1. 2.4 Complex Numbers What is an imaginary number What is a complex number How to add complex numbers How to subtract complex numbers How to multiply complex numbers How to rationalize the denominator How to plot complex numbers

2. Imaginary Numbers i is an imaginary number and is the solution to the quadratic equation: x 2 = - 1. Any number in the form b i is an imaginary number. Here is the i multiplication table. i 13 i 27 i 42

3. Complex Numbers A complex number is in the form a + b i where a is the real part and b is the imaginary part. Every real number is complex, & every imaginary number is complex. Examples

4. Adding Complex Numbers Adding complex numbers is really easy. Add the real to real and the imaginary part to the imaginary part. Examples

5. Subtracting Complex Numbers Subtracting complex numbers is easy. Simply subtract the real from the real and the imaginary from the imaginary. Examples

6. Multiplying Complex Numbers Multiplying complex numbers is just like multiplying binomials. Examples ( 3 – 2 i ) ( 5 + 2 i ) = 15 – 10 i + 6 i – 4 i 2 but i 2 = -1 so we get 15 – 10 i + 6 i + 4 = ? ( 2 + i ) ( - 2 + i ) = - 4 + 2 i - 2 i + i 2 = ? 2 ( -3 + 5 i ) = ? 3 i ( 4 – 2 i ) = 12 i – 6 i 2 = ? 19– 4 i - 5 - 6 + 10 i 6 + 12 i

7. Dividing Complex Numbers (rationalizing the denominator) For some reason we dont like is in the denominator. So we rewrite the fraction by multiplying by the complex conjugate. Example

8. Plotting Complex Numbers (on the Argand plane) i R -5 – 2 i 2 – 2 i 2 + 3 i -5 -2 + 3 i -3 i