An n-tangle is an embedding of a disjoint union of $n$ unoriented arcs in a 3-ball $B³$ with the endpoints of the arcs lying in the ball's boundary $S²$. Addition and multiplication of tangles is defined by  surgery on the boundaries of pairs of balls that connects an endpoint of each arc in one ball with an endpoint in the other ball. With these operations, we can construct iterated tangles that fall into several distinct classes. There is a one-to-one equivalence between the class of rational tangles (up to isotopy) and continued fractions, and some very surprising connections between finite knots and links and the rational and algebraic numbers, and between wild knots (knots with an infinite number of crossings) and the complex numbers.