The analysis in this chapter illustrates Temple’s observation regarding the necessity for creative imagination in mathematics. A simple expression is all that is needed to develop the theory of continued fractions which leads to a deep theorem of Lagrange and also leads to an optimal way to approximate real numbers as rational fractions.

1. 1.

   These results have been seen before, see equations (5.19) and (5.27).

2. 2.

   Later in equation (10.50) on page 152 integersn_i andd_i are shown to be co-prime.

3. 3.

   A quick proof establishes this fact. Suppose that\(\sqrt {\beta } = c/d\) for integersc andd. This implies thatd²β = c². Sinceβ is not square, there must be a primep with an odd exponent in its factorization. All of the exponents in the prime factorizations ofc² andd², however, are even. This implies thatp has an odd exponent ind²β and an even exponent inc² which means they cannot be equal. This contradicts the claim that\(\sqrt {\beta }\) is rational.

4. 4.

   Such numbers are calledquadratic irrational numbers.

5. 5.

   This follows from the fact that if they had a common multiple, so thatn_i = am andd_i = bm, thenn_id_i−1 − n_i−1d_i = m(ad_i−1 + bn_i−1) = ±1. This implies thatm must divide 1 forcingm = 1.

Authors and Affiliations

1. (Home address), Beverly, MA, USA

   Randolph Nelson

Reprints and permissions

© 2020 Springer Nature Switzerland AG

Policies and ethics