Surds and the pursuit of happiness

My son Arthur mentioned that he had learned about surds in maths, so we asked him what one was. The definition I vaguely recalled from my school days was that a surd is an irrational number, but of course that's not the whole story, since it would seem that not every irrational number is a surd. Arthur did not know an exact definition, and it would seem that no-one else has tried very hard to pin it down precisely. Higher GCSE Mathematics for Edexcel by Alan Smith, p492, states:

Some quantities in mathematics can only be written exactly using a square root symbol.

For example, if x²=5, then the exact value of x is √5 (or -√5).

Quantities like these, written using roots, are called surds.

Based on discussions and exercises on the following pages, it appears that a number like 1+√2 is a "surd expression" rather than just a surd, but neither was it ruled out as being a legitimate surd. The book gave no hint about whether, for example, the cube (as opposed to square) root of 2 is a surd.

Other sources are similarly imprecise. Wikipedia indicates that a surd in an N-th root (presumably, an N-th root of a positive integer, where N is also a positive integer). It says here that

An unresolved root, especially one using the radical symbol, is often referred to as a surd.

Based on the usage of the word in that web page (which also explains its origin) it looks like it's supposed to be a (real-valued) positive integer root of a positive integer.

This web page states the most restrictive definition: "A surd is a square root which cannot be reduced to a whole number." Presumably they mean: a square root of a positive integer, and not a number like √(9/4) = 3/2. Wiktionary says: "An irrational number, especially one expressed using the √ symbol." (which would appear to allow 1+√2).

With a view to inducting my sons into the family trade, I thought that it would be a worthwhile mathematical exercise to discuss what should be the right definition. (The definition itself will not be interesting mathematically, but the pursuit of one is of great value; by analogy, the chap who coined the phrase "Life, liberty and the pursuit of happiness" clearly figured out that pursuit of happiness, rather than happiness itself, was the point.) It's a topic that touches on all sorts of issues, such as which if any, of the alternative definitions are equivalent, and why. More fundamentally, it addresses the issue of what constitutes a genuine mathematical definition, as opposed to some general guidelines on usage. Finally, the alternative definitions will have various different merits, such as being a set of numbers that is closed under addition. In the event the discussions did not get very far, but looks like a good one to have in high school math lessons.

(added later: Mark Jerrum pointed out this link on mathematical terminology; in the case of surds, it contains more historical detail than wikipedia's page.)