When considering the usual absolute value, rational numbers can be extended to real numbers. If we were to take any p-adic absolute value on rational numbers instead of the usual absolute value, we can extend rational number to p-adic numbers. This master’s thesis is an expansion of the undergrad thesis titled »p-adic norms and p-adic numbers«. In my diploma thesis absolute values on rational numbers were introduced, Ostrowski's theorem was proven, p-adic numbers were constructed and their representation was briefly discussed. This master’s thesis focuses on comparing real numbers with p-adic numbers. Decimal representations of p-adic and real numbers are compared. It can be seen that the representation of p-adic numbers is analogue to the representation of decimal real numbers, although p-adic numbers have unique representations while representations of real numbers are sometimes not unique. Topology of real numbers and p-adic numbers is compared and the connection between Cantor’s set and p-adic numbers is described. Afterwards, a comparison between the real and the p-adic analysis is made. p-adic numbers (same as real numbers) are a complete normed field in which similar analytical problems can be solved as in real numbers. We finish the thesis with discussions about arithmetic operations in p-adic numbers, sequences, series, logarithmic and exponential functions.
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