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• we're gonna do randomized Fibonacci sequences.

• So we're gonna take a Fibonacci sequence.

• We're gonna randomize it with this coy, I think.

• Do a recap of Fibonacci sequences.

• First has to see what the original Fibonacci sequences.

• So you're Fibonacci Sequence starts with a one and a one, and then the next term is the sum of the previous two.

• We're gonna add those together.

• So one plus one makes two one plus two makes 32 and three make 58 on or 13 21 and so on.

• So you want to write that down a little formula, you'd write it like this.

• You want to find the next Fibonacci number?

• You just at the previous two.

• Okay, so a couple of well known facts about Fibonacci numbers as well if you take two consecutive Fibonacci numbers will divide the big one by the small one.

• Then you'll see.

• It's been proven for a long time that this will tend to a number which is 1.618033988 something something something on people will know that this is a special number.

• This is called the golden ratio, which is this symbol, the golden ratio.

• But you can use that to estimate.

• Let's say you've got a really big Fibonacci number and you don't know what it is.

• You can use the golden ratio.

• So the idea is every time was sort of multiplying by the golden ratio.

• To get the next one up less true at the beginning becomes Maur true.

• The bigger these numbers are so you can kind of reverse engineer your numbers.

• Yeah.

• Yes, you can work out the millions, Fibonacci number right.

• So if you wanted to work out the 1,000,000 Fibonacci number, what you're doing is multiplying by the golden ratio a 1,000,000 times.

• So let's say approximately equal to golden ratio, which is this symbol to a 1,000,000.

• And I could tell you what it is.

• I've actually worked this out 4.4 times 10 and it's gonna be a huge number.

• 208,987.

• So this is a really big Fibonacci number, but this is an estimate because it is a bit rough if you want to see what it actually is.

• The actual 1,000,000 Fibonacci number is 1.95 times 10 to the power 208,987.

• They're so not quite there.

• But the size of these numbers, This spot on sounds just a little recap off, Fibonacci.

• Because what I want to do now is randomize this Ralph Fibonacci we know everything about it are boring.

• So predictable.

• So now let's use my coin to randomize this a little bit with this idea in a creative random Fibonacci sequence, the next one is going to be the sum or the difference off the previous two numbers.

• So we're gonna start in the same way with a one and a one.

• Now, what is the next value?

• We're gonna use my coin to find out.

• Normally, it would have been, too.

• So this is this.

• If it was that the ordinary Fibonacci sequence, we would add these together.

• We get a two, I'm gonna flip my coin.

• I might take the son.

• I might take the difference.

• So let's see what it's going to be.

• You got negative on one side.

• Uh, okay, so I've prepared.

• I came prepped.

• Yep.

• Alright.

• Says, do.

• Is it a sum or difference?

• And I've got difference difference Okay, so it's north is the next night.

• Zero is the next number.

• Let's do the next number, is it?

• Give me a sum or difference.

• It's gonna be a sum.

• So I've got one now at the previous two to give me one.

• Let's see why.

• Get next.

• Try it, Whip.

• Okay, I got some.

• So we got back to another one there.

• Okay.

• Got Plus two.

• Let's keep going.

• Is there a song or a difference?

• Okay, now got different.

• So we can do to minus 1 51 Let's do a few more.

• Got Plus now.

• So that's going to be a 3 201 Oh, and a difference here.

• So three minus one is a too.

• So it's unpredictable.

• I don't know what's going to happen next on.

• If you don't mind, I'm gonna really make that point.

• If I did it again, I'm going to get a different sequence.

• But is there anything I can say about them?

• So I don't know if these are going to go off to infinity like the Fibonacci sequence just gets bigger and bigger beer goes off to infinity.

• I don't know if he's gonna maybe just goes off to minus infinity.

• I don't know that.

• Maybe it kind of hovers around zero.

• Maybe the pluses and minuses.

• Cancel out or maybes unpredictable majors chaotic.

• And I don't know what's going to happen next.

• So the surprises.

• I can make some predictions.

• I could make some long term predictions.

• For example, I can tell you what the millionth random Fibonacci number is going to be A because, like the original Fibonacci sequence, there is a growth ratio just the same way I want to show you that.

• So it's one point 1319882487943 something something.

• Something.

• Okay, so this is a constant and it's going on forever, Which means we can use this to protect the 1,000,000 random number.

• So let's do it.

• Random Fibonacci number one million.

• Same kind of thing I did with the Fibonacci numbers.

• It's gonna be approximately this constant.

• Oh, I'll just do 1.319 You get the idea to the power one million, which is some big number 8.3 times 10 to the 53,841.

• I will tell you this is a statement about the size of the number, not the sign of the number s.

• Oh, I don't know if it's a plus or minus number, so it might be a possible minus.

• So this might be going off to positive Infinity might.

• There's no reason why you can't go down to negative infinity, but they are growing, and the size of these numbers are growing on DDE.

• In long term, we can predict how much it will grow by.

• If you want to put it another way, you could say it like this.

• We take the 10th random Fibonacci numbers and make it positive we took the end fruit.

• This is tending to our magic number here, 1.1319 on dhe stuff as end tends to infinity, although, well, I'll put a little thing here.

• Almost surely that's a funny expression I've slipped in.

• This tends to this constant almost surely, because the original Fibonacci sequence would count as a randomized Fibonacci sequence if I was just getting plus, every time I fit my coin plus plus plus plus, so that counts, and that has a growth rate, which is the golden ratio.

• It's not this value, but the chances of me getting plus plus plus over and over again.

• So I've got the fibbing artist sequence becomes increasingly small.

• And so if you look at all the possible random Fibonacci sequence, things like the original Fibonacci sequence almost never happens.

• I'm not going to say it's impossible, but it's almost never gonna happen.

• You did this for real almost 100% of the time.

• You're getting a properly random sequence of process and minuses, which tends towards this growth constant.

• Another example of what wouldn't work is if I had a pattern I had to be like plus minus minus as a pattern, and it went plus minus minus.

• So I started with a one on one.

• It would be plus minus.

• Minus is a pattern did go.

• Plus would make it too than a minus.

• B two minus one says I actually won, Then a minus again, one minus two, minus one.

• Then we get for plus, which gives me easy, right.

• They will do a minus, which is going to give me a one, and I know that.

• Myers, give me a one and I get back to the original two numbers and then that pattern will continue.

• So that's not growing at all.

• That's just oscillating opened down, so that doesn't have a growth rate, because it's not growing anywhere.

• You won't really see this coming out, though, until you're using until you've done this for about 200,000 times.

• So you have to go quite far, and then you start to see that this starts to tend to these numbers.

• You don't think someone watching is gonna make some computer simulator that would do this \$200,000 on That's what they did.

• That's exactly what they've done.

• So when they were trying to calculate these numbers, they simulated it on.

• They could see that in the long term.

• These Miss Watts, the growth rate.

• So in the 19 sixties, they showed that these randomized Fibonacci sequences do grow.

• I think that's a surprise in itself because, you know, we've got pluses and minuses.

• Don't we cancel out?

• But it turns out we don't.

• So these sequences are growing so they could prove that it was growing, but they didn't know what the grave of eight waas so that top are till 1999 so we're talking 30 years later on it was calculated what?

• This growth rate Woz.

• And that was really hard.

• Proper trans maths, proper advanced computing.

• But what they did calculate waas these this part of it one point 13198824 And that's the stars they could get because it's really hard to calculate on.

• That was 20 years ago on.

• In the 20 years since we've worked out five more digits on that's a start we've got.

• So I know people.

• Audrey, hang on, Audrey.

• You want to hear their applications about even RTC cause you're not supposed to be in here.

• Out.

• Come on, actually.

• Come on.

• Don't you go.

• He was interested in you.

we're gonna do randomized Fibonacci sequences.

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# Random Fibonacci Numbers - Numberphile

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林宜悉 posted on 2020/03/28
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