1. Solve the quadratic equation x 2 + 1 = 0. Solving for x, gives x 2 = – 1 We make the following definition: Bell Work #1

2. Complex Numbers Note that squaring both sides yields: therefore and so and And so on…

3. Real Numbers Imaginary Numbers Real numbers and imaginary numbers are subsets of the set of complex numbers. Complex Numbers

4. Definition of a Complex Number If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form. If b = 0, the number a + bi = a is a real number. If a = 0, the number a + bi is called an imaginary number. Write the complex number in standard form

5. Addition and Subtraction of Complex Numbers If a + bi and c +di are two complex numbers written in standard form Sum: Difference:

6. Perform the subtraction and write the answer in standard form. ( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i 4

7. Multiplying Complex Numbers Multiplying Complex Numbers Multiplying complex numbers is similar to multiplying polynomials and combining like terms. Perform the operation and write the result in standard form.( 6 – 2i )( 2 – 3i ) FOILFOIL 12 – 18i – 4i + 6i 2 12 – 22i + 6 ( -1 ) 6 – 22i

8. Consider ( 3 + 2i )( 3 – 2i ) 9 – 6i + 6i – 4i 2 9 – 4( -1 ) 9 + 4 13 This is a real number. The product of two complex numbers can be a real number.