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Complex Numbers OBJECTIVES •Use the imaginary unit i to write complex numbers •Add, subtract, and multiply complex numbers •Use quadratic formula to find complex solutions of quadratic equations Consider the quadratic equation x2 + 1 = 0. What is the discriminant ? a = 1 , b = 0 , c = 1 therefore the discriminant is 02 – 4 (1)(1) = – 4 If the discriminant is negative, then the quadratic equation has no real solution. (p. 114) Solving for x , gives x2 = – 1 2 x 1 x 1 We make the following definition: i 1 Note that squaring both sides yields i 2 1 Real numbers and imaginary numbers are subsets of the set of complex numbers. Real Numbers Imaginary Numbers Complex Numbers Definition of a Complex Number If a and b are real numbers, the number a + bi is a complex number written in standard form. If b = 0, the number a + bi = a is a real number. If b 0, the number a + bi is called an imaginary number. A number of the form bi, where b 0, is called a pure imaginary number. Write the complex number in standard form 1 8 1 i 8 1 i 4 2 1 2i 2 Equality of Complex Numbers Two complex numbers a + bi and c + di, are equal to each other if and only if a = c and b = d a bi c di Find real numbers a and b such that the equation ( a + 6 ) + 2bi = 6 –5i . a+6=6 2b = – 5 a=0 b = –5/2 Addition and Subtraction of Complex Numbers, If a + bi and c +di are two complex numbers written in standard form, their sum and difference are defined as follows. Sum: (a bi) (c di) (a c) (b d )i Difference: (a bi ) (c di ) (a c) (b d )i Perform the subtraction and write the answer in standard form. Ex. 1 ( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i Ex. 2 8 18 4 3i 2 8 i 9 2 4 3i 2 8 3i 2 4 3i 2 4 Properties for Complex Numbers • Associative Properties of Addition and Multiplication • Commutative Properties of Addition and Multiplication • Distributive Property of Multiplication Multiplying complex numbers is similar to multiplying polynomials and combining like terms. #28 Perform the operation and write the result in standard form. ( 6 – 2i )( 2 – 3i ) F O I L 12 – 18i – 4i + 6i2 12 – 22i + 6 ( -1 ) 6 – 22i Consider ( 3 + 2i )( 3 – 2i ) 9 – 6i + 6i – 4i2 9 – 4( -1 ) 9+4 13 This is a real number. The product of two complex numbers can be a real number. Complex Conjugates and Division Complex conjugates-a pair of complex numbers of the form a + bi and a – bi where a and b are real numbers. ( a + bi )( a – bi ) a 2 – abi + abi – b 2 i 2 a 2 – b 2( -1 ) a2+b2 The product of a complex conjugate pair is a positive real number. To find the quotient of two complex numbers multiply the numerator and denominator by the conjugate of the denominator. a bi c di a bi c di c di c di ac adi bci bdi 2 2 c d 2 ac bd bc ad i 2 2 c d Perform the operation and write the result in standard form. (Try p.131 #45-54) 6 7i 1 2i 6 7i 1 2i 1 2i 1 2i 6 14 5i 6 12i 7i 14i 2 2 1 4 1 2 2 20 5i 5 20 5i 5 5 4 i Principle Square Root of a Negative Number, If a is a positive number, the principle square root of the negative number –a is defined as a ai i a . Use the Quadratic Formula to solve the quadratic equation. 9x2 – 6x + 37 = 0 a = 9 , b = - 6 , c = 37 What is the discriminant? ( - 6 ) 2 – 4 ( 9 )( 37 ) 36 – 1332 -1296 Therefore, the equation has no real solution. 9x2 – 6x + 37 = 0 a = 9 , b = - 6 , c = 37 x 6 6 4937 29 2 6 36 1332 x 18 6 1296 6 1296 x i 18 18 18 1 36i x 3 18 1 2i 3