On the Origin and Precise Definition of the Term 'Surd'

Question

So, in the course of last week's class work, I ran across the Maple function surd() that takes the real part of an nth root. However, conversation with my professor and my own research have failed to produce even an adequate definition of the term, much less a good reason for why it is used in that context in Maple. Various dictionaries indicate that it refers to certain subsets (perhaps all of?) the irrationals, while the Wikipedia reference link uses it interchangeably with radical. However, neither of those jive with the Maple interpretation as $\mbox{Surd}(3,x) \neq\sqrt[3]{x}\;\;\;\;\;\;\;x<0$.

So, the question is: what is a good definition for "surd"?

For bonus points, I would be fascinated to see an origin/etymology of the word as used in mathematical context.

Answer 1

Further following up the source given by Wikipedia actually answers the second part of your question as well.

The source (Earliest Known Uses of Some of the Words of Mathematics (S)) says

The Arabic translators in the ninth century translated the Greek rhetos (rational) by the Arabic muntaq (made to speak) and the Greek alogos (irrational) by the Arabic asamm (deaf, dumb).

This was translated as surdus ("deaf" or "mute") in Latin.

It has more, but the interesting fact here is that the Greek for "irrational" got literally translated into Arabic for "dumb" and then literally into Latin as surd, which again is used for irrational numbers! (This reminds me of the story of the word sine, originating in Sanskrit jiva, turning into Arabic jiba, being written as jb, being read by Latin translators as the Arabic word jaib meaning bay, and being translated into Latin sinus for bay.)

It goes on to answer the second part of your question:

According to Smith (vol. 2, page 252), there has never been a general agreement on what constitutes a surd. It is admitted that a number like sqrt 2 is a surd, but there have been prominent writers who have not included sqrt 6, since it is equal to sqrt 2 X sqrt 3. Smith also called the word surd "unnecessary and ill-defined" in his Teaching of Elementary Mathematics (1900).

G. Chrystal in Algebra, 2nd ed. (1889) says that "...a surd number is the incommensurable root of a commensurable number," and says that sqrt e is not a surd, nor is sqrt (1 + sqrt 2).

So there's no clear definition. This is clear from looking at various other sources:

Wiktionary:

(arithmetic) An irrational number, especially one expressed using the √ symbol.

Wolfram MathWorld (emphasis mine):

An archaic term for an irrational number.

There's even a Language Log post called "Ab surd" about this and other meanings of surd.

I think we'd all be better off if the word stopped being used altogether, or at least was always used with an accompanying precise definition.

Answer 2

(Note that this answers only addresses the "an origin/etymology of the word as used in mathematical context" part of the question.)

Wikipedia claims:

The term surd traces back to al-Khwārizmī (c. 825), who referred to rational and irrational numbers as audible and inaudible, respectively. This later led to the Arabic asamm (deaf, dumb) for irrational number being translated as surdus (deaf or mute) into Latin.

— and cites the page "Earliest Known Uses of Some of the Words of Mathematics (S)" as its source. The Arabic word in question would seem to be أصم. Note that a number of Latin-based languages have words like surd to mean "deaf", including Romanian, in which the word is surd.

A hat tip to lhf (comment on the question) for this source.

Answer 3

I studied math in an Indian language, so it is easy for me to join some of the dots here. The word for hypotenuse in Sanskrit (and hence in many Indian languages) is karNa, which also means ear. When the length of a hypotenuse can be calculated easily, as when the sides are, say, 4 and 3, you get 5, it is a good hypotenuse - that is good karNa; but when the sides are, say, 4 and 4, you can only approximately get the length of the hypotenuse - that is bad karNa.

Al Khwarismi seems to have transliterated these two types into Audible numbers and Inaudible numbers, seemingly mixing numbers up with the other meaning of the word karNa, an ear. Possibly, he thought of Bad Ear = Inaudible.

The onward translation to Latin and English is clear enough. Inaudible to Deaf. I hope that makes things clear.