Complex Numbers (𝑖)LESSON 7.8

Objective

Evaluate the square root of a negative real

number

Add or Subtract complex numbers

Multiply or divide complex numbers

Evaluate the powers of 𝑖

Imaginary Numbers

The imaginary number, denoted by 𝒊 (not a ‘j’…) is the number whose square equals −1.

𝑖2 = −1 or 𝑖 = −1Complex numbers are the numbers in the form 𝑎 + 𝑏𝑖 where the real number is ‘𝑎’ and the imaginary part is ‘𝑏𝑖’

Imaginary Numbers

Evaluate the radicals

1. −25 2. −2

Imaginary Numbers

Evaluate the radicals

3. −48

Complex Numbers

Write in standard form 𝑎 + 𝑏𝑖

4. 3 − −16 5. 5 + −12

Complex Numbers

Write in standard form 𝑎 + 𝑏𝑖

6. 15− −75

5

Add, Subtract Complex Numbers

1. Write in standard form 𝑎 + 𝑏𝑖

2. Combine like terms

real combines with real

Imaginary combines with imaginary

3. Simplify if needed

Add, Subtract Complex Numbers

Add or Subtract

7. 2 + 3𝑖 + (−6 + 7𝑖) 8. 5 + −36 − (2 − −49)

Multiply Complex Numbers

1. Write in standard form 𝑎 + 𝑏𝑖

2. Multiply using standard distribution

3. Simplify if necessary

REMINDER: 𝑖2 = −1

Multiply Complex Numbers

Multiply

9. −49 ⋅ −4 10. 2𝑖(5 − 3𝑖)

Multiply Complex Numbers

Multiply

11. (5 − 2𝑖)(−1 + 3𝑖) 12. (3 + 2𝑖)(3 − 2𝑖)

Divide Complex Numbers

1. Write in standard form 𝑎 + 𝑏𝑖

2. Multiply numerator and denominator by the

conjugate of the denominator (just like with

radicals)

3. Simplify if necessary

REMINDER: 𝑖2 = −1

Divide Complex Numbers

13. 6+5𝑖

3𝑖14.

2−𝑖

4+3𝑖

Powers of 𝑖

𝑖0 = 𝑖4 =

𝑖1 = 𝑖5 =

𝑖2 = 𝑖6 =

𝑖3 = 𝑖7 =

Powers of 𝑖

Simplify

15. 𝑖27 16. 𝑖38

17. 𝑖401 18. 𝑖4003