Part IV Branches of Mathematics beyond the fact that its square is 2 (a viewpoint taken toward algebraic numbers by kronecker [VI.48], as we shall see below). We can write 2 in various forms, e.g., 2 = |1 - i|, (1) and we can think of 1-i = 1-e 2 i/4 as the world's sim- plest trigonometric sum; we shall see generalizations of this for all quadratic surds below. We can also view 2 IV.1 Algebraic Numbers Barry Mazur The roots of our subject go back to ancient Greece while its branches touch almost all aspects of contemporary mathematics. In 1801 the Disquisitiones Arithmeticae