Binary representation of 2-adic integers I would like some examples of the binary representation of 2-adic integers that are not standard integers. What is the 2-adic expansion of $1/3$? Of $-1/3$? What number does $...010101$ represent?

To answer your last question first, $$...010101 = 1 + 4 + 16 + \cdots + 4^n + \cdots = 1/(1-4) = -1/3.$$ This also answers your second question. As for your first question, $$1/3 = \dfrac{1}{2}( 1 - 1/3) = ...0101011.$$