1. 4.8 C OMPLEX N UMBERS Part 1: Introduction to Complex and Imaginary Numbers

2. R EAL N UMBERS See Page 12 in Textbook

3. C OMPLEX N UMBERS The set of Real Numbers is a subset of a larger set of numbers called Complex Numbers The complex numbers are based on a number whose square root is –1 The imaginary unit i is the complex number whose square root is –1.

4. S QUARE R OOT OF A N EGATIVE R EAL N UMBER For any real number a,

5. E XAMPLE : S IMPLIFY EACH NUMBER BY USING THE IMAGINARY NUMBER

8. R EAL AND I MAGINARY N UMBERS An imaginary number is any number of the form a + bi, where a and b are real numbers and b ≠ 0. If b = 0, then the number is a real number. If a = 0 and b ≠ 0, then the number is a pure imaginary number a + bi ↑↑ Real Part Imaginary Part

9. C OMPLEX N UMBERS Imaginary numbers and real numbers make up the set of complex numbers

10. POWERS OF IMAGINARY NUMBERS

11. EVALUATING POWERS Divide the exponent by 4 and determine the remainder. Equivalent power depends on the remainder Remainder of 1 Remainder of 2 Remainder of 3 Remainder of 0

12. T RY THESE

13. G RAPHING C OMPLEX N UMBERS In the complex number plane, The x – axis represents the real part The y – axis represents the imaginary part The point ( a, b ) represents the complex number a + bi The absolute value of a complex number is its distance from the origin in the complex plane.

14. E XAMPLE : W HAT ARE THE GRAPH AND ABSOLUTE VALUE OF EACH NUMBER ?

16. A DDING AND S UBTRACTING COMPLEX NUMBERS To add or subtract, combine like terms

17. A DDING AND S UBTRACTING 2 Add or subtract

18. M ULTIPLYING C OMPLEX N UMBERS

20. T HE Q UADRATIC F ORMULA Every quadratic equation has complex number solutions (that are sometimes real numbers). We can use and the quadratic formula to solve all quadratic equations.

21. F IND ALL SOLUTIONS TO EACH QUADRATIC EQUATION

23. H OMEWORK P253 #1, 2, 8 – 17 all, 39 – 44 all