Usually we think that we are living in the real world, namely the world of real numbers. Sometimes it is useful to move to the larger and more complex world of complex numbers, since there we have various powerful tools including complex analysis and Euler's identity. Also we usually think that integers and rational numbers are parts of real numbers and complex numbers. However, it turned out that they are not enough to reveal mystery in number theory.
In the last century, we humans found parallel universes to our real world which exist for every prime number p. In the universe for p, p-adic numbers and p-adic periods play comparable roles of real numbers and complex numbers. In this talk, I will explain what they are like, and how they have contributed to developments of number theory.