In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The <EM>p-adic numbers contain the <EM>p-adic integers <EM>Z<SUB>pwhich are the inverse limit of the finite rings <EM>Z/p<SUP>n. This gives rise to a tree, and probability measures w on <EM>Z<SUB>p correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space <EM>L<SUB>2(<EM>Z<SUB>p,w). The real analogue of the <EM>p-adic integers is the interval [-1,1], and a probability measure w on it gives rise to a special basis for <EM>L<SUB>2([-1,1],<EM>w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For special (gamma and beta) measures there is a "quantum" or "<EM>q-analogue" Markov chain, and a special basis, that within certain limits yield the real and the p-adic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group <EM>GL<SUB>n(<EM>q)that interpolates between the p-adic group <EM>GL<SUB>n(<EM>Z<SUB>p), and between its real (and complex) analogue -the orthogonal <EM>O<SUB>n (and unitary <EM>U<SUB>n )groups. There is a similar quantum interpolation between the real and p-adic Fourier transform and between the real and p-adic (local unramified part of) Tate thesis, and Weil explicit sums.