Suppose A is an n × n matrix with real elements, and that λ = λ1 is a dominant double real eigenvalue. We will assume for simplicity that none of the order eigenvalues are repeated. Then there are two possibilities

• (a) matrix is non-defective, and we can find two linearly independent eigenvectors corresponding to the double root λ1,

• (b) matrix is defective, and we have only one linearly independent eigenvectors corresponding to the double root λ1.

Mensuration involves the construction of measures and the methods of using them and the calculation of various quantities from the simplest possible measurements. It is in this last sense that the term is usually employed, and herein problems in heights and distances are solved, the lengths of curves and the areas of plane or other figures and the volumes and surfaces of solids are measured.

This number arises naturally, of course, as the hypotenuse of a right triangle with legs of length 1. Notoriously, √2 was found to be irrational by the Pythagoreans in around 500 BC. Their mathematics and philosophy was based upon natural numbers and small number ratios, and thus this discovery would have been very troubling: the Pythagoreans wouldn’t be the only ones to react badly to the imposition of the irrational.

The earlier sections of this account are concerned with some standard properties of pure recurring decimals. These include the cyclic property, illustrated for example by the decimals representing the six proper fractions with denominator 7, each of which has a period using a cyclic permutation of the digits 142857 It was a question concerning this property, raised by a student, which led to the writer's interest in the properties considered in the later sections. References are given in Dickson's History of the Theory of Numbers, Vol. I, Ch. 6.