On 19/12/06 16:35, "jansymello" <jansy@AETERN.US> quoted from VN’s Bend Sinister:
The square root of I is I.
JM: another sweet tease to add to my growing folder “VN and Mathematics.”
The lowercase I used in expressing the complex number (x + iy) represents the square root of minus 1, to which the term ‘imaginary’ was misleadingly applied to contrast them with ‘real’ numbers — alas, these terms have persisted, greatly confusing the general public. VN’s little joke seems to play on i-the-imaginary-number and I-the-possibly-unreal-EGO, as posited by the Viennese Delegation!
The first thing that would attract a mathematician’s attention is the leading definite article THE, implying only ONE SQUARE ROOT for I. Taking a square root normally gives TWO solutions. E.g., if I is indeed a square root of I, so is –I (minus I). I and –I (minus I) are ONLY equal if I = 0 (zero). So, if we take VN’s “THE square root of I is I” ‘literally’ (or rather, mathematically!), it reduces to the POSSIBLY INTENDED proposition that “I is zero.” At least, it’s a very arcane way of ‘eliminating the authorial I.’
Supposing VN meant “A square root of I is I”: this allows the solution I = 1 (one) since 1-squared is certainly 1! This is also plausibly related to VN’s musings on undivided-monism, solipsism and ‘ego (I) as NUMBAH ONE’ (using the college sports spelling.)
If VN had in mind i-the-imaginary-square-root-of-minus-one (it does occur in general, non-technical writing, e.g., Einstein’s own popular introductions to Relativity), then the statement is false. Since i-squared is minus 1, i’s-square-roots cannot be +-or-minus-i. If I haven’t lost you, getting the FOUR square-roots-of-(i-the-square-root-of-minus-one) means solving the quadratic equation x^4 + 1 = 0;
using EULER’s MOST beautiful & mysterious formula e^(i*pi) = -1, we obtain
x = +/- 1/(2^(1/2)(cos pi/4 +/- i*sin pi/4).
Stan Kelly-Bootle