Question

Z1 = 2 + 3i

Z2 = 3 + 4i
Find Z1/Z2
(a) (2 + 3i)/(3 + 4i)
(b) (5i - i)/(1 + 2i)
(c) (5i + 5)/(5 + 5i)
(d) None of these

Answer 1

To find Z1/Z2 for Z1 = 2 + 3i and Z2 = 3 + 4i, we multiply both numerator and denominator by the conjugate of the denominator to get Z1/Z2 = 18/25 + i/25.

Step-by-step explanation:

The student is asking how to divide two complex numbers. Given are two complex numbers Z1 = 2 + 3i and Z2 = 3 + 4i, and we need to find Z1/Z2. To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. The conjugate of Z2 (3 + 4i) is (3 - 4i).

Here is how to calculate Z1/Z2:

1. Multiply both the numerator and the denominator by the conjugate of the denominator: (2 + 3i)(3 - 4i) / (3 + 4i)(3 - 4i).
2. Expand the numerator: (2*3 + 2*(-4i) + 3i*3 + 3i*(-4i)) which simplifies to (6 - 8i + 9i - 12i^2).
3. Since i^2 = -1, this further simplifies to (6 + i - 12*(-1)) which gives (6 + i + 12).
4. Finally, the numerator becomes (18 + i).
5. Expand the denominator: (3*3 + 3*(-4i) + 4i*3 + 4i*(-4i)), which simplifies to (9 - 12i + 12i - 16i^2).
6. Since i^2 = -1, this further simplifies to (9 - 16*(-1)), giving (9 + 16).
7. Finally, the denominator becomes 25.
8. The result is (18 + i) / 25 or 18/25 + i/25.

So, Z1/Z2 = 18/25 + i/25.