[MUSIC] Welcome back. I'm standing next to what I think is a very beautiful Renaissance painting of Friar Luca Pacioli. Luca Pacioli was a friar, a religious man, but he was also a mathematician. And because of his standing as a mathematician he became the teacher and good friend of Leonardo Da Vinci, who is, of course, the most famous of all the Renaissance men. Luca Pacioli is now known as the Father of Accounting and Bookkeeping, but our interest in him is because he was the author of a book called the De Divina Proportione or On the Divine Proportion. The Divine Proportion is a number. It's a number that we now call the Golden Ratio. And Luca Pacioli obviously thought this number was divine, was a godly number. We're going to see that this number has a very close relationship to the Fibonacci Sequence. So now what I want to do, is to introduce to you, the Golden Ratio. So what is the Golden Ratio? You can understand what the Golden Ratio is by starting with a line segment And dividing the line segment into two segments, one of length x and one of length y. And we're going to assume that the length of x is larger than the length of y. And of course, they're both positive because they're both lengths of a line segment. So what is the Golden Ratio? We're going to call the Golden Ratio capital Phi, the Greek letter Phi. It's the ratio of the larger line segment x, to the smaller line segment y, x over y. Where these lines segment satisfy a particular equality, that the larger segment over the smaller, x/y, is the same as the total length of the line segment x+y divided by the larger segment x. So golden ratios are these ratios, x/ y, or x+y/ x, but we require that x/ y equals x+y/ x. So we can do some mathematics on this equation. So let's try and do that. So, x / y is Phi, so we have x/y as our Phi. And then we can manipulate x + y / x, so we can divide x by x, x/ x is 1. And then we can divide y by x. But y divided by x is just the reciprocal of x divided by y. So y divided by x is 1/Phi. So the Golden Ratio satisfies the equation, Phi = 1+1/Phi. Okay? We can take this equation and we can multiply both sides by Phi. And then bring everything to the left hand side of the equation. So what do we get? So we multiply Phi x Phi. So we get Phi squared. And then Phi squared is equal to 1. We multiply 1 x Phi and bring it to the left side. So we get minus Phi. Equal to 1+1/PHI. We multiply 1/Phi x Phi. That becomes 1. We bring it to the left side, so we get minus 1. And this is equal to 0. So the Golden Ratio satisfies a rather simple equation, it's called the quadratic equation, Phi squared- Phi- 1 = 0. We can use the quadratic formula to solve for Phi, right? If you have an equation, ax squared plus bx plus c equals 0, you know that x is equal to negative b plus or minus the square root of b squared minus 4ac divided by 2a. So we can apply the quadratic equation to solve for Phi. There are two roots, but one is negative and we know that Phi is the ratio of two lengths, so Phi has to be positive. So the positive root, if you just use the quadratic formula, you can show that this is equal to the square root of 5+1 /2, and that's the golden Ratio. And this has a numerical value. It's an irrational number but we can write it, approximate it 1.618 something. And that's the famous Golden Ratio. Just a number. Okay? We'll see why this Golden Ratio is divine later on in this course. And we'll also see what the connection is between the Golden Ratio and the Fibonacci numbers. In addition to the Golden Ratio, it's useful for us to define another number which is called the Golden Ratio conjugate. Rather than using big Phi, we use a small letter phi, also phi, but a small case phi, to define the Golden Ratio conjugate. This one we define as square root of 5-1 / 2. What is the difference between the Golden Ratio and the Golden Ratio conjugate? Well, the Golden Ratio conjugate, you can see, is 1 less than the Golden Ratio. If we subtract 1 from capital Phi, then the 1/2- 1 becomes -1/2. So 1 less than the Golden Ratio is just going to be 0.618 something. So it's just the fractional part of the Golden Ratio. So we can write phi = the Golden Ratio- 1. But phi also has another relationship to the golden Ratio. So we had the Golden Ratio of phi is equal to 1+1 over the Golden Ratio. The Golden Ratio of phi is equal to 1+1 over the Golden Ratio. So 1 over the Golden Ratio is equal to little phi, the Golden Ratio conjugate. So the Golden Ratio conjugate is also equal to 1 divided by the Golden Ratio. So the Golden Ratio is already a special number that the fractional part of the Golden Ratio is actually just the reciprocal of the Golden Ratio. The property will lead to this Golden Ratio being special in a very specific way that will be important in its appearance in nature. Okay, so that's the golden Ratio, the divine proportion. I'll see you next time.