[MUSIC] Welcome to my course on the Fibonacci sequence and the golden ratio. I'm standing next to the iconic portrait of the greatest mathematician of the Middle Ages, Fibonacci. Fibonacci was born almost 850 years ago in Pisa, around the same time they started building what later became the leaning tower of Pisa. In the year 1202, Fibonacci finished his influential book Liber Abaci. The book of calculation, that brought the Arabic numerals to Europe, the zero, one, two, three that we use today. Fibonacci also presented a problem about a growing rabid population. And derived the now famous sequence of numbers that is named after him, the Fibonacci Sequence. In this video, I want to explain to you about the rabbit problem and also derive Fibonacci's numbers. So what is the rabbit problem? We start with one newborn rabbit pair say, placed in an enclosed field. This newborn rabbits take one month to mature to adults, the pair is a male and a female. Then they mate and the female becomes pregnant, and the female is pregnant for one month and then gives birth to a newborn rabbit pair. And then the rabbit pair mates again and so on, continuing to give birth every month. Also we make a simplification in this problem either it's a very short time interval or we just assume that no rabbits will die. So the problem posed by Fibonacci was how many rabbit pairs would there be after one full calendar year? So what we need to do then is we need to count rabbit pairs. So we can do that be constructing a table, so here's our table. The first row, we put the month, January, February, March, April. Let's say, we start on January 1st and the asset to our problem will be after one calendar year say, on January 1st of the next year, how many rabbit pairs will there be? We can count rabbit peers by counting the newborn rabbits at the beginning of every month. I call them juveniles, and then we can count the number of adults at the beginning of every month. And then sum the juveniles and adults to get the total number of rabbit peers. So in the first month, let's say January 1st, we introduce one newborn rabbit pair into the population. So we have one juvenile, no adults and one rabbit total. After one month to February 1st, the juvenile care of rabbits mature into an adult care of rabbits. So we have no juvenile now, we have one adult and still one total rabbit pair. One rabbit pair total, and then what happens in the next month? Well those adult rabbits mated, the male and female mated. The female was pregnant for one month, and then gave birth to a newborn pair of rabbits on March 1st. So then we have one juvenile on March, one juvenile peer one adult peer this adult peer is still the same adult peer we had in February and two total rabbit peers. So in April then this adult peer will again give birth to a juvenile pair of rabbits, and this juvenile pair then will mature into adults. So we have one juvenile pair, two adult pairs, and then three total rabbits, and the population precedes like this. So the adult pairs always give birth on this the female of the adult pair of rabbits. Always gives birth to new born rabbits and the new born rabbits the month always mature into adults. So if we continue, we can fill up the table and the answer to Fibonacci's problem is on January 1st of the next year, we'd already have 233 rabbit pairs. So in 1 calendar year, we've gone from 1 rabbit pair to 233 rabbit pairs. The population is growing very fast, we use the term breeding like rabbits. Because rabbit population can grow very fast because you can have new born rabbits every month. So let's look at the numbers, let's look at the total number of rabbit pairs. This is a sequence down here in the last row. The sequence goes like, 1, 1, 2, 3, 5, 8, 13, 21, 34, 35. This is a number sequence. This is called the Fibonacci Sequence. The Fibonacci Sequence has a very characteristic pattern. If we look at starting with this number 2, we see that this number 2 is 1 plus 1. It's the sum of the preceding two numbers, and if we continue, the number 3 is 1 plus 2. The number 5 is 2 plus 3, the number 8 is 3 plus 5, 13 is 5 plus 8. So every number then is the sum of the preceding two numbers. We can write that as an equation. So if we call capital F sub n, the nth Fibonacci number, then the equation is that the F sub n, the nth Fibonacci number is equal to F sub n-1, the n-1 Fibonacci number + F sub n-2, the n-2 Fibonacci number. This is called a recursion relation and is the basis of the Fibonacci sequence. But it's not enough to generate all of the Fibonacci numbers, because you have to start somewhere with this equation. You have to have some starting value, some initial values of the Fibonacci numbers. It's traditional to choose F sub 1 and F sub 2 as the starting values of the Fibonacci number. So we say, F sub 1 = 1, that means we're starting with the newborn rabbit pair. And we say, F sub 2 = 1, which means that it takes 1 month for the newborn rabbit pair to mature into adults. So in this video, we derive the Fibonacci Sequence. The rest of the course we'll look at some interesting properties of this sequence. And also it's relationship to a famous number which is called the golden ratio. I'll see you next time.