A pocket calculator is needed for this and the following chapter.

Irrational square roots

1 If a and b are positive real numbers and a/b = √2, prove that (2b -a)/(a-b) = √2 and that b > a-b > 0. Use Fermat's method of descent to prove that a and b cannot both be integers.

2 If a and b are positive real numbers and a/b =√7, prove that (7b-2a)/(a-2b) = √7 and that b>a-2b>0. Deduce that a and b cannot both be integers.

3 Construct a proof, similar to the ones you have used in q 1 and q 2, which establishes that √57 is not a rational number.

4 Find integers a, b, c, d such that 2520/735 = 2a3b5c7d.

5 If p1 p2, P3,… is the sequence of distinct prime numbers, explain why every non-zero rational number can be expressed in the form for some n and uniquely defined integers a, a2, …, an.

6 If what can be said about the indices for r2?

7 Can 2, 3, 5 or 6 be the square of a rational number?

Convergence

The notion of a continued fraction emerges from an attempt to find rational approximations to irrational square roots.

8 Prove that, and deduce that

9 The five numbers

are also conventionally exhibited in the form

and in the form

[1], [1,2], [1,2,2], [1,2,2,2], [1,2,2,2,2].

• (i) Express each of these five numbers as a quotient of integers.

• (ii) If a/b and c/d are adjacent terms in (i), find ad-be for the four possible cases.

• (iii) Use a calculator to express these five numbers in decimal form, and notice whether the sequence increases, decreases or oscillates, and whether it appears to approach √2.

10 For real numbers a1, a2, a3,…, an, the simple continued fraction is denoted by [a1, a2, a3, …, an]. Convention demands that a, when ai≥2 .