2 4 Complex Numbers Examples Operations with Complex

2. 4 – Complex Numbers •

Examples •

Operations with Complex Numbers •

Simplifying RADICALS • Write number under radical as product of prime factors and bring PAIRS through the “magic door” • MULTIPLY RADICALS: – Multiply numbers out front to get new coefficient – Multiply numbers underneath to get new # – Write result as prime factors & bring pairs through door. • ADD/SUBTRACT RADICALS – Add/subtract numbers out front ONLY IF #’s under are SAME – Simplify if necessary to make #’s under the same (prime factors and pairs through the door)

Examples •

Dividing Complex Numbers • Quotient of Complex Numbers • NO IMAGINARIES in DENOMINATOR • Steps to dividing 1. Find complex conjugate of denominator Same terms, opposite signs separating them a + bi and a - bi 2. Make a fraction with this conjugate as numerator and denominator (= 1) 3. Multiply original by this fraction (mult. across: distribute or FOIL as necessary) 4. Simplify (i² = -1, combine like terms) 5. Write final answer as “ a + bi “

Examples •

Graphing Complex Numbers • Complex Plane: – x-axis = real axis – Y-axis = imaginary axis – Ex: Graph : 2 + 3 i , 3, -5 i • Fractal Geometry = application of complex #’s – Mandelbrot set = most famous fractal • Pattern: number, pervious² + previous, previous² + previous, … • A number is in this set if BOUNDED ( alternate of get closer & closer to a number: smaller) • Not if the set if get infinitely large or small: Unbounded • EX: c = -1 find first 5 terms • EX C = I find first 5 terms