perform arithmetic operations with complex numbers …€¦ · perform arithmetic operations with...
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Analytic Geometry EOCT UNIT 4: EXTENDING THE NUMBER SYSTEM
115 Copyright © 2013 by the Georgia Department of Education • All Rights Reserved
PERFORM ARITHMETIC OPERATIONS WITH COMPLEX NUMBERS
KEY IDEAS
An imaginary number is a number whose square is less than zero. An imaginary number 1.can be written as a real number multiplied by the imaginary unit, i, where
21 and 1i i � � . Examples:
25 25 1 5i<� �
� �48 48 1 4 3 1 4 3i� � � <
The powers of i form a repeating pattern as shown. 2.0
1
2
3 2
4 2 2
5 4
6 4 2
1
111 1 1
11 1 1
ii iii i i i ii i ii i i i ii i i
� � � � � � �
<<<<
<<
<<
# #
A complex number is the sum of a real number and an imaginary number, in the form 3.a + bi, where a and b are real numbers and i is the imaginary unit.
To add (or subtract) complex numbers, add (or subtract) the real parts and add (or 4.subtract) the imaginary parts.
� � � � � � � �a bi c di a c b d i� � � � � �
This is similar to combining like terms when adding or subtracting polynomials. Example:
� � � � � � � � � � � �2 3 4 5 2 4 3 5 2 4 3 5 6 8i i i i i i� � � � � � � � � �
Analytic Geometry EOCT UNIT 4: EXTENDING THE NUMBER SYSTEM
116 Copyright © 2013 by the Georgia Department of Education • All Rights Reserved
To multiply complex numbers, use the Distributive Property. Multiply each term of the 5.first complex number by each term in the second complex number.
� �� �
� � � �
2
( 1) ( ) ( )
a bi c di ac adi bci bdiac adi bci bdac adi bci bdac bd adi bciac bd ad bc i
� � � � �
� � � � � � �
� � �
� � �
REVIEW EXAMPLES 1) Subtract: (5 + 7i) – (8 – 4i). Identify the real part and imaginary part of the difference.
Solution:
First rewrite the expression:
(5 7 ) (8 4 ) 5 7 8 4i i i i� � � � � � Distributive Property. 5 8 7 4i i � � � Commutative Property.
3 11i � �
For complex number a + bi, the real part is a and the imaginary part is b. For –3 + 11i, the real part is –3 and the imaginary part is 11.
2) Rewrite the expression i2(3i – 7) in the form a + bi, and justify each step.
Solution: Use properties and math operations to rewrite the expression.
i2(3i – 7) = –1(3i – 7) Substitute –1 for i2. = –3i + 7 Distributive Property. = 7 – 3i Commutative Property of Addition.
Analytic Geometry EOCT UNIT 4: EXTENDING THE NUMBER SYSTEM
117 Copyright © 2013 by the Georgia Department of Education • All Rights Reserved
3) Rewrite the expression (11i4 + 2i3) – (2i4 – 6i3) in the form a + bi, and justify each step.
Solution:
Use properties and operations to rewrite the expression. (11i4 + 2i3) – (2i4 – 6i3) = 11i4 + 2i3 + (–1)(2i4) + (–1)(–6i3) Distributive Property.
= 11i4 + 2i3 – 2i4 + 6i3 Multiply.
= 11i4 – 2i3 + 2i4 + 6i3 Commutative Property of Addition.
= 9i4 + 8i3 Combine like terms.
= 9(1) + 8(–i) Substitute 1 for i4 and –i for i3.
= 9 – 8i Rewrite in a + bi form.
4) Multiply (6 + 4i)(8 – 3i).
Solution: Use the Distributive Property to find the product.
� �� � � � � �6 4 8 3 6 8 6 3 4 8 4 3< < < <i i i i i i� � � � � � � Distributive Property. 2(6 4 )(8 3 ) 6 8 6 3 4 8 4 3i i i i i� � � � �< < < < Commutative Property.
= 48 – 18i + 32i – 12(–1) Multiply.
= 48 – 18i + 32i + 12 Multiply.
= 48 +12 –18i + 32i Commutative Property of Addition.
= (48 +12) + (–18 +32)i Associative Property of Addition and Distributive Property.
= 60 + 14i Add.
Analytic Geometry EOCT UNIT 4: EXTENDING THE NUMBER SYSTEM
118 Copyright © 2013 by the Georgia Department of Education • All Rights Reserved
EOCT Practice Items 1) Which has the same value as 5 3 ?i i� �
A. –2i B. –2 C. 2 D. 2i
[Key: A]
2) Let 4r i � and 1 .s i � What is the value of 2 ?r s�
A. 14 i� B. 15 i� C. 14 7i� D. 14 9i�
[Key: D]
3) Which has the same value as (5 – 3i)(–4 + 2i)?
A. –26 – 2i B. –26 + 22i C. –14 – 2i D. –14 + 22i
[Key: D]