everything you said right there makes sense to me. I think it was just the context on which I was getting confused.
RE: Theory of Relative Squares - Guy1234567890 - 07-01-2013
XD yeah probably