1. Complex Numbers Or I’ve got my “ i ” on you.

2. Real Numbers Imaginary Numbers Rational Numbers Irrational Numbers COMPLEX NUMBERS

3. Standard Form of a Complex Number a + b i REAL PART IMAGINARY PART

4. i 2 = -1 i 2 = -1 is the basis of everything you will ever do with complex numbers. i 2 = -1 is the basis of everything you will ever do with complex numbers. Simplest form of a complex number never allows a power of i greater than the 1 st power to be present, so ……… Simplest form of a complex number never allows a power of i greater than the 1 st power to be present, so ………

5. Simplifying Powers of i Simplification Simplest Form i None needed i i2i2 By definition- 1 i3i3 i 2 x i = -1 x i =- i i4i4 ( i 2 ) 2 = ( -1) 2 = 1 i5i5 ( i 2 ) 2 x i = ( -1) 2 x ii i6i6 ( i 2 ) 3 = ( -1) 3 = - 1 i7i7 ( i 2 ) 3 x i = ( -1) 3 x i- i i8i8 ( i 2 ) 4 = ( -1) 4 = 1

6. Simplification Examples i 42 = Divide exponent by 2 (42 ÷ 2 = 21 R 0) Quotient is exponent; Remainder is extra power of i Quotient is exponent; Remainder is extra power of i Write as power of i ( i 2 ) 21 Simplify = (-1) 21 = - 1

7. Simplification Examples i 27 = Divide exponent by 2 (27 ÷ 2 = 13 R 1) Quotient is exponent; Remainder is extra power of i Write as power of i ( i 2 ) 13  i Simplify = (-1) 13  i = - 1  i = - i

8. Adding/Subtracting Complex Numbers Adding and subtracting complex numbers is just like any adding/subtracting you have ever done with variables. Simply combine like terms. (6 + 8 i ) + (2 – 12 i ) = 8 – 4 i (7 + 4 i ) – (10 + 9 i ) = 7 + 4 i – 10 – 9 i = -3 – 5 i

9. Multiplying Complex Numbers This will be FOIL method with a slight twist at the end. An i 2 will ALWAYS show up. You will have to adjust for this. (4 + 9i)(2 + 3i) = 8 + 12 i + 18 i + 27 i 2 = 8 + 30 i – 27 = -19 + 30 i (7 – 3 i )(6 + 8 i ) = 42 + 56 i – 18 i – 24 i 2 = 42 + 38 i + 24 = 66 + 38 i

10. Binomial Squares and Complex Numbers You can still do the five-step shortcut, or you can continue to do FOIL. You will still have to adjust for the i 2 that will show up. (7 + 3 i ) 2 = 49 + 42 i + 9 i 2 = 49 + 42 i – 9 = 40 + 21 i (8 – 9 i ) 2 = 64 – 144 i + 81 i 2 = 64 – 144i – 81= -17 – 144 i

11. D2S and Complex Numbers Situations that in the real numbers would have been differences of two squares (D2S) demonstrate in the complex numbers what are known as conjugates. (3 + 4 i )(3 – 4 i ) = (3) 2 – (4 i ) 2 = 9 – 16 i 2 = 9 + 16 = 25 When conjugates are used, there will be no i in the answer.

12. Things Not Allowed in a Denominator 1. 1. Negative sign 2. 2. Radical 3. 3. Fractional Exponent 4. 4. Complex Number Each one of these must be adjusted out of the problem.

13. Clearing Complex Numbers from the Denominator If there is a pure imaginary number in the bottom, multiply by i with the opposite sign. Example: If denominator contains 3 i, multiply both sides by - i. – –Why?This also takes care of a negative in the denominator. 3 i  ( - i ) = -3 i 2 = 3 Example: If denominator contains -2 i, multiply both sides by i. -2 i  i = -2 i 2 = (-2)  (-1) = 2

14. Clearing the Denominator (continued) If the denominator is of the form a + b i, then multiply both sides of the fraction by the conjugate. If denominator contains (7 + 3 i ), multiply both sides by (7 – 3 i ). (7 + 3 i )(7 – 3 i ) = 49 – 9 i 2 = 49 + 9 = 58 If denominator contains (8 – i ), multiply both sides by (8 + i ). (8 – i )(8 + i ) = 64 – i 2 = 64 + 1 = 65

15. Simplifying Square Roots of Negative Numbers √ – 9 does not exist in the reals because there is no number that can be squared to give a negative answer. Therefore, you must use i 2 to replace the negative. √ – 9 = √ 9 i 2 = 3 i √ – 20 = √ 20 i 2 = √ 4  5  i 2 = 2 i√ 5

16. Multiplying Square Roots of Negative Numbers Any time multiplication of square roots involves the square root of a negative number, you MUST replace the negative with i 2 before doing any computation. √ 6  √ – 3 = √ 6  √ 3 i 2 = √ 18 i 2 = √ 9  2  i 2 = 3 i√ 2 √ – 2  √ – 8 = √ 2 i 2  √ 8 i 2 = √ 16 i 4 = 4 i 2 = – 4

17. Solving Equations in the Complex Numbers x 2 + 4 = 0 Remember this equation that we used to show why a sum of two squares never factors in the reals? x 2 = - 4  √x 2 = √-4 x =  √-4=  √ 4 i 2 =  2 i Complex solutions always come in conjugate pairs.