 Alright, let's continue to find rational points on elliptic curves, but this time let's look at the problem from an algebraic standpoint. So if you trace back through the geometric steps for adding two points, we can translate these into an algebraic procedure as follows. So first off, if I'm trying to add the point at infinity, what I find is that adding the point at infinity to any rational point just gives you the point itself. In other words, this point at infinity serves the same purpose as the additive identity zero does in the addition of the integers. And again, you might draw a few diagrams to convince yourself this must in fact be the case. Now let's say I have two distinct points on the elliptic curve, and again we'll assume that these two are rational points. Now what I can do is I can calculate lambda, the slope of the line between these two points, then the sum of these two points is going to be given by the coordinates x3, y3, where I can compute x3 by this formula, lambda squared minus x1 minus x2, and y3, lambda x1 minus x3 minus y1. And again, this algebraic formula emerges from applying the tools of algebra to the geometry of adding two points on an elliptic curve. And then finally, if p and q are the same point, then what I'm going to do is I'm going to use the slope of the tangent line, which I can find using the implicit function derivative, and I can use the slope of the tangent line, and then that gives me the slope of the line that I need, and the algebra is essentially the same, and my coordinates are going to be given by the preceding formulas. For example, let's take a rational point on the curve, y squared equals x cubed plus 5x minus 2, and suppose we already know one rational point to 4, and let's see if we can find a set. So we have our equation that gives us the coordinates of a third rational point given 2. So for any two points on an elliptic curve, my third point has to satisfy x3, lambda squared minus the x coordinates, y3, lambda x1 minus x3 minus y1, when lambda is the slope of the relevant line. If the two points are distinct, then the relevant line is going to be the line in between the two points. If I only have one point, then I'm just going to use the slope of the tangent line. So here I only have one point initially, so the slope of the tangent line is going to be the relevant slope. So I'll use the implicit differentiation method, and I can find the slope of the tangent line. I have my values of x and y, x is 2, y equals 4. Substituting those in, I get lambda equals 17 over 8 as the slope of the line. Then I have my x coordinate, lambda squared minus x1 minus x2. I'll substitute in my values and get my x coordinate, my y coordinate, again, lambda x1 minus x3 minus y1, substituting in, and that gives me the coordinates x3, y3 of the point that is defined as the sum of point p plus itself, 2p in other words.