Exploring the Mathematical Landscape of p-adic Numbers: Unveiling Uniqueness and Ostrowski’s Theorem

In this paper, we investigate on the p –adic number, metric and absolute value over the words more précises, we shall construct the uniqueness theorem and ostrowki’s theorem by using p – adic number. We explain a p – adic absolute value, non-archimedean axioms, after that we study their metric space and completion of X to XP. Introduction: The p-adic numbers, where p is any prime number, originate from an alternate way of defining the distance between two rational numbers. Kummer first discovered p-adic numbers in 1850. The p-adic numbers are useful because they provide another toolset for solving problems, one which is sometimes easier to work with than the real numbers. Methods: the strong inequality of p-adic numbers, uniqueness theorem for p- adic number and also Ostrowski’s theorem for p-adic numbers. Results: the absolute value p –adic metric is non-Archimedean, all triangles are isosceles in a non - Archimedean metric space, a non –Archimedean metric space theorem and completion of X to XP. Conclusions: we have defined a p – adic number and p – adic metric. After that, we give their uniqueness and Ostrowski’s theorem.