Complex Operations

Lesson #6 of Unit 1: Quadratic Functions

and Factoring Methods (Textbook Ch1.6)

Learner Goal

Solve quadratic equations with complex solutions and perform operations with complex numbers.

Imaginary Number, i

has no real-number solutions because the square of any real number x is never a negative number.

Imaginary unit , is defined as , and it’s the solution for . In other words, . If r is a positive real number, then .

Solve each equation using the imaginary unit.

Text p.41

Complex Number

Text p.42

Place the following numbers in their correct positions in the Venn Diagram below:

5, √5, 15/3, 5.2, -2/3, 0.3333333......., 0.272727........., -3, i, 4+3i

COMPLEX NUMBER

NATURAL NUMBER

INTEGERS

RATIONAL NUMBER

REAL NUMBER

Place each of these numbers into the appropriate sets below:

List of Numbers: 3, 0, 2 + 7i , 4 + 0i , -5 + 7i , + 5i , i , 9

Complex Number

Real Number Imaginary Number

Pure Imaginary Number

COMPLEX NUMBER

REAL PART

IMAGINARY PART

+ 5i 5i

2 + 7i

4 + 0i

-5 + 7i

3

0

i

Plotting Complex Number

Text p.44

Imaginary Number

Real Number a

b a+bi

Plot following complex numbers on a complex plane. 1) 2) 3) 4)

Group Discussion 1

Simplify following using the imaginary unit, .

Identify the pattern that emerge from the above, and then use the pattern to simplify .

Suggested Problems 1

Workbook p.13: Solve the equation.

1) 2)

3) 4)

7) 8)

Workbook p.14: Simplify the expression.

32) 33)

34) 35)

40)

Adding and Subtracting Complex Numbers

To add (or subtract) two complex numbers, add (or subtract ) their real parts and their imaginary parts separately.

Sum of complex numbers: (a +bi ) + (c + di) = (a + c) + (b + d)i

Difference of complex numbers: (a + bi) – (c + di) = (a – c) + (b – d)i

1. (3 + 4i ) + (6 + 7i )

2. (14 + 16i ) – (7 + 3i )

3. (-12 – 4i ) + (-10 – 3i )

4. (-2 + 15i ) +( 2 – 15i )

5. (-24 – 6i ) – (-28 + 6i )

6. (-12 + 4i ) – (-12 + 4i )

Text p.42

Perform each operation, and then write the expression in standard form.

Multiplying Complex Numbers

To multiply two complex number, use the distributive property or the FOIL method just as you do when multiplying real numbers or algebraic expressions.

Multiplying complex numbers: (a +bi ) × (c + di) = ac + adi + bci + bd i 2

1. 4i ( -6 + i )

2. (9 – 2i ) (-4 + 7i )

3. (8 – i ) + (5 + 4 i )

4. (3 + i ) (5 – i )

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Perform each operation, and then write the expression in standard form.

Dividing Complex Numbers

When you divide two complex numbers, change it to the quotient. When the quotient has an imaginary number in the denominator, rewrite the denominator as a real number by the complex conjugates.

* Complex conjugates : a + bi and a – bi

Dividing complex numbers: (a + bi) ÷ (c + di) = x

Text p.44

Perform each operation, and then write the expression in standard form.

Group Discussion 2

Plot following complex numbers on a complex plane.

Perform each operation.

Plot the results of addition and subtraction on a complex plane. Explain how addition and subtraction can be demonstrated visually using your plotting.

Suggested Problems 2

Workbook p.13: Write the expression as a complex number in standard form.

12)

13)

14)

16)

20)

21)

Workbook p.14: Plot the numbers in a complex plan. 29) 30) 31)

22) 23)