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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