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5.4 Complex
Numbers
Algebra 2
Learning Target
• I can simplify radicals containing negative
radicands
• I can multiply pure imaginary numbers, and
• I can solve quadratic equations that have pure
imaginary solutions
• I can add, subtract, and multiply complex
numbers.
Definition of a complex number
• A complex number is any number that
can be written in the form a + bi
where a and b are real numbers and i
is the imaginary unit. a is called the
real part , and bi is called the
imaginary part.
Definition of a complex number
• A real number is also a complex
number. For example, 3 can be
expressed as 3 +0i. The imaginary
part is 0. A complex number is real
only if the imaginary part is zero.
• NOTE: the set of complex numbers
has two independent subsets—the
real and the imaginary numbers.
The diagram below shows the relationship
between the various sets of numbers we have
studied and the complex numbers.
The Complex Numbers
R = reals
Reals, R
Non-reals
Q
Z
W
N
Pure
Imaginary
Pure
Imaginary
Two complex numbers are equal if and only if their
real parts are equal and their imaginary parts are
equal. That is,
a + bi = c + di if and only if a = c and b = d
I = irrationals
Q = rationals
Z = Integers
W = wholes
N = Naturals
Ex. 1: Find values for x and y such
that 3x + 4yi = 12 + 8i
3x + 4yi = 12 + 8i
3x = 12 and 4y = 8
x=4
y=2
Check:
3x + 4yi = 12 + 8
?
3(4) + 4(2)i = 12 + 8i
12 + 8i = 12 + 8i
Ex 2: Simplify (2 + 5i) + (4 – i)
• To add or subtract complex numbers, we must combine like terms,
that is, combine the real parts and combine the imaginary parts.
(2 + 5i) + (4 – i) = (2 + 4) + (5i - i)
= 6 + 4i
Ex 2: Simplify (8 - 2i) – (6 – 4i)
• To add or subtract complex numbers, we must combine like terms,
that is, combine the real parts and combine the imaginary parts.
(8 - 2i) – (6 – 4i) = (8 - 2i) + (-6 + 4i)
= (8 – 6) + (-2i + 4i)
= 2 + 2i
The complex plane is also known as the
Gaussian plan or an Argand diagram
• The complex numbers can also be
graphed on a complex plane, where
the horizontal axis represents the real
part of the complex number and the
vertical axis represents the imaginary
part. The complex numbers are
represented by segments whose
endpoint are the origin and a point
whose coordinates are the real part
and the imaginary part of the complex
number.
The complex plane is also known as the
Gaussian plan or an Argand diagram
• Addition of complex numbers can also be
represented by graphing. First, graph the two
numbers to be added. Then complete the
parallelogram that has two sides represented by
the segments. The segments from the origin to
the fourth vertex of the parallelogram represents
the sum of the two original numbers.
Ex. 4: Graph -4 + 3i and 5 + 2i on the complex
plane. Find their sum geometrically.
• Graph each complex
number using the real
part as the x- coordinate
and the imaginary part as
the y – coordinate.
Connect each point to the
origin.
• Next complete the
parallelogram. The fourth
vertex has coordinates (1,
5) or 1 + 5i
• Check algebraically.
6
1 + 5i
4
-4 + 3i
2
5 + 2i
0, 0
-2
-4
-6
-8
5
Ex. 5: Simplify (9 – 3i)(2 + 2i).
• You can multiply complex numbers using the
FOIL method.
• (9 – 3i)(2 + 2i)
= 18 + 18i – 6i – 6i2
= 18 + 12i –(6)(-1)
= 18 + 12i + 6
= 24 + 12i
Ex. 6: Simplify (-2 + 3i)(3 – i).
• You can multiply complex numbers using the
FOIL method.
• (-2 + 3i)(3 – i)
= -6 + 2i + 9i – 3i2
= -6 + 11i – (3)(-1)
= -6 + 11i + 3
= -3 + 11i
Summary Chart
For any complex numbers a + bi and c + di
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) – (c + di) = (a – c) + (b – d)i
(a + bi)(c + di) = (ac – bd) + (ad + bc)i
The Assignment
• pp. #29 - 63 (even)