1. Section 2-5
Complex Numbers

2. Section 2-5 complex numbers and i
operations with complex numbers (+,-, x)
complex conjugates and division
solving quadratic equations with complex solutions
plotting complex numbers
absolute value of complex numbers

3. Complex Numbers
we learned back in Algebra 1 that the square root of a negative number is not a real number
there is a way to work with these numbers using the imaginary unit, i
we use this simple definition:
for example:

4. Complex Numbers
all numbers we work with are part of the set of complex numbers
this set consists of all real numbers and all imaginary numbers (contain i)
all complex numbers can be written in the form a + bi
a is the real part, b is the imaginary part

5. Operations With Complex Numbers
to add complex numbers, add their like parts (same for subtraction)
to multiply complex numbers, use FOIL
use the fact that

6. Division of Complex Numbers
if a + bi is a complex number, then its complex conjugate is a – bi
in order to simplify the division of two complex numbers, multiply the top and bottom of the fraction by the conjugate of the denominator
use FOIL on both top and bottom; the bottom will no longer contain i

7. Solving Quad.’s
now, when you solve a quadratic equation for which the discriminant is negative, you can find its complex solutions
the solutions will be complex conjugates

8. Plotting Complex Numbers
complex numbers cannot be plotted on a single number line because they have both a real and imaginary part
instead, we plot them on a complex plane which looks a lot like a coordinate plane we use for ordered pairs
the axes of this plane are the real axis and the imaginary axis

9. Plotting Complex Numbersa
3 – 6i

10. Absolute Value of Complex Numbers
remember that absolute value means distance from 0 on a number line
for complex numbers, it’s the distance from the origin
we use the distance formula to compute it