Name______________________________
Date: _________________ Block: _______
Warm – up: Can you match the following properties with its example?
(given that a, b, and c are real numbers)
1. Commutative
A. (a + b) + c = a + (b + c)
2. Associative
B. a(b + c) = ab + ac
3. Identity
C. a + (– a) = 0
4. Inverse
D. (a + b) + c = c + (a + b)
5. Distributive
E. a + 0 = a
Notes Complex Number System and Field Properties
Complex Numbers are in the form _______________, where a is the _____________________
and b is the_____________________
Imaginary Numbers
i2
i = −1
= ____
Example One Simplifying Complex Numbers
1.
−81
2.
−8
3.
−32
4. (8i)(3i)
Example Two Adding and Subtracting Complex Numbers
(Use the commutative and associative properties to combine like terms with the real parts and the
imaginary parts)
1. (4 – i) + (3 + 2i) = ______________
2. (7 – 5i) – (1 – 8i) = ______________
You Try Simplify the Complex Numbers
1.
−24 = ______
2. (8 – 3i) + (–4 + 10i) = ______________
3. 3(1 + 2i) – (5 – i) + (8 – 4i) = ______________
Example Three Multiplying Using Complex Numbers (remember: i2 = –1)
1. 5 (-2 + )
2. (-8 + 2 )(4 – 7 )
You Try Multiplying Using Complex Numbers (remember: i2 = –1)
1. – 3 (1 – 6 )
2. (4 – 2 )(1 + 5 )
Name______________________________
Date: _________________ Block: _______
Conjugate A binomial formed by negating the second term. When you multiply a
complex number in the form a + bi by its conjugate, you get a real number.
Complex Number
4 + 3i
10 – 5i
–5 + 12i
–1 – 6i
Conjugate
How do we use conjugates? __________________________________________________.
EXAMPLE ONE Multiply by the conjugate. Verify that the answer is a real number.
(b) –5 – 12i
(a) 4 + 3i
EXAMPLE TWO Rationalize the denominator by multiplying by the conjugate.
(a)
(b)
(c)
Powers of i  As we raise i to increasing exponents, we can see a pattern develop.
Power of i
i1
I2
I3
I4
I5
I6
I7
I8
I9
i10
i11
Equivalent Value
NEVER LEAVE
AN
EXPONENT
ON i
Describe any patterns you see in the exponents
i1, i5, i9, i13, i17, i21, i25, … = i
i2, i6, i10, i14, i18, i22, i26, … = –1
i3, i7, i11, i15, i19, i23, i27, …= –i
i4, i8, i12, i16, i20, i24, i28, … = 1
EXAMPLE THREE Simplify the exponents
(a) i232
(b) i493
(c) i55
Name______________________________
Date: _________________ Block: _______
Identify the property that the statement illustrates.
1. (4 + 9) + 3 = 4 + (9 + 3)
4. 15 · 1 = 15
2. 6 · 4 = 4 · 6
5. 5 + (–5) = 0
3. 7(2 + 8) = 7(2) + 7 (8)
6. (6 · 5) · 7 = 6 · (5 · 7)
Simplify the expression and write your answer in standard form (a + bi)
7. (6 – 3i) + (5 + 4i)
10. (10 – 2i) – (–11 – 7i)
13. (–2 + 5i)( –1 + 4i)
8. (9 + 8i) + (8 – 9i)
11. 6i(3 + 2i)
14. (5 – 7i)(4 – 3i)
9. (–1 + i) – (7 – 5i)
12. –i(4 – 8i)
15. (4 – 3i)(4 + 3i)
Name:
__________________________________
Date:
_________________
Block:
______
Simplify
each
of
the
following
expression.
Write
each
answer
as
i,
–1,
–
i,
or
1.
1.
4.
2.
5.
3.
6.
7.
8.
9.
Simplify
the
expression
by
multiplying
the
complex
conjugate
10.
11.
12.
13.
14.
15.