1. 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

2. Consider the quadratic equation x 2 + 1 = 0. What is the discriminant ? a = 1, b = 0, c = 1 therefore the discriminant is 0 2 – 4 (1)(1) = – 4 If the discriminant is negative, then the quadratic equation has no real solution. (p. 114) Solving for x, gives x 2 = – 1 We make the following definition:

3. Note that squaring both sides yields Real Numbers Imaginary Numbers Real numbers and imaginary numbers are subsets of the set of complex numbers. Complex Numbers

4. 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, the number a + bi is called an imaginary number. A number of the form bi, where, is called a pure imaginary number. Write the complex number in standard form

5. 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 Find real numbers a and b such that the equation ( a + 6 ) + 2bi = 6 –5i. a + 6 = 62b = – 5 a = 0 b = –5/2

6. 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: Difference:

7. Perform the subtraction and write the answer in standard form. Ex. 1 ( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i Ex. 2 4

8. 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 ) FOILFOIL 12 – 18i – 4i + 6i 2 12 – 22i + 6 ( -1 ) 6 – 22i

9. Consider ( 3 + 2i )( 3 – 2i ) 9 – 6i + 6i – 4i 2 9 – 4( -1 ) 9 + 4 13 This is a real number. The product of two complex numbers can be a real number.

10. 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 ) a 2 + b 2 The product of a complex conjugate pair is a positive real number.

11. To find the quotient of two complex numbers multiply the numerator and denominator by the conjugate of the denominator.

12. Perform the operation and write the result in standard form.(Try p.131 #45-54)

13. 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

14. Use the Quadratic Formula to solve the quadratic equation. 9x 2 – 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.

15. 9x 2 – 6x + 37 = 0 a = 9, b = - 6, c = 37