Download Skill 1-Complex Numbers

 Skill 1 Completed Notes Complex Numbers Skill 1a: Definition of Imaginary Unit and Complex Numbers Skill 1b: Operations on Complex Numbers Skill 1c: Plot Complex Numbers Skill 1d: Find the Absolute Value of Complex Numbers Skill 1a: Definition of imaginary unit and Complex Numbers Some quadratic equations have no real solutions, such as x2 = -­‐1. To overcome this deficiency, the imaginary unit, 𝒊, is introduced. 𝑖 = −1 Powers of imaginary units: Since 𝑖 = −1 , the following powers can be determined: 𝑖 ! = 1 𝑖 ! = 𝑖 ! ∙ 𝑖 ! = −𝑖 ∙ 𝑖 = −𝑖 ! = − −1 = 1 𝑖 ! = 𝑖 ! ∙ 𝑖 ! = −𝑖 ∙ 𝑖 = −𝑖 ! = − −1 = 1 𝑖 ! = 𝑖 𝑖 ! = 𝑖 ! ∙ 𝑖 ! = 1 ∙ 𝑖 = 𝑖 𝑖 ! = 𝑖 ! ∙ 𝑖 ! = 1 ∙ 𝑖 = 𝑖 𝑖 ! = −1 𝑖 ! = 𝑖 ! ∙ 𝑖 ! = 𝑖 ∙ 𝑖 = 𝑖 ! = −1 𝑖 !" = 𝑖 ! ∙ 𝑖 ! = 𝑖 ∙ 𝑖 = 𝑖 ! = −1 𝑖 ! = −𝑖 𝑖 ! = 𝑖 ! ∙ 𝑖 ! = −1 ∙ 𝑖 = −𝑖 𝑖 !! = 𝑖 !" ∙ 𝑖 ! = −1 ∙ 𝑖 = −𝑖 Notice the pattern that develops. When dividing the exponent of 𝑖 by 4, the remainder can be used to determine the value: Remainder of 0, value is 1 Remainder of 1, value is 𝑖 Remainder of 2, value is −1 Remainder of 3, value is −𝑖 Determine the value of each imaginary power: 1. 𝑖 !" = −𝑖 !"
!
has a remainder of 3 4. 𝑖 !"# = 1 !"#
!
2. 𝑖 !! = −1 !!
5. 𝑖 !" = 𝑖 has a remainder of 0 !
!"
!
has a remainder of 2 has a remainder of 1 3. 𝑖 !" = 𝑖 !"
6. 𝑖 !"# = −1 !"#
!
has a remainder of 1 !
has a remainder of 2 If 𝑎 and 𝑏 are real numbers, the number 𝑎 + 𝑏𝑖 is a complex number written in standard form. If 𝑏 = 0, the result is a real number. If 𝑎 = 0, the result is an imaginary number. Simplify the following radicals using the imaginary unit 7. −25 8. −63 = 𝑖 25 = 5𝑖 = 𝑖 9 ∙ 7 = 3𝑖 7 9. −125 = 𝑖 25 ∙ 5 = 5𝑖 5 Skill 1b: Operations with Complex Numbers Adding and Subtracting Complex Numbers: 1. (6 + 4i) + (-­‐17 + 3i) 2. 5 – (12 + i) + 8i = 6 − 17 + 4𝑖 + 3𝑖 = 5 − 12 − 1𝑖 + 8𝑖 = −7 + 7𝑖 3. (3 – 5i) – 9i + (12 + 4i) – 10 4. 6(8 -­‐ 3i) = 3 + 12 − 10 − 5𝑖 − 9𝑖 + 4𝑖 = 48 − 18𝑖 = −11 + 3𝑖 = 5 − 10𝑖 5. 7i( -­‐12 + 8i) 6. (5 – i)(6 + 2i) = −84𝑖 + 56𝑖 ! = 30 + 10𝑖 − 6𝑖 − 2𝑖 ! = −84𝑖 + 56 −1 = 30 + 4𝑖 − 2 −1 = 32 + 4𝑖 7. (9 + 5i)(9 – 5i) 8. State the complex conjugate a) 6 – 4i b) 3 + i = −56 − 84𝑖 = 9! + 5! = 106 6 + 4𝑖 3 − 𝑖 !!!!
(9 – 5i) is the complex conjugate of (9 + 5i)_ 9. (-­‐4 + 11i)2 10. = −4 + 11𝑖 −4 + 11𝑖 =
= 16 − 44𝑖 − 44𝑖 + 121𝑖 ! =
= 16 − 88𝑖 + 121 −1 =
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= −105 − 88𝑖 =
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= 1 − 𝑖 !!!
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𝑖 Skill 1c: Plotting Complex Numbers Complex Numbers can be plotted on a coordinate system called a complex plane. The vertical axis is called the imaginary axis and the horizontal axis is called the real axis. A complex number, 𝑎 + 𝑏𝑖, is graphing the ordered pair (𝑎, 𝑏). Plot each complex number in the complex plane: 1. -­‐5 + 3i 2. 4i 3. 2 + i Write the complex number that is represented by the point on the complex plane: 5. 6. −5 − 2𝑖 4. -­‐2 −2𝑖 Skill 1d: Find the Absolute Value of a Complex Number The absolute value of a complex number is equal to the length of the line drawn from the origin to the point plotted on the complex plane. Using the Pythagorean Theorem, 𝑎 + 𝑏𝑖 = 𝑎 ! + 𝑏 ! 1. 6 − 3𝑖 = 6! + −3 ! = 45 = 9 ∙ 5 = 3 5 2. −7𝑖 = 0! + −7 ! = 49 = 7 3. −12 = −12
= 144 = 12 !
+ 0! 4. !!!
!!!!
First, 3 − 𝑖 1 − 3𝑖
1 + 3𝑖 1 − 3𝑖
3 − 9𝑖 − 𝑖 + 3𝑖 !
1! + 3!
3 − 10𝑖 + 3 −1
=
10
−10
=
𝑖 10
= −𝑖 Now, 3−𝑖
1 + 3𝑖
= −1𝑖 = 0! + −1 ! = 1