Rational and Irrational Numbers Putnam Practice Summary: Rational and Irrational Numbers Putnam Practice September 7, 2004 A rational number is one that can be expressed in the form a/b, where a, b are integers and b = 0. To represent a given non-zero rational number, we can choose a/b such that a is an integer, b is a natural number, and (a, b) = 1. We shall say then that the representative fraction is in lowest terms. An easy consequence of the definition is that any rational number has a periodic decimal expansion. Real numbers with non-repeating decimal expansions cannot be expressed in the form a/b and are called irrational. A number that satisfies an equation of the form c0xn + c1xn-1 + ... + cn-1x + cn = 0; where c0, ..., cn are integers and c0 = 0 is called algebraic. A number that is not algebraic is called transcendental. It is known that and e are transcendental numbers. Theorem 1 (Rational Root Theorem) If c0, c1, ...cn are integers, a/b is in lowest terms and x = a/b is a root of the equation c0xn Collections: Mathematics