Complex numbers and arithmetic. recall that the imaginary number i is defined to be v-1 and a complex number is any quantity of the form a + bi where a, b are real numbers. we often use symbols c- a + bi, z-x + iy, and w -u + iv for complex numbers. arithmetic with complex numbers follows the same rules as real arithmetic - just remember that any time you have i times you replace it with-1. (a) for specific illustration, let c 1+2i and w 3- 4i and calculate cw. the compler conjugate of a complex number c- a is always real and nonnegative for any complex number c; in fact we define the magnitude of c to be cvc*c-va2 +b2 (the nonnegative square root) (b) calculate lcl, i 삐, ic삐 and check that ic삐 = elle). - 41 and calculate cw. 叶bi i̇s c* a bi. note that c*c = cc* a2 + b2 calculate c, w, cw and check that cw (c) using conjugates we can define complex division: w/c check that lav/d- /le- (we*)/(cc ). calculate w/c, w/d and (d) find the two complex roots r+-r-of the quadratic function q(z)一~2 +4: + 13. observe that these roots are complex conjugates of each other. check that the sum of the roots equals the coefficient of the linear term, and the product of the roots equals the constant term, just as when roots are real. these facts are true of all quadratic functions with real coefficients (and leading coefficient - 1