Download 4-4 Complex Numbers 10-20

Five-Minute Check (over Lesson 4–3)
CCSS
Then/Now
New Vocabulary
Example 1: Square Roots of Negative Numbers
Example 2: Products of Pure Imaginary Numbers
Example 3: Equation with Pure Imaginary Solutions
Key Concept: Complex Numbers
Example 4: Equate Complex Numbers
Example 5: Add and Subtract Complex Numbers
Example 6: Real-World Example: Multiply Complex Numbers
Example 7: Divide Complex Numbers
Over Lesson 4–3
Solve x 2 – x = 2 by factoring.
Solve c 2 – 16c + 64 = 0 by factoring.
Solve z 2 = 16z by factoring.
Solve 2x 2 + 5x + 3 = 0 by factoring.
Write a quadratic equation with the roots –1 and 6
in the form ax 2 + bx + c, where a, b, and c are
integers.
In a rectangle, the length is three inches greater
than the width. The area of the rectangle is
108 square inches. Find the width of the rectangle.
Over Lesson 4–3
Solve x 2 – x = 2 by factoring.
A. 2, –1
B. 1, 2
C. 1, 1
D. –1, 1
Over Lesson 4–3
Solve c 2 – 16c + 64 = 0 by factoring.
A. 2
B. 4
C. 8
D. 12
Over Lesson 4–3
Solve z 2 = 16z by factoring.
A. 1, 4
B. 0, 16
C. –1, 4
D. –16
Over Lesson 4–3
Solve 2x 2 + 5x + 3 = 0 by factoring.
A.
B. 0
C. –1
D.
–1
Over Lesson 4–3
Write a quadratic equation with the roots –1 and 6
in the form ax 2 + bx + c, where a, b, and c are
integers.
A. x2 – x + 6 = 0
B. x2 + x + 6 = 0
C. x2 – 5x – 6 = 0
D. x2 – 6x + 1 = 0
Over Lesson 4–3
In a rectangle, the length is three inches greater
than the width. The area of the rectangle is
108 square inches. Find the width of the rectangle.
A. 6 in.
B. 8 in.
C. 9 in.
D. 12 in.
Content Standards
N.CN.1 Know there is a complex number i
such that i2 = –1, and every complex number
has the form a + bi with a and b real.
N.CN.2 Use the relation i2 = –1 and the
commutative, associative, and distributive
properties to add, subtract, and multiply
complex numbers.
Mathematical Practices
6 Attend to precision.
You simplified square roots.
• Perform operations with pure imaginary
numbers.
• Perform operations with complex numbers.
• Imaginary unit
 𝑖 2 = −1
• imaginary number pattern
 𝑖 = −1
2
2
 𝑖 = −1 = −1
 𝑖 3 = 𝑖 2 • 𝑖 = −1 • 𝑖 = −𝑖
 𝑖 4 = 𝑖 2 • 𝑖 2 = (−1) • (−1) = 1
• complex number
 a + bi
• complex conjugates
 a + bi and a - bi
Square Roots of Negative Numbers
A.
Answer:
Square Roots of Negative Numbers
B.
Answer:
A.
A.
B.
C.
D.
B.
A.
B.
C.
D.
Products of Pure Imaginary Numbers
A. Simplify –3i ● 2i.
–3i ● 2i = –6i 2
= –6(–1)
=6
Answer: 6
Products of Pure Imaginary Numbers
B.
Answer:
A. Simplify 3i ● 5i.
A. 15
B. –15
C. 15i
D. –8
B. Simplify
A.
B.
C.
D.
.
Equation with Pure Imaginary Solutions
Solve 5y 2 + 20 = 0.
5y 2 + 20 = 0
5y 2 = –20
y 2 = –4
Answer: y = ±2i
Solve 2x 2 + 50 = 0.
A. ±5i
B. ±25i
C. ±5
D. ±25
Equate Complex Numbers
Find the values of x and y that make the equation
2x + yi = –14 – 3i true.
a + bi = c + di iff a = c and b = d
Real Part
2x = –14
x = –7
Answer: x = –7, y = –3
Imaginary Part
y = -3
Find the values of x and y that make the equation
3x – yi = 15 + 2i true.
A. x = 15
y=2
B. x = 5
y=2
C. x = 15
y = –2
D. x = 5
y = –2
Add and Subtract Complex Numbers
A. Simplify (3 + 5i) + (2 – 4i).
(3 + 5i) + (2 – 4i) = (3 + 2) + (5 – 4)i
=5+i
Answer: 5 + i
Add and Subtract Complex Numbers
B. Simplify (4 – 6i) – (3 – 7i).
(4 – 6i) – (3 – 7i) = (4 – 3) + (–6 + 7)i
=1+i
Answer: 1 + i
A. Simplify (2 + 6i) + (3 + 4i).
A. –1 + 2i
B. 8 + 7i
C. 6 + 12i
D. 5 + 10i
B. Simplify (3 + 2i) – (–2 + 5i).
A. 1 + 7i
B. 5 – 3i
C. 5 + 8i
D. 1 – 3i
Multiply Complex Numbers
ELECTRICITY In an AC circuit, the voltage V,
current I, and impedance Z are related by the
formula V = I ● Z. Find the voltage in a circuit with
current 1 + 4j amps and impedance 3 – 6j ohms.
V=I●Z
= (1 + 4j)(3 – 6j)
= 1(3) + 1(–6j) + 4j(3) + 4j(–6j)
= 3 – 6j + 12j – 24j 2
= 3 + 6j – 24(–1)
= 27 + 6j
Answer: The voltage is 27 + 6j volts.
ELECTRICITY In an AC circuit, the voltage V,
current I, and impedance Z are related by the
formula V = I ● Z. Find the voltage in a circuit with
current 1 – 3j amps and impedance 3 + 2j ohms.
A. 4 – j
B. 9 – 7j
C. –2 – 5j
D. 9 – j
Divide Complex Numbers
A.
Answer:
Divide Complex Numbers
B.
Answer:
A.
A.
B. 3 + 3i
C. 1 + i
D.
B.
A.
B.
C.
D.