Imaginary Numbers Mastery

Imaginary Numbers Mastery

Read each question carefully and provide your answer in the space provided. Show all your work for full credit.

1. The imaginary unit, denoted by 'i', is defined as the square root of  .

2. A complex number is typically written in the form  , where 'a' is the real part and 'b' is the imaginary part.

3. The complex conjugate of a complex number a + bi is  .

4. Which of the following is equivalent to i^27?

5. Simplify the expression: (3 + 2i) + (1 - 4i)

6. Simplify the expression: (5 - 3i) - (2 + 6i)

7. Multiply the complex numbers: (2 + i)(4 - 3i)

8. Divide the complex numbers: (5 + 2i) / (1 - i)

9. Plot the complex number 3 - 4i on the complex plane below.

10. The impedance Z (in ohms) in a series circuit is given by Z = R + jX, where R is the resistance and X is the reactance. If a circuit has a resistance of 10 ohms and a reactance of 5 ohms, and another circuit has a resistance of 8 ohms and a reactance of -3 ohms, find the total impedance if the two circuits are connected in series. (j is used instead of i in electrical engineering).