MATT PARKER: Okay, so, Lucas Numbers are named after a guy called Edward Lucas Although, he was French, so ‘Loo-cah’, but I’m going to say ‘Lu-cas’ because I am lazy And if it wasn’t for Lucas, we wouldn’t really care about Fibonacci, because Fibonacci did some things he wrote those numbers down, didn’t generalize them, didn’t analyze them He did introduce some interesting number systems to Europe, and he did some other cool bits and pieces but these are more of a footnote to the story of his life and Lucas went, “Hey, these are really
interesting”, and he’s the guy who promoted them and he went with his own sequence which is better than the Fibbonacci Numbers Then you think, okay, if Fibbonacci starts 1,1 what is our next option? So Fibonacci starts 1, 1 and then it goes 2 and then 3 and then 5 and then 8 and then 13 and then onwards Well, what if we start — instead of starting 1, 1 what if we start 1, 2? but we can’t start 1, 2 because we’re just starting here All we’ve done is move along slightly and so we’re going to get the same numbers if we start 1, 2 Lucas numbers, you start 2, 1 — so you just swap them around and then you repeat exactly the same pattern So 2 + 1 then gives us 3 and then we get 4 and then you get 7 and then you get 11 and 18 And these are the Lucas numbers! And to my mind this sequence of numbers is far more interesting then the Fibonacci sequence of numbers, and these have a stronger link to the golden ratio So if you take the golden ratio Φ (phi), which is equal to 1.61803398… and then a 9, and so on Okay, right, so we know for any of these sequences they’re going to approach that ratio But what if we start with the ratio, what if we want to take the ratio and then get a sequence of numbers? So if you calculate Φ² — Φ²=2.6180…, which if you round to the nearest whole number is 3 (approximately) Okay, so I’m going to do this column as rounding to the nearest whole number if you do Φ³ — Φ³=4.23606…, which if you round that to the nearest whole number is 4 Φ⁴=6.85410…, and rounded to the nearest whole number is 7 Φ⁵ — and if I carry on and take each of these and round them to the the nearest whole number, what do I get? Sure enough, these are the Lucas numbers appearing up here, and that will carry on all the way down Every power of the golden ratio beyond the nearest whole number gives you the Lucas numbers and so that’s not the fact that the Fibonacci numbers are linked to the golden ratio in a way that every other similar sequence is. If you start with the golden ratio and then take powers — is because what you’re doing is multiplying each term by the golden ratio to get the next term which is that wonderful property that everyone bangs on about But these are the numbers it generates, if you start with that and then go to the sequence and that’s why I think the Lucas numbers are vastly superior I’ve been very careful, I’ve never said Lucas sequence, which is a very good point, because Lucas sequence is, well, a family of other things. So Lucas did some amazing research into all the weird properties you get from these various sequences and these numbers were just one specific example of a Lucas sequence but they’re the most common one and the most famous one, which is why they are called the Lucas numbers but there are all sorts of amazing things you can do these numbers You can — everything from obviously wonderful ratios but you can use them to test if a number is prime and, I mean, there is a whole amazing area of math Lookup Lucas sequences, there is some incredible stuff in there BRADY: One thing about this that strikes me though is we often talk about how mathematics is very precise and perfect and this great thing This sequence does match, but it only matches by kind of roughly rounding, and it seems like each time it kind of slightly misses the target but it misses by such a small amount that it doesn’t matter This doesn’t seem very much like a lot
of the other mathematics I see it seems kind of a bit fuzzy and “almosty” DR. PARKER: I think rounding — people rain hate on rounding because you do it at primary school and people view it as maths you did very early on, and very simple and there’s kind of a false — People link mathematics with unnecessary precision and they link mathematics with doing things that are pointless and over the top and so, for some reason, one of the first things people would do when they’re learning math at high school and want to look smart is write down an unnecessary number of decimal places Right? because they think it’s more maths-y — the more numbers you write down, the more maths we’re doing and I think it’s a shame, there are some amazing things that drop out of rounding and so I think it’s nice that the Lucas numbers you do get these wonderfully precise answers out of the powers then you round to the nearest whole number and that’s where the beauty is, I like that.
]MATT PARKER: 8,229 + 17,399=45,628 so I’ll do it for a few of these and you can zip through it obviously BRADY: So is this the Brady sequence? This is the Brady sequence! We’re going to call these the Brady numbers.