Q5. Find real x and y if (3x − 7) + 2iy = −5y + (5 + x)i.
Real parts: 3x − 7 = −5y. Imaginary parts: 2y = 5 + x, so x = 2y − 5. Then 3(2y − 5) − 7 = −5y, so 11y = 22: y = 2 and x = −1.
Q6. Write z = (2 − 3i) + (−4 + 5i) as a + ib, and give Re z and Im z.
Add the real parts and the imaginary parts separately: z = −2 + 2i, so Re z = −2 and Im z = 2.
Q7. Is 0 a real number, a purely imaginary number, or both?
0 = 0 + i0 has imaginary part 0, so it is real, and real part 0, so it is purely imaginary: it is the only complex number that is both. Books that require b ≠ 0 for purely imaginary count 0 as real only.
Q8. If x + iy = 4 − 3i with x, y real, find x2 + y2.
x = 4 and y = −3, so x2 + y2 = 16 + 9 = 25.