1. Let : 1 +2i, w 2-i, and 4+3i. Compute (a) : + 3w;(b) -2w+ Use the quadratic formula to solve these equations; express the answers as (d) zz-z = 1; (e) z-22. 3. Sketch the locus of those points w with (a)|w 3:(b)lw-21- 1:(e)|w +22 4. Find Re(1/2) and Im(1/z) if z = x + iy, 0. Show that Re(iz)--Im z and 5. Give the polar representation for (a)-1+ i; (b) I + iV3; (c)-|; (d) (2-i) 6. Give the complex number whose polar coordinates (r, 0) are (a) (V3, x/4); 7, Let a, b, and c be real numbers with a Ï 0 and b2 4ac. Show that the two roots 8. Suppose that A is a real number and B is a complex number. Show that (g) Re[zl1 +] Im(iz)- Re z. (b) (1 y 2, Ï); (c) (4,-Ï/2); (d) (2,-24); (e) (1,4ç §(f) (v/2, 9-4). of ax + bx c0 are complex conjugates of each other. and 9. Show that |z 1 if and only if 1/z- 10. Let z and w be complex numbers with zw 0. Show that either z or w is zero. 11. Show that lz + w12-12-w12 = 4 Re(zi) for any complex numbers z, w. 12. Let z1, z2 z, be complex numbers. Establish the following formulas by mathematical induction: (b) Re(zit zit , , , + z.) = Re(z.) + Reiz) + + Re(%) 13. Determine which of the following sets of three points constitute the vertices of 14. Show that cos θ-cos Ï and sine-sin Ï if and only if θ-Ï is an integer 15. Show that the triangle with vertices at 0, z, and w is equilateral if and only if a right triangle: (a) 3+5i, 2+2i,5 +i; (b)2+ i, 35i,4 + i; (c) 6+4i,7+Si, 8 + 4i. multiple of 2 10 Chapter 1 The Complex Plane 16. Let zo be a nonzero complex number. Show that the locus of points tzo. -ao
