*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, ifx^{2}=5, then the exact value ofxis √5 (or -√5).

Quantities like these, written using roots, are calledsurds.

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 asurd.

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.)

## 3 comments:

and did the boys take to the idea of defining surds properly...? I look forward to surd-related discussions with Lily & Tom :-)

Not really. Although I think the success of this kind of discussion is mainly down to timing (catch'em when they're in a receptive mood).

Are you serious ?? surds must be intriguing! i can see exactly when i'm going to need surds in later life.i can just imagine my cambridge admissions test now.

"

question_1

a) solve the following

(1) sin x(1-2cos x)=0 for 0 degrees is < or = to x which is < or = 360 degrees

(2)2sin x cos + cos(squared)x=0 for 0 degrees is < or = to x which is < or = 360 degrees

6 marks

b) what is the exact denotation (dictionary definition) of a surd

78 marks

c)what is the exact denotation (dictionary definition) of a ...

a) a graph homomorphism

b) a random coloured tree

c) ferro magnetic space potts

d) unstable marriages

16 marks

total marks 100

(isaac)

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