Diophantine Approximations

Using \(\pi \) as an example, it is explained that even bad approximations can be very accurate, and useful for their conversion into good approximations. The condition for an approximation of a number to be a convergent of the continued fraction is established. Liouville’s theorem on the approximation of algebraic numbers is proved, thereby showing the existence of real transcendental numbers. The theorems of Hermite and Lindemann on the transcendence of e and \(\pi \) are given. The continued fraction for e, and the proof of the irrationality of \(\zeta (3)\) are also included.