2.4 – Complex Numbers Imaginary unit = i = −1 𝑖 2 = -1 −# = i # Standard form of a complex number a + bi ex.: (3 + 2i) 0 + bi = Pure imaginary ( 0+ 3i ) or (3i) a + 0i = real number (3 + 0i) or (3) Complex # equality: if a + bi = c + di then a = c and b = d

Examples 1. Find a & b so that the equation is true. (a-1) + (b+3)i = 5 + 8i 2. Write in standard form a) 5 + −16 b) -5i + i² c) ( −50 )²

Operations with Complex Numbers First turn −# into i # if possible Properties of complex numbers same as integers. Add or Subtract Complex Numbers Add or Subtract LIKE terms Write result in STANDARD form (a + bi) Multiply Complex Numbers Mult., distribute, FOIL, extended distrib. Etc. Turn i² into -1 Simplify & write as “ a + bi”

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 Multiply and write in standard form 3. −6 • −2 4. −8 + −50 3. −6 • −2 4. −8 + −50 5. (1 + i) (3 – 2i) 6. (4 + 5i)² - (4 – 5i)²

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

Examples 7. 2+𝑖 2 − 𝑖 8. 2 1+𝑖 - 3 1 −𝑖 Find common denom., subtract, → 1 fraction, then treat like # 7.

Graphing Complex Numbers Complex Plane: x-axis = real axis Y-axis = imaginary axis Ex: Graph : 2 + 3i , 3, -5i 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 = 1 find first 5 terms EX C = I find first 5 terms