 Hello again to another screencast where we're exploring more methods of proof. The last technique of proof we're going to look at in this section is a general umbrella term called a constructive proof. So what is a constructive proof? Well, in a constructive proof what we're dealing with are theorems that claim one of two things possibly. First of all, a theorem that claims that something exists, or a theorem that claims that if you're given a certain set of objects then a certain calculation holds. Every one of these two theorems involve things that we could build. We could actually build the thing that we claim exists, or I could actually do some computations and find out that the calculation holds. If we do that in a proof that's called a constructive proof, we simply build the thing that the theorem is claiming to either exist or build the mathematical calculation that the theorem claims to hold true. We're going to look at two examples of this. The first one is the following conjecture. This is actually taken from an early in class activity for section 1.2. It has to do with the solutions to a quadratic equation. So let A be a real number not equal to zero. And suppose that this equation AX squared plus BX plus C equals zero has two distinct or different real number solutions. Then we claim that the sum of those solutions always equals negative B divided by A, that would be the negative of the coefficient on X divided by the coefficient on A. So this is a conjecture that gives us an object. It gives us this quadratic equation right here. And asks us to suppose something about the solutions to that equation. And then show that the solution satisfy a certain calculation. If you take the two solutions and add them together to equal something. So the constructive approach to proving this theorem would be to say, first of all, let's go get the solutions. Let's just find what those solutions are and have them concretely in front of us. And then to simply add them together and show that once I take these two solutions, I found them, if I just simply add them together, they equal what the conjecture says they equal. So that's the constructive proof. We're going to build the object, the item that the conjecture pertains to, the solutions that is, and then build the calculation and show that it comes out to equal what we say it does. So let's go over here and write a proof. This is not going to be done in a no show table. I'm just going to kind of sketch an outline of a proof here. Just to remind us, we have this equation in front of us, AX squared plus BX plus C equals zero and A is not equal to zero. Okay, well the solutions to that equation we know quite well from the quadratic formula, there are two of them. The solutions to this equation are X1 equals negative B plus radical B squared minus 4AC over 2A. And since A is not equal to zero, this dividing by 2A is legitimate. I'm not dividing out by zero. And the other solution, of course, is the X2, let's call it. And that would be negative B minus the square root of B squared minus 4AC. And again, that is because of the quadratic formula. Okay, this is a prior known mathematical fact. We don't really need to show why the quadratic formula is true. This is just something we know is true. So here are the solutions. Let's just simply add them together and see what we got. So I've got, let's take what is X1 plus X2. We hope it comes out to equal negative B over A. But let's just kind of work with it. And substituting in, I have this for X1. So it's negative B plus radical B squared minus 4AC over 2A. And plus X2 is this thing over here. That's negative B minus radical B squared minus 4AC over 2A. Now here, I can just do a direct calculation here. I have common denominators, so why don't I combine the numerators? That would be negative B plus radical B squared minus 4AC. Plus another negative B, I'll parenthesize that. Minus radical B squared minus 4AC. And this is all over 2A again. Okay, so so far this is just saying, look, here are the solutions. I'm claiming something about what happens when these solutions are added together. I'm just going to add the solutions together and follow the math here. Now what you see in the next step here is that the radicals cancel out. And I have a couple of negative B. So this is equal to negative 2B divided by 2A. And if I just divide off the 2s, I get negative B over A. And that's the end of the proof. I've sketched it, not really formal, but that's the end of the proof. That's it. I've just gone through and said, okay, look, if you give me the information that you say you're going to give me, here are the solutions. And I know this from high school algebra. And then if I just add them together, follow the math and it comes out to be this quantity right here, which we claim to be true. That's a constructive proof. It's quite satisfying, isn't it? We just simply take what we're given and build what we say actually happens. So here's another example of a constructive proof. This one involves proving that something exists. Okay, so this is quite a common usage of a constructive proof. Let X1, X2, and X3 be any three real numbers you like. Then there exists a fourth real number I'm calling Y such that the average of X1, X2, X3, and Y equals X2. And average, of course, just from your basic stats knowledge, would be the sum of X1, X2, X3, and Y divided by four. Okay, so I'm claiming that something exists. The one excellent way to claim that something exists is just to simply build it and produce it. And if you have it right there in front of you, then it obviously exists. So here's a little sketch of a proof down here. Let's just set up what we want to be true. The average of those four numbers equaling X2. So I'm going to set up, here's what the average would be. X1 plus X2 plus X3 plus Y divided by four equals X2. Now if I can work backwards and actually find the Y, prove that there exists such a Y that makes this work, I'm done. I'm just going to build or construct the Y. All right, so let's see if I can get there. If I cross multiply here and get the multiply both sides by four. Okay, just a little arrow here to indicate a step. I'd have X1 plus X2 plus X3 plus Y equals four X2. Okay, and that's just my algebra. And now if I just simply solve for Y, I end up with Y equals, subtract X2 from both sides and up with three X2. And then subtract everything else from both sides and up with this. So there it is, I've proven that there is such a Y that makes this average work out to be X2. And that, friends, is the end of the proof. I claimed that given any three real numbers, then there exists this number that makes this happen. And here it is, okay? You can't deny that the thing doesn't exist because all I'm doing is taking the three numbers I was given and combining them using arithmetic. So that Y certainly exists, you're looking at it. So that makes it exist. Okay, so that's constructive proof. When we're working with conjectures or theorems that claim that either something exists or that a certain calculation works out to be what it says it works out to be. Oftentimes we're going to just build it and the proof, as they say, will come.