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Math 102 0.8 "Complex Numbers" Objectives: * Express the square root of a negative number in terms of i: * Add, subtract, multiply and divide complex numbers. Adding and Subtracting Complex Numbers We need to study the complex numbers because there are some very simple equations that do not have real solutions. For example, x2 + 1 = 0 has no real solutions. To solve such equations, we need to extend the real number system. In this section we will introduce the de…nition of complex numbers. De…nition: "Complex Numbers " A complex number is any number that can be expressed in the form p where a and b are real numbers, and i is the imaginary unit i = 1 ; : Example: The form a + bi is called the standard form of a complex number. The real number a is called the real part of the complex number, and b is called the imaginary part. (b is a real number even though it is called the imaginary part). Adding/Subtracting Complex Numbers: (a + bi) + (c + di) = (a + bi) (c + di) = Note: The set of complex numbers is closed with respect to addition; that is, the sum of two complex numbers is a complex number. Moreover, the commutative and associative properties of addition hold for all complex numbers. The additive identity element is 0 + 0i, or simply the real number 0. Example 1: (Adding and subtracting complex numbers) Add or subtract as indicated. a) ( 9 + 3i) + (4 + 5i) c) 5 8 + 12 i 7 8 + 51 i b) (5 + 3i) + (7 2i) + ( 8 i) d) (5 2i) 2i) Page: 1 7i) (6 ( 1 Notes by Bibiana Lopez College Algebra by Kaufmann and Schwitters 0.8 Multiplying and Dividing Complex Numbers Recall that i = roots (i and p 1 =) : Therefore, in the set of complex numbers, i) : This is expressed symbolically as : 1 has two square Let’s extend the de…nition so that in the set of complex numbers, every negative real number has two square roots. For any positive real number b; p Therefore, the principal square root of b is denote by b and de…ne it to be : p 2 b = For this reason we obtain that : : Example 2: (Using the de…nition of i) Write each expression in terms of i and simplify. p p a) 49 b) 8 We must be careful with the use of the symbol c) p q 4 9 p d) 6 8 b; where b > 0. Some properties that are true in the set of real numbers involving the square root symbol do not hold if the square root symbol does not represent a real number. For example, does not hold if a and b are both negative numbers. Correct Incorrect Example 3: (Using the de…nition of i) Write each in terms of i, perform the indicated operations, and simplify. p p p p 25 9 b) 7 9 a) p 64 c) p 16 Note: p 54 d) p 9 The two complex numbers and are called conjugates of each other. The product of a complex number and its conjugate is a real number. Page: 2 Notes by Bibiana Lopez College Algebra by Kaufmann and Schwitters 0.8 Example 4: (Multiplying complex numbers) Find each product and express the answer in standard form. 2 a) (4 + 5i) (2 9i) b) (4 2i) c) (5 + 3i) (5 3i) d) (7 + 3i) (8 + 4i) Example 5: (Dividing complex numbers) Find each quotient and express the answer in standard form. 3i 3 5i a) b) 6 + 2i 4i c) 1 i 2 3i d) Page: 3 3 + 9i 4 i Notes by Bibiana Lopez