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Analytic Geometry EOCT UNIT 4: EXTENDING THE NUMBER SYSTEM
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
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.