That's a New One on Me:
Reading a book about the history of math, I came across the word surd. Never heard of it before, despite my many years of math education. I probably won't use it, precisely because if it's obscure to a fairly math-savvy person like me, it's probably obscure to others, too. But it's good to know, if only for Boggle purposes.
"That's not a surd; it's a ___!"
I'm a newbie.
But I want the mathematical antonym!
Maybe consonants sounds are voiceless? Like "K K K K atie, b, b, b, beautiful Katie", "Shhhh" or "Phhht!" I think some languages use consonant sounds with no associated vowels.
2. Somehow I thought "surd" was something much more general than "irrational." I think it was defined in Lewin &Lewin's Introduction to Real Analysis, but I don't have it in front of me right now. Thinking back on it, I think I had the impression at the time it meant something more like "algebraic," but I'm not exactly.
3. Lewis Carroll wrote the following poem:
And what mean all these mysteries to me
Whose life is full of indices and surds?
x^2 + 7x + 53
= 11/3.
4. "Absurd" comes from "ab" + "surdus"; "surdus" means deaf, silent, or stupid -- remember the Romans weren't terribly PC about these things. That explains both the linguistic meaning of "surd" (a voiceless sound) and its mathematical meaning (irrational!). The "ab" prefix in absurd, I suspect, is not the Greek prefix of negation, but the Latin preposition "from," like "that's something so senseless it could come from a stupid deaf person (ab surdo)!"
The Nothing That Is: A Natural History of Zero -- by Robert Kaplan, Ellen Kaplan
On the surd front, the use is not uncommon, although you will not find it in new books written since the 1960s. Some math profs still use the term, but mostly in correspondence, not in classes or normal conversations (since when to math profs have normal conversations??).
I'm assuming that Sasha has some inside info. Otherwise, Boyer's 2nd ed. would not have been my choice for a guess. In fact, it would not be my choice as a good history of math source either, but it still offers a fairly good overview. Boyer's 1st edition had more character, in part because of the anecdotal parts that have been reduced and replaced in the 2nd. It's not quite as close to fiction or folklore as Bell's bios, but it does have that element. For better math and more historical accuracy, Katz is the source of choice, but it is a textbook. Still, I'll give him a plug for good work. Some of the more specialized recent publications have been quite entertaining as well.
Therefore, the best strategy is to not play Boggle or Scrabble except with people with either similar knowledge or a similar ethos about the game.
There are indeed - Eugene V. knows one.
A written or spoken word which fails to convey a meaning has failed in its primary purpose. In my more energetic moments I try, purely for the hell of it, to use unusual words but in contexts which make their meanings obvious even to listeners unfamiliar with them. The idea is to entertain rather than to befuddle. Sometimes it even works.
But "surd"? Sorry, five years at MIT and I never heard such a thing. If all it means is "irrational", I'd prefer, well, "irrational".
So is there any difference between a surd and a transcendental number?
At some point, though it may not be in the book just cited, Kline observes that mathematics and law have at least the following in common: laypersons expect each to rigorous and certain, whereas the professional practicioners realize those properties are not to be had, at least in no ultimate sense.
Any budding Bourbakis of law out there?
absurd-a. Fr. absurde, ad. L. absurd-us inharmonious, tasteless, foolish, f. ab off, here intensive + surdus deaf, inaudible, insufferable to the ear.
surd-ad. L. surdus (in active sense) deaf, (in pass. sense) silent, mute, dumb, (of sound, etc.) dull, indistinct.
The mathematical sense ‘irrational’ arises from L. surdus being used to render Gr. {alenisacu}{lambda}{omicron}{gamma}{omicron}{fsigma} (Euclid bk. x. Def.), app. through the medium of Arab. açamm deaf, as in ja{edh}r açamm surd root.
You're being too literal/analytical about the definition of "surd" as a "voiceless sound". They're not an abberation. They're a common part of every known language.
Sample surds in English:
f, h, k, p, s, t, x, ch, sh...
Voiceless vowels, in contrast, are not found in standard English, but are found very uncommonly in other languages, for example Japanese. As DRJ indicated, voiceless vowels may also be produced by those with certain types of speech disorders, either organic or pragmatic. But people with those disorders also produce other ill-formed speech sounds as well; analysis of disordered speech is a specialized study of linguistics and the term "surd" is a more general one.
But this disordered voiceless or unusually voiced speech is not properly "surds", which are specifically voiceless consonants and are part of normal speech.
The comment about "I think some languages use consonant sounds with no associated vowels" is correct (and yes, Russian, Polish, Slovak, and other languages do have many such words). But that does not a surd make. Some such consonants are surds, and some are not. In Polish, the preposition "w" is a surd only before a word which itself starts with a surd, and a voiced consonant otherwise.
It should be noted that "surd" is a pretty old and deprecated term in linguistics as well, and is only useful when you are doing research from old texts which use the term.
'ab' in this case doesn't mean literally "off" or "from" but is used as an intensifier, which means it keeps the meaning of the root word but intensifies it. Kind of like how the 'in' in "inflammable" doesn't mean "not".
So it means basically "really dumb".
Mark
The "voiced" sounds in linquistics *might* refer, if I remember correctly, to something other than silence. Some consonants involve simple explusion of air, others creation of a manner of hum, and I think the latter were known as voiced consonants. The tongue and lip positions for t and d, or p and b, are identical. But the first of each is unvoiced and the latter one voiced with a hum.
Voiced sounds simply refer to "sounds which are produced with simultaneous vibration of the vocal chords". It means this regardless of whether the sound produced is a consonant or vowel.
You are correct (in English at least) about t/p being the voiceless equivalants of d/b. You can also have voiceless vowels.
Technically in linguistics this distinction is known as "phonation". There are other types of phonation besides voiced and voiceless, actually.
Can't be a root of a polynomial over which field? I thought that √2 is a root of the polynomial (x2-2) over the real numbers. Over the rationals, though, such a polynomial doesn't have a root.
The distinctions between rationals, irrationals, and transcendentals appear simple to me, once I understood how algebraists defined such things. However, I am also puzzled about how surds fit into that relationship.
I'm affraid you are just wrong. I have seen surds refer to algebraic numbers. In fact, I have seen "surd" refer to ONLY numbers whose expression contains radicals (even if not irrational). The latter, however, makes little sense in its breadth.
So, it's easier to consult a semi-authoritative source.
PlanetMath gives the definition I suspected, but is more specific--it's all numbers that can be expressed as combinations of rational powers (of rational numbers). This, however, seems to be as odd as the definition that you gave (which is really just the transcendental numbers).
Wolfram says, "An archaic term for an irrational number." That certainly includes both transcendental and algebraic numbers.
A number of number theory books--and Wolfram's site as well--also include the variant "quadratic surd", which refers specifically to numbers that include a square root part (that would be a + b*sqrt(c), with a and b rational and c not a perfect square). There are more contemporary terms for this, but this is the one that survived longer than the term "surd" alone. In fact, some books that start out with "quadratic surd", eventually drop the adjective for brevity.
It seems that people often confuse the quadratic surds specifically with surds in general, so that the narrow definition becomes generalized in a somewhat odd way. I can trace the PlanetMath definition to flawed adoption of a legitimate source. What PlanetMath refers to is the "surd form", which is a part of British engineering slang. PlanetMath, however, makes no such disclaimers. So, we are back to the general Wolfram definition. The question remains, why would we need another term for an irrational number. Let's not forget that the use is likely old British, picked up from Latin as a calque from Arabic. The term actually predates the formal definition of "irrational". So, if you are keeping score at home, my conclusion is that "surd" is simply another term for irrational, even if it is occasionally used for other purposes.
"Surd" is basically an older term for a radical, used in practice to refer to integral roots of integers. The application was primarily to the manipulation of surd forms--that is, linear combinations of surds with rational coefficients. As we now know, these not only strictly contain the rationals, but are strictly contained in the algebraics.
To one schooled in modern mathematics, terms like algebraic and transcendental seem much more natural. However, these are concepts that require a great degree of abstraction to comprehend; our current understanding rests on a great edifice of mathematical achievement. To anyone prior to Galois and Abel, certainly, surds would have seemed natural and obvious.
My early 60's high school algebra book uses the term and defines it as the real nth root of a rational number, when the root is irrational. The surd is of order n, so the fourth root of 2 is a fourth-order surd, etc.
"Surds are number that can be built by using rational numbers and the operations +, -, *, / and [nth root] (where n can be any natural number)."
So "surd" is not simply another word for irrationals (since all rationals are clearly surds, though not all surds are rational). Some irrationals are surds, but not all (e.g., the solutions of 8x^3 - 6x - 1 = 0 are irrational but not surds). In fact, by this definition, surds form a countable set, so cannot contain all of the irrationals.
The term surd arises in the classical quadrivial liberal art of "harmonics" (sometimes identified as "music"). Harmonics is the study of rational proportions, typically expressed as ratios of string lengths on the canon (or monochord). These ratios were considered pleasing to the ear while the irrational intervals could only come "from deafness".
The late American composer Lou Harrison was an advocate for just intonation (a musical tuning practice where the primary intervals between tones are ratios of small whole numbers, as opposed to the irrational intervals of a musical temperament) and frequently used the term surd in his teachings, in his Music Primer (1974), for example.
James Joyce also appears to have been aware of this. In Finnnegans Wake, he wrote "my herrings the surdity of it all".
My other favorite 4-letter old math word is LUNE, which were certain crescent shapes studied by the greeks. One of the chapters in my personal favorite history of math book (William Dunham's Journey Thorugh Genius) is titled "The Quadrature of the Lune."
Yet what are all such gaieties to me
Whose thoughts are full of indices and surds
x^2 + 7x + 53
= 11/3
Also, it's not a poem, it's the third stanza of the first part of "Four Riddles", a set of puzzle-poems Carroll wrote.
While 'surd' may have meant 'irrational' in the past, today it seems to not have as precise a meaning as would be desired in mathematics. Perhaps that is why it is not in common use among mathematicians, but instead relegated to elementary school where 'nitpicky' details are overlooked in favor of getting across the essential idea.
Generally speaking it excludes transcendental numbers and refers to algebraic irrationals. Wikipedia gives a pretty good description, under "Radical mathematics", which sounds rather like one of those 60s innovations.
http://en.wikipedia.org/wiki/Radical_%28mathematics%29
"... home of the Surds and Cosines, fanatical Muslim warrior sects."
So, have we heard yet what book inspired this thread? My money's still on Wallace, as it's still relatively new and "hip". Kline I'd be surprised to find he hasn't been through already, or that he'd be interested at this point in that hoary tome. As for Boyer and Merzbach, possibly...
Even though I was a Lingustics major in college, I never encountered "surd" in that sense. We just used "voiceless".