COMPLEX NUMBERS AS VECTORS OF REAL NUMBERS -- [r1, r2], i.e., a + bi -> [a,b] -- TO TOTALIZE ROOT EXTRACTION: BELOW, MULTIPLY [0,1] BY [0,1].   (W. R. HAMILTON (1805-65) CREATED THESE VECTORS & WORD "VECTOR".) CRITICAL SQUARE ROOT OF NEGATIVE ONE IS ACHIEVED BY MODIFYING MULTIPLICATION RULE FOR INTEGERS AS VECTORS OF NATURAL NUMBERS. UNLIKE INTEGERS & RATIONALS, VECTOR FORM OF COMPLEX NUMBERS CANNOT BE HIDDEN, SINCE (VIDE OYSTEIN ORE ) THEY FORM A BIMODUL. (Thanks to M. Kazmierczak.) VECTOR A=[a1,a2] VECTOR B=[b1,b2] RESULT C=[c1,c2] TOTAL OPERATIONS: A + B = [a1+b1,a2+b2]; A - B = [a1-a2,a2-b2]; A * B = [a1*b1-a2*b2,a1*b2+a2*b1] A / B = [(a11*b11+a12*b12)/(b11b11*b12*b12),(a12*b11-a11*b12)/(b11b11*b12*b12)]. Aconj: [a11,a12] -> [a11. -a12]; Bconj: [b11,b12] -> [b1,-b12]