In this video, we introduce the Theorem of Pythagoras, possibly familiar to many of you from school. We'll look at it in the context of an interesting historical perspective relating to Babylonian Mathematics. An application of this theorem is a geometric interpretation of the square root of two, a very important real number in the history of Mathematics. We prove that the square root of two is irrational, that is, cannot be expressed as a fraction. A result that was known to the ancient Greeks. We begin the story by considering a unit square with side lengths one unit. I'll reproduce the unit square, you start off lets say, one corner and travel to the opposite corner. You could for example, move along the sides of the square, but that's not the most direct path, the most direct path from one point to another is a straight line. If we go along the diagonal of this square, that's the shortest path, and it has a certain length we'll call it x. So, what I'm going to do is play around with this square, this is a trick that goes back to Babylonian times. So, here we have a unit square, now I've just taken a copy of it and they form the stack with two squares. I will just put another stack of two squares next to the original two squares to create a two by two square. So we've got four unit squares, making a larger square with side lengths two units. Now, I'm going to focus on the midpoints of these sides of this square, and then join up these midpoints by forming diagonals in this way. Okay. Now, notice something in the middle of this two-by-two squares is another square whose side length is x, the length of the diagonal of the unit square, and that inner square has an area which is x squared. I can count the area by adding up the areas of certain triangles in this diagram. If I shade that triangle then that has area which is a half of a unit square, then I add the area of this triangle, another half, the area of this triangle and the area of this triangle. So, I'm adding a half, four times which gives you two. So, the area of the inner square, is all shaded now in green, is two units. So, x is a number whose square is two. So, x is the square root of two. Now, I'm going to do something, I'm going to copy this figure. I'm just going to rub out everything outside that green inner square, and then I'm going to get some clay. So, I've got some soft clay and I have imprinted on that clay an image of that inner square that we've been discussing. I put it in the oven and bake it, and then a few thousand years later this is what we get. Okay. This is called the Yale Tablet and it's the oldest recorded proof of the theorem of Pythagoras. In fact predates Pythagoras by about a thousand years. Now, what's this got to do with the theorem of Pythagoras? Well, let's go back to the unit square and its diagonal which is the square root of two, which we demonstrated by that method. This is what's called an Isosceles triangle, where the shorter side lengths are equal in this case to one. I can draw another isosceles triangle which could be much bigger or much smaller with common side length a. We scale up the length of the smaller triangle. So, the hypotenuse of this new isosceles triangle is root two times a. If we call that c, then c squared is root two a, all squared, which is two a squared, which is a squared plus a squared. Now, perhaps you can see that this starts to resemble a familiar result. The square of the hypotenuse is the sum of the squares of the other two sides. This is a special case of a general theorem, the theorem of Pythagoras, which says, if you take any right-angle triangle, the side lengths ab and hypotenuse c, and a and b can be different now. If you form squares on each side of the triangle, then the square of the hypotenuse is the sum of the squares of the other two sides. So, we get this famous equation, c squared equals a squared plus b squared, and this is the theorem of Pythagoras. What we discussed earlier was in fact, a proof of this theorem in the special case where, a and b are equal to one, which clearly scales up to a and b being equal. Now, you can modify that proof very easily to get a general proof, and that modification appears in the notes that accompany this video. Now, if we have a look, c squared equals a squared plus b squared, implies that c is the square root of a squared plus b squared. Now, when you have an expression involving a square root sign, it's called a Surd expression, okay. You don't expect it to simplify. Occasionally, it will, for example, if I take a right angle triangle where the shortest side is three units long, the other short side is four units long, then the hypotenuse turns out to be five units long. So, you have what's called a three, four, five triangle. The five is predicted by this theorem because three squared plus four squared which is nine plus 16 is equal to 25, which is indeed five squared. Now, that's very rare that the hypotenuse simplifies so simply. Typically, the surd expression doesn't simplify and if we go back to the unit square, triangle associated with the unit square, the hypotenuse has length root two, and it turns out that root two cannot be expressed as a fraction. Now, of course the length, the distance between two points is very important and ancient mathematicians wanted the mathematics to be sufficiently powerful to model distances between points, okay. So, studying the square root of two the diagonal of the unit square was a very important mathematical entity and now when you develop mathematics you start off, for example, with counting numbers and then you form fractions of counting numbers, they're called rational numbers and it was an important question whether the mathematics of fractions was sufficiently powerful to model distances like the square root of two, and it turned out the answer is no. There is no fraction which can be used to express the square root of two exactly. The square root of two is not a fraction. Again I'm going to try and explain why this is the case. Now in mathematics, proofs or demonstrations are a little bit like criminal investigations, it's typical to make some kind of supposition and see if it leads to nonsense or a contradiction. Now if you're investigating a crime and you want to know who's the murderer or who was the thief, if a suspect can produce an alibi that contradicts the possibility of them committing the crime, then you know that they're innocent. We're going to use that technique in this proof. We're going to suppose to the contrary, that root two is a fraction and see if it produces nonsense, okay. So, I'm going to suppose root two is a divided by b, where a and b are counting numbers or integers and I can assume that we have canceled off any common divisors, in the numerator and the denominator and this is in reduced form, and that will turn out to be important. We will square both sides of this equation, on the left-hand side we get two, on the right-hand side we get a squared divided by b squared. So, if we multiply both sides by b squared, we get two b squared is equal to a squared. So a squared is two times a counting number two b squared, so, it's even, okay. What follows, you can think about why this is the case, that a itself must be an even counting number. So, if it's even, means is two times something, say a is equal to two times c, for some counting number c. So, if I go back to the previous line, you've got two b squared is equal to a squared, but a is two c. So, that's two c, all squared which is four c squared, okay. So, now we divide both sides by two and we conclude that b squared is two c squared. So, b squared is two times something which is even, then as before, if the square of a number, counting number's even, the number must be even. So, therefore b is even. Now, have a look, b is even and a is even. So, they're both divisible by two, they've both got a common divisor of two and that contradicts the assumption that a over b is expressed in reduced form. So, this is actually nonsense, it can't be true. So, therefore, root two is not a fraction. Again we've covered a lot of ground in just a few minutes much more details provided in the accompanying written notes. Please take some time to read and digest them carefully and then try the exercises that follow. Thank you for watching and I look forward to seeing you next time.
