5.4 Complex Numbers
As you know, not all quadratic equations have real-number solutions – some have
imaginary solutions. Because we cannot take the square root of a negative number,
mathematicians created an expanded system of numbers using the imaginar
imaginary
y unit i,
defined as i = −1 .
Ex: Solving a Quadratic Equation
x 2 + 64 = 0
b) 2x + 26 = −10
c) 3x 2 + 10 = –26
d) 2( x − 1) = −40
a)
2
e) −6( x + 5) = 120
2
2
f) −
1
( x + 1)2 = 5
2
A complex number written in standard form is a number a + bi, where a is the “real
part” and bi is the “imaginary part.” Provided b ≠ 0, this number can be called an
number.
imaginary number
Every real number corresponds to a point on the coordinate plane. Similarly, every
complex number corresponds to a point in the complex plane.
real axis: the horizontal axis in the complex plane
imaginary axis: the vertical axis in the complex plane
Plotting Complex Numbers of the form
form a + bi
1.) Start at the origin and move a units right or left
2.) From there, move b units up or down
Ex: Plot the complex numbers in the complex plane
imaginary
g) 2 – 3i
h) –3 + 2i
i) 4i
real
j) – 4 –
i
k) 5
l) 1 + 3i
Ex: Write the expression as a complex number in standard form
m) (4 – i ) + (3 + 2i )
n) (7 – 5i ) – (1 – 5i )
o) (–1 + 2i ) + (3 + 3i )
p) (2 – 3i ) – (3 – 7i )
q) 6 – (–2 + 9i ) + (– 8 + 4i )
r) 2i – (3 + i ) + (2 – 3i )