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Complex Numbers
Lesson 5-6
Algebra 2
Additional Examples
Simplify
–54 =
=
–1 • 54
–1 •
=i•
54
=i•3
= 3i
6
6
54
–54 by using the imaginary number i.
Complex Numbers
Lesson 5-6
Algebra 2
Additional Examples
Write
–121 – 7 in a + bi form.
–121 – 7 = 11i – 7
= –7 + 11i
Simplify the radical expression.
Write in the form a + bi.
Complex Numbers
Lesson 5-6
Algebra 2
Additional Examples
Find each absolute value.
a.
|–7i|
–7i is seven units from the origin on the imaginary axis.
So |–7i| = 7
b.
|10 + 24i|
|10 + 24i| =
=
102 + 242
100 + 576 = 26
Complex Numbers
Lesson 5-6
Algebra 2
Additional Examples
Find the additive inverse of –7 – 9i.
–7 – 9i
–(–7 – 9i)
Find the opposite.
7 + 9i
Simplify.
Complex Numbers
Lesson 5-6
Algebra 2
Additional Examples
Simplify the expression (3 + 6i) – (4 – 8i).
(3 + 6i) – (4 – 8i) = 3 + (–4) + 6i + 8i
= –1 + 14i
Use commutative and
associative properties.
Simplify.
Complex Numbers
Lesson 5-6
Algebra 2
Additional Examples
Find each product.
a. (3i)(8i)
(3i)(8i) = 24i 2
Multiply the real numbers.
= 24(–1)
Substitute –1 for i 2.
= –24
Multiply.
b. (3 – 7i )(2 – 4i )
(3 – 7i )(2 – 4i ) = 6 – 14i – 12i + 28i 2
Multiply the binomials.
= 6 – 26i + 28(–1)
Substitute –1 for i 2.
= –22 – 26i
Simplify.
Complex Numbers
Lesson 5-6
Algebra 2
Additional Examples
Solve 9x2 + 54 = 0.
9x2 + 54 = 0
9x2 = –54
Isolate x2.
x2 = –6
x = ±i
6
Find the square root of each side.
Check: 9x2 + 54 = 0
9(i
6)2 + 54
0
9(6)i 2 + 54
0
54(–1)
–54
54 = 54
9x2 + 54 = 0
9(i(–
6))2 + 54
0
9(6i 2) + 54
0
54(–1)
–54
–54 = –54
Complex Numbers
Lesson 5-6
Additional Examples
Algebra 2
Find the first three output values for f(z) = z2 – 4i. Use z = 0 as
the first input value.
Use z = 0 as the first input value.
f(0) = 02 – 4i
= –4i
f(–4i )= (–4i )2 – 4i
First output becomes second input.
Evaluate for z = –4i.
= –16 – 4i
f(–16 – 4i )= (–16 – 4i )2 – 4i
Second output becomes third input.
Evaluate for z = –16 – 4i.
= [(–16)2 + (–16)(–4i ) + (–16)(–4i) + (–4i )2] – 4i
= (256 + 128i – 16) – 4i
= 240 + 124i
The first three output values are –4i, –16 – 4i, 240 + 124i.