17. Complex Numbers

Question 1

Imaginary Unit

Answer 1

i = √(–1)

Question 2

• i*¹ =
• i*² =
• i*³ =
• i*⁴ =
• i*⁵ =

Answer 2

i

–1

–i

1

i

Question 3

i^<em>n</em> =

Answer 3

i^{<em>n </em>mod 4}

where n mod 4 is the remainder when n is divided by 4.

Question 4

Imaginary numbers

Answer 4

In the form bi,

where b is a real number.

Question 5

√(–n) =

where n is a postive number

Answer 5

i √(n)

Question 6

Standard form of a complex number

Answer 6

a + bi

where a and b are real

• a* = “real part”
• bi* = “imaginary part”

Question 7

To add or subtract complex numbers…

(a + bi) + (c + di) =

Answer 7

add or subtract their real and imgainary parts.

(a + c) + (b + d)i

Question 8

To multiply complex numbers…

(a + bi)(c + di) =

Answer 8

multiply like you would any two binomial expressions, using FOIL.

(ac – bd) + (ad + bc)i

Question 9

To find the quotient of two complex numbers…

Answer 9

multiply the denominator and numerator by the conjugate of the denominator. Then simplify.

Question 10

Graphical representation of a complex number

Answer 10

• x*-coordinate is the real part
• y*-coordinate is the imaginary part

Question 11

Modulus of a complex number…

Answer 11

is its distance to the origin,

which is √(a² + b²)

Question 12

Conjugate of the complex number a + bi =

Answer 12

a – bi

The graphs of conjugates are reflections about the real (x) axis.

Question 13

The product of a complex number and its conjugate…

Answer 13

is the square of the modulus.

(a + bi)(a – bi) = a² + b²