1. 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

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

3. 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”

4. 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)

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

6. 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 “

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

8. 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