Golden Proof – Numberphile

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The Fibonacci numbers again start: 1, 1, 2, 3, 5 and the rule is: if each number here equals the one before it So, if this was the nth number and we’re gonna call that x n for the nth number it equals the one before it plus the one before that so that equals x n minus one plus x n minus two So that’s our finding thing and what we’re trying to find ultimately is what is the ratio between the two of the numbers and importantly, because we’re looking for the ratio it is approaching the limit it doesn’t matter which two terms it should be the same ratio, give or take and if you do infinitely many the same ratio for all of them but, rrrrhhhhh So. Uh. For any two of the terms so what we can say now is the ratio between x to the n and the one before that which was x to the n minus one. The one before. should equal the same as the ratio between x n minus one to the one before that which is x n minus two and we’re gonna call it What letter would you like Brady? awa aya a any any Greek letter? Do you have a favorite? Brady: uhhh…. delta? Guy: DELTA! ok, so lets do a capital delta because its quicker to draw so we’re trying [to come up] with this ratio, delta is. and we know its the ratio between any two consecutive terms now we know according to our relationship which generates this sequence that x n equals the two before it added together so I’m gonna put that in intstead of so all I’ve done is substitute that in but now I could start simplifying because I’ve got the same term there twice so, in fact, if I divide through that’s one plus x to the n minus two over x to the n minus one which still equals x to the n minus one over x to the n minus two which still equal to the ratio delta we’re looking for so now you’ll go hang on a second that is the same as this but upside down and in fact that equals that ratio delta so I can now change this to be one plus well that’s the inverse of this ratio so that’s one over delta equals and that there is the delta and so the relationship for the ratio of any sequence where you get the next term from adding the two previous ones one plus the inverse of that ratio equals the ratio Now what ratio could that possibly be? well let’s work it out! because we can turn this into a quadratic? If I multiply through by triangle it equals that plus one equals triangle squared and so delta squared minus delta minus one equals zero So, all I’ve done is rearrange that formula now we can solve this using our friend the quadratic equation cuz we know that if you have a times x squared plus b x plus c equals zero then x equals negative b plus or minus the square root of b squared minus four a c over two a so we can substitute in to use the quadratic formula So we now know delta equals negative b, which is the coefficient there which will be one plus or minus the square root of b squared –is going to be one minus four times one, times negative one, so it’s going to be plus four. Two times one is two. So that’s going to equal… Now I’m only going to use the positive for now. One plus root 5 divided by two, which is the golden ratio. If you actually work that out, that’s where 1.6180… comes from. And in fact, that is one of the definitions of the golden ratio And so that’s why, for any starting term, it always approaches the same sequence If you use the negative, you get the negative inverse of the golden ratio; another wonderful property of it. It I’d use the negative… If I’d used equals one minus root five over two and that equals -0.6180… So they’re the two solutions to that [particular] equation So, that’s why it works of any sequence where you add two terms to get the next one Brady: So the fact… The Fibonacci numbers meant nothing there? Guy: The Fibonacci numbers are one of [an] infinite family of such numbers because for any two whole numbers you want to start with, the exact same thing works out In fact, why use whole numbers? Go bananas! At no point did I specify whole numbers here. It’s just, whatever you start with, add two terms, you get the next one. You always get the golden ratio That’s not some special property Fibonacci numbers have. All of them have got that property. We want a better link. What we really want is a sequence of numbers which have a link the the golden ratio that other sequences don’t get. We want something above this. Brady: The Brady numbers! Guy: Well, the Brady numbers are wonderful, but I can go slightly better So, I’m going to show you the Lucas numbers. I, personally, am a massive golden ratio septic. I think it’s hugely overrated…

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