Complex Numbers

Question

Q1)

 

Q2)

 

Q3)

Answer 1

(1).(a).

The complex numers are w = 3 + 4i and z = - 2 + 4i.

(i).

w - z = (3 + 4i) - (- 2 + 4i)

         = 3 + 4i + 2 - 4i

         = 5 + 0i.

(ii).

w * z = (3 + 4i) * (- 2 + 4i)

         = (3)(- 2 + 4i) + (4i)(- 2 + 4i)

         = 3(- 2) + 3(4i) + 4i(- 2) + 4i(4i)

         = - 6 + 12i - 8i + 16i².

         = - 6 + 4i + 16( - 1)                    [ Since i² = - 1]

         = - 22 + 4i.

 

(b).

2z = 2(- 2 + 4i)

     = 2(- 2) + 2(4i)

     = - 4 + 8i.

w + 2z = (3 + 4i) + (- 4 + 8i)

            = (3 - 4) + (4i + 8i)

            = - 1 + 12i

= - 1 - 12i.

The imaginary part of is - 12.

 

(c).

w = 3 + 4i.

= (3 - 4i)

z = (3 - 4i) * (- 2 + 4i)

        = (3)(- 2 + 4i) - (4i)(- 2 + 4i)

        = (3)(- 2) + (3)(4i) - (4i)(- 2) - (4i)(4i)

        = - 6 + 12i + 8i - 16i²

        = - 6 + 16i - 16(-1)                    [ Since i² = - 1]

        = 10 + 16i.

The real part of z is 10.

Answer 2

(2).

The complex numers are z = 5 + 3i and w = - 4 - 3i.

The absolute value of the complex number z = a + bi is | z | = | a + bi | = √(a² + b²).

(a).

| z | = | 5 + 3i |

       = √[(5)² + (3²)

       = √(25 + 9)

         = √34.

(b).

| w | = | - 4 - 3i |

       = √[(-4)² + (-3)²]

       = √(16 + 9)

       = √(25)

         = 5.

(c).

z + w = (5 + 3i) + ( - 4 - 3i)

         = (5 - 4) + (3i - 3i)

         = 1 + 0i

| z + w | = | 1 + 0i |

       = √[(1)² + (0)²]

         = 1.

Answer 3

(3).

(a). 6 - z = 2z + (15 - 18i)

Add 2z to each side.

6 - z - 2z = 2z + (15 - 18i) - 2z

6 - 3z = 15 - 18i

Subtract 6 from each side.

6 - 3z - 6 = 15 - 18i - 6

- 3z = 9 - 18i

Multiply each side by negative 3.

(-3z)/(-3) = 9/(-3) + (-18i)/(-3)

z = - 3 + 6i.

The complex number is z = - 3 + 6i.