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Warm Up Simplify 1. 3 12 2. 6 2 11 Solve 3. 2 x 2 1 17 4. 7 10 x 2 1 SAT Review question Dana walks from home to school at a rate of 5 mph. It takes her 2 hours longer to walk home from school than it did to walk to school. If her total walking time to and from school was 8 hours, what was Dana’s rate of speed walking home from school? a) 3 b) 4 c) 5 d) 8 e) 15 Section 5.4 Complex Numbers Essential Question • What is an imaginary number? Imaginary Unit • Until now, you have always been told that you can’t take the square root of a negative number. If you use imaginary units, you can! • The imaginary unit is i. • i= 1 • It is used to write the square root of a negative number. Property of the square root of negative numbers • If r is a positive real number, then r i r Examples: 3 i 3 4 i 4 2i If i - 1, then i i 5 i 1 2 i i 3 i 1 4 i 1 6 i i 7 i 1 8 etc. *For larger exponents, divide the exponent by 4, then use the remainder as your exponent instead. Example: i ? 23 23 5 with a remainder of 3 4 3 So, use i which -i i i 23 Examples 2 1. (i 3 ) i 2 ( 3)2 1( 3 * 3 ) 1(3) 3 2. Solve 3x 10 26 2 3 x 36 2 x 12 2 x 12 x i 12 x 2i 3 2 Complex Numbers • A complex number has a real part & an imaginary part. • Standard form is: a bi Real part Example: 5+4i Imaginary part Adding and Subtracting complex numbers 1. add or subtract the real parts 2. add or subtract the imaginary parts Ex: (1 2i) (3 3i) (1 3) (2i 3i ) 2 5i Ex: (2 8) (5 50) You try! (13 2i ) (5 6i) (8 18) (4 3i 2) Multiplying 1. Treat the i’s like a variable 2. Change any that are not to the first power Ex: 5 * 10 i 5 * i 10 i 2 50 1(5 2) 5 2 Ex: (2 3i )( 6 2i ) 12 4i 18i 6i 2 12 22i 6(1) 12 22i 6 6 22i You try! (6 2i )(2 3i) 8i (9 4i ) Conjugates • The conjugate of a complex number has the same real part and the opposite imaginary part • Ex. Find the conjugate of 5 + 3i 5 – 3i Ex. Find the conjugate of 3 – 2i 3 + 2i Imaginary numbers in the denominator • i’s cannot be in the denominator (like radicals) • To get rid of the i’s, multiply numerator and denominator by the conjugate • If there is only an imaginary part in the denominator, multiply by the same imaginary number Example 14 2i 14 2i * 2i 2i 28i 2 4i 28i 4 7i 3 11i Ex : 1 2i 3 11i 1 2i * 1 2i 1 2i (3 11i )( 1 2i ) (1 2i )( 1 2i ) 3 6i 11i 22i 1 2i 2i 4i 2 3 5i 22(1) 1 4(1) 3 5i 22 1 4 2 25 5i 5 25 5i 5 5 5 i You try! 5 i 5 1 i Assignment Pg. 277: #17-21(odd), 29-33(odd), 37-41(odd), 47-51(odd), 57-61(odd), 65-69(odd), 92 Assessment • Concept circles