Unit 11 Math-2 (Honors) 11.1: Dividing Square root numbers Dividing Complex numbers and and Graphing in the complex plane

Review (simplify)  

Dividing complex numbers NOT allowed to have imaginary numbers in the denominator! Identity property of multiplication simplify

Dividing square root numbers NOT allowed to have Square root numbers in the denominator! Identity property of multiplication simplify

Dividing complex numbers NOT allowed to have imaginary numbers in the denominator! Identity property of multiplication simplify Standard form of a complex number

Your turn divide (really means simplify) 1. 2.

Multiply complex conjugates Complex conjugate pairs Multiply complex conjugate pairs and the “i” term disappears!!!!

Dividing Complex Numbers NOT allowed to have imaginary numbers in the denominator! Multiply by “1” (Ratio of Complex conjugate)

What about square root numbers Multiply complex conjugate pairs and the “ ” term disappears!!!!

Your Turn simplify 3.

Your Turn simplify 4.

Your Turn simplify 5.

The “Real” Plane y = x + 1 rule f(x) x x + 1 2 3 -1 -2 -3 1 -4 2 + 1 3 2 3 -1 -2 -3 1 -4 2 + 1 3 3 + 1 4 - x + x -1 + 1 -2 + 1 -1 -3 + 1 -2 1 + 1 2 -4 + 1 -3 - y 0 + 1 1

2 + 3i (2, 3i)

We call this vector: “Z” z = 2 + 3i The red arrow is called a “vector” (it goes from the origin directly to the point) We call this vector: “Z”

Your turn: 5. Graph (-3 + 4i) on the complex plane (draw the vector also) Graph (5 – 2i) on the complex plane (draw the vector also) 6. -3 + 4i + real + imaginary - imaginary - real 5 – 2i

Now watch this: Make a parallelogram What is this complex number? (-3 + 4i) + (5 – 2i) = ? -3 + 4i + real + imaginary - imaginary - real 5 – 2i

Vectors are added “head to tail” (-3 + 4i) + (5 – 2i) = ? -3 + 4i + real + imaginary - imaginary - real 5 – 2i

Add and Then Graph the following 7. (2 – 3i) + (2 + 3i) 8. (-3 + 4i) + (-3 – 4i)

Graph the two complex numbers 9. (2 –i) + real + imaginary - imaginary - real 10. (2 + i) 2 + i 2 –i

Multiply the two complex numbers 11. (2 –i) (2 + i) + real + imaginary - imaginary - real (4 + 1) = 5 2 + i 12. Graph the result. 5 + 0i 2 – i