Rationals are Everywhere: Proving the Density of Q скачать в хорошем качестве

Rationals are Everywhere: Proving the Density of Q

We often think of the Real number line as a solid, unbroken stream. But where do the fractions live? It turns out they are everywhere. In this video, we visualize the rigorous proof of the Density of Rational Numbers. This theorem states that between any two Real numbers, no matter how close they are, you can always find a Rational number. To prove this, we use the Archimedean Property. We visualize a magnification technique: We start with a tiny gap between two numbers. We multiply by a large integer to stretch that gap until it is wider than one unit. This allows us to trap an integer inside the gap. We divide back down to find our rational fraction sitting perfectly in the middle. This simple but powerful logic connects the discrete world of integers to the continuous world of real numbers. #realanalysis #math #calculus #education #visualization