 In this video, we will prove the impossibility of the classic geometric construction of squaring the circle. So what does that mean specifically? So to square the circle means that given a circle of constructable diameter, it is not always possible to construct with a straight edge and compass alone the edge of a square that has the same area as the original circle. And so our counter example is gonna be the following. Take the unit circle, the circle whose length, whose radius is equal to one. One is a constructable number. The diameter would then be two, which is also a constructable number. By the classic formula area equals pi r squared, we see that the area of the unit circle would be pi in that situation. We wanna argue that you cannot construct a square whose area is pi. So if we did that, we had our square like this, the area of a square is its side length squared. That's why we call it squared, right? So let's say this square is S by S. Well, if this was equal to pi, then that means taking the square root of both sides, the side length would have to be the square root of pi. And so I then claim that the square root of pi, which is the necessary side length for this triangle is not a constructable number. Because if it were, if the square root of pi was a constructable number, then if you square the square root of pi, you get pi. And the field of constructable numbers is a field. So I should, if the square root of pi is constructable, then the square root of pi squared is likewise constructable, which is then pi. But pi is a transcendental number. And therefore no algebraic extension of Q contains pi. The field of constructable numbers is in fact an infinite algebraic extension of the rational numbers. And therefore it contains no transcendental numbers. And that's where we get our contradiction. Pi is not a constructable number. And therefore it is impossible to construct a square for every circle.