Saturday, January 30, 2010

`i, the Square Root of Minus-One'

From a math teacher I’ve learned an absurd new word – “surd,” a phonic mingling of “surly” and “curd.” The students are learning to calculate square roots, a satisfying enterprise though slightly less so when the root is a surd – an irrational root, in the mathematical sense. Irrational numbers are decimals that neither repeat nor end. The square root of four is two, a rational number, not a surd. The square root of three, 1.732050808…, is definitely a surd.

The etymology of this silly-sounding word is culturally revealing. In English it dates from the 16th century, from the Latin surdus, “unheard, silent, dull.” The mathematical sense derives from the use of surdus to translate the Arabic (jadhr) asamm – “deaf (root)” – a loan-translation of the Greek alogos, meaning “speechless, without reason” (as used by Euclid). One monosyllable recapitulates our Greek, Arabic, Roman inheritance.

Lewis Carroll (who, as Charles Lutwidge Dodgson, was a lecturer in mathematics at Oxford University) composed a poem including this quatrain, which scans and rhymes perfectly, even the equation:

“And what are all such gaieties to me,
Whose thoughts are full of indices and surds?
x2 + 7x + 53
= 11/3”

Tom Disch writes in “The Dot on the i” (in About the Size of It):

“When it comes to the sense
Of beauty we are all Pythagoreans,
Transfixed upon the ineffable and inexplicable
Significance of a number; for instance
(Or especially?), i, the square root of minus-one.”

The square root of minus-one is, of course, a surd. Go here to learn that “i” was chosen to represent this irrational number because it’s short for “imaginary” – that is, absurd. Disch probably knew Pythagoras and his followers were shocked by the discovery of irrational numbers. For them, numbers were as real as poets and mathematicians, and irrational numbers represented a flaw in nature, in the world’s texture and design. The 20th-century Hungarian mathematician Paul Erdős thought otherwise:

"Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is.”

May said...

All the different branches of mathematics - pure and applied - are equally fascinating. Numbers are just a taste.

Ed Kane said...

X2 should be read as x-squared. The verse may be Carroll's whimsical portrayal of himself, but no possible value of x solves the quadratic equation. This can be corrected by changing the sign in front of 53 from plus to minus; then the equation has two possible irrational solutions. Perhaps this is an old printing error, but it is also possible that Carroll fully intended the equation to be nonsense. Surds, infinitely repeating fractions, are largely forgotten today; this is a pity, as they frequently express irrational numbers such as pi and the square root of two with evident patterns, whereas infinitely long decimals expressing irrationals often appear random and meaningless. As a teacher of both, I have often enjoyed the fusion of mathematics and language in Carroll's work.

Ray Girvan said...

no possible value of x solves the quadratic equation

Although once imaginary numbers are on the table, complex solutions exist, in thise case:

x = -7/2 - i*sqrt(1335)/6
x = -7/2 + i*sqrt(1335)/6

The full poem, number I of Four Riddles, is interesting in itself: Caroll wrote that it was a double acrostic composed for a Miss Keyser, but it has never been fully solved.