Abstract: We all know one metric on the rationals: the one induced by absolute value. Did you know there are others? For each prime p there is an associated metric on ℚ, induced by what’s known as the p-adic norm. These appear naturally when one attempts to classify the norms that can be placed on ℚ and are the only others that exist up to some equivalence. The rabbit hole doesn’t end there! Remember that ℝ is the completion of ℚ, i.e. what appears when you “fill in the holes”. There’s nothing stopping us from completing ℚ when equipped with these new fancy metrics, and we obtain fields known, each known as the p-adic numbers. At this talk, we will discover how the p-adic norms appear as we classify all norms on ℚ, a result due to Ostrowski. We’ll then discuss one method of constructing ℝ from ℚ with the absolute value and use that same construction to produce the p-adic numbers.