Hawkes Learning Systems: College Algebra

1. Hawkes Learning Systems: College Algebra Section 1.6: The Complex Number System

2. Objectives • The imaginary unit and its properties. • The algebra of complex numbers. • Roots and complex numbers.

3. The Imaginary Unit and its Properties Consider the following question: For a given non-zero real number a, how many solutions does the equation have? As we have noted before, this equation has two solutions: if a is positive. However, if a is negative,it has no (real) solutions . The following definitions of complex numbers explain this situation in more detail.

4. The Imaginary Unit and its Properties The Imaginary Unit The imaginary unit is defined as . In other words, has the property that its square is : . Square Roots of Negative Numbers If a is a positive real number, . Note that by this definition, and by a logical extension of exponentiation, .

5. Example 1:The Imaginary Unit and its Properties Write the following radicals in simplified form in terms of . a. b. We write a constant, such as 3, before letters in algebraic expressions, even if the letter is not a variable. Remember: is NOT a variable: . We write the radical factor last.

6. Example 2: The Imaginary Unit and its Properties Write the following expressions in simplified form. c. d. e.

7. The Imaginary Unit and its Properties Complex Numbers For any two real numbers a and b, the sum is a complex number. The collection is called the set of complex numbers. Ex: The number is a complex number, where a = 2 and b = –3.

8. The Imaginary Unit and its Properties Complex Numbers The number a is called the real part of , and the number b is called the imaginary part. If the imaginary part of a given complex number is 0, the number is simply a real number. Ex. If the real part of a given complex number is 0, the number is a pure imaginary number. Ex.

9. The Algebra of Complex Numbers Some notes about Complex Numbers: Every real number is a complex number with 0 as the imaginary part. Furthermore, the set of real numbers is a subset of the complex numbers. Real Numbers ( ) Rational Numbers ( ) Complex Numbers ( ) Irrational Numbers Integers ( ) Whole Numbers Natural Numbers ( ) The field properties (closure, commutate, associative, distributive and the existence of an identity and inverse), discussed in Section 1.2, still apply to complex numbers.

10. The Algebra of Complex Numbers When faced with a complex number, the goal is to write it in the form . Simplifying Complex Expressions Step 1: Add, subtract, or multiply the complex numbers, as required, by treating every complex number as a polynomial expression. Remember, though, that is not actually a variable. Treating as a binomial in is just a handy device. Step 2: Complete the simplification by using the fact that .

11. Example 3: The Algebra of Complex Numbers Simplify the following complex number expression. a. b. Treating the two complex numbers as polynomials, we combine the real and then the imaginary parts and simplify.

12. Example 3: The Algebra of Complex Numbers Simplify the following complex number expressions. The product of two complex numbers leads to four products via the distributive property.

13. Example 4: The Algebra of Complex Numbers Simplify the following complex number expression. Squaring a complex number leads to four products via the distributive property.

14. The Algebra of Complex Numbers Division of Complex Numbers: In order to rewrite a quotient in the standard form , we make use of the following observation: Given any complex number , the complex number is called its complex conjugate. We simplify the quotient of two complex numbers by multiplying the numerator and denominator of the fraction by the complex conjugate of the denominator.

15. Example 4: The Algebra of Complex Numbers Simplify the quotient. a. Find the complex conjugate of the denominator and multiply that by both the numerator and the denominator. Remember that , which makes solving the product in the denominator much easier than using the distributive property. This would be an acceptable answer however, we have stated that all simplified complex numbers should be in the form .

16. Example 5: The Algebra of Complex Numbers Simplify the quotients. a. b. Find the complex conjugate of the denominator.

17. Roots and Complex Numbers We earlier said that if are real numbers, then If are not real numbers, then these properties do not necessarily hold. For instance: In order to apply either of these two properties, first simplify any square roots of negative numbers by rewriting them as pure imaginary numbers.

18. Example 6: Roots and Complex Numbers Simplify the following expressions. a. b.