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College Algebra Lecture Notes Section 1.4 Page 1 of 4 Section 1.4: Complex Numbers Big Idea: .Calling the square root of negative one “i” allows us to state the solution of many algebra equations that would otherwise be unsolvable with real numbers. Big Skill: You should be able to perform arithmetic (add, subtract, multiply, and divide) with complex numbers. A. Identifying and Simplifying Imaginary and Complex Numbers Imaginary Numbers and the Imaginary Unit Imaginary Numbers are those of the form k , where k is a positive real number. The imaginary unit i represents the number whose square root is -1: i2 = -1 and i 1 Rewriting imaginary numbers For any positive real number k, Practice: 1. Evaluate k 1 k i k . 25 . 2. Evaluate 2 . 3. Evaluate 48 . 4. Evaluate 12 36 . Complex Numbers Complex numbers are numbers that can be written in the form a + bi, where a and b are real numbers and i 1 is the imaginary unit. The real number a is called the real part of the complex number. The real number b is called the imaginary part of the complex number. The standard form of a complex number is a + bi. Practice: 5. Write in standard form: 18 2 25 . 10 College Algebra Lecture Notes Section 1.4 Page 2 of 4 The Subset Hierarchy of Numbers: B. Add and Subtract Complex Numbers Adding, subtracting, multiplying, or dividing complex numbers results in an answer that is also a complex number. To add complex numbers, add “like terms”: a bi c di a c b d i To subtract complex numbers, subtract “like terms”: a bi c di a c b d i Practice: 6. 2 3i 6 7i 7. 5 8. 4 2i 2 7i 36 2 49 College Algebra Lecture Notes Section 1.4 Page 3 of 4 C. Multiply Complex Numbers and Find powers of i To multiply complex numbers, use the distributive property, then combine “like terms” (like using FOIL): One thing to keep in mind: the product property of radicals does not apply when the radicand is a complex number, so evaluate the square root of the negative number first, then multiply the radicals. Correct: 25 4 25i 4i Incorrect: 25 4 25 4 5i 2i 100 10i 2 10 10 Product of Complex Conjuagtes For a complex number a + bi and its conjugate a – bi, theor product (a + bi)( a – bi), is the real number a2 + b2. To compute powers of i, use an exponent that is the remainder of dividing the original exponent by 4. This works because the powers of i repeat themselves every four factors of i, as can be seen from making a list of powers of i. Practice: 100 81 9. 10. 2i 5 3i 11. 5 2i 1 3i 2 2 2 2 i i 12. 2 2 2 2 College Algebra Lecture Notes Section 1.4 Page 4 of 4 13. i 27 14. i 38 D. Divide Complex Numbers To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator, then multiply and simplify. Note: the denominator will always multiply out to a difference of squares, which will be a real number. Example: 4 3i 4 3i 1 2i 1 2i 1 2i 1 2i 4 3i 1 2i 1 2i 1 2i 4 8i 3i 6i 2 12 2i 4 11i 6 1 4i 2 2 11i 1 4 2 11i 5 5 Practice: 6 5i 15. 7i 16. 2i 4 3i 2