Download Complex number system

Name
May 13, 2016
Math 4
Complex number system page 1
Complex number system
The complex numbers are expressions of the form a + bi, where a and b are real numbers, and
i has the special properties i = − 1 and i2 = –1. In the set of complex numbers, one can use
various algebraic operations (addition, multiplication, etc.) and all of the familiar algebra
properties (distributive, commutative, etc.).
Complex numbers and the plane
One of the important things about the set of real numbers is that it can be visualized as a line,
called the real number line. Every real number has a point on the line, and every point on the line
is labeled with a number.
In a similar way, the complex numbers can be visualized as a plane, the complex plane. Here’s
the simple way that complex numbers and points are associated with each other:
The complex number a + bi corresponds to the point (a, b).
You try it
1. Write the coordinates of the point
corresponding to each of these
complex numbers. Then draw all the
points on the grid.
a. 3 + 4i
b. –2 + 6i
c. 1 – 7i
d. 5i
2. The real numbers are a part of the complex numbers, because any real number r can be
written as r + 0i.
a. What point in the complex plane corresponds to real number r ?
b. Where does the real number line lie within the complex plane?
Name
May 13, 2016
Math 4
Complex number system page 2
Review: Multiplying complex numbers
You can multiply complex numbers using the distributive property (written using a
multiplication table or whatever other distributing method is easiest for you) and remembering
the fact that i2 = –1. It always turns out that one of the four boxes of the multiplication table uses
the i2 = –1 fact; the resulting term ends up being a real number.
Example: Multiply (3 – 4i)(7 + 2i) = 21+ 6i − 28i − 8i 2 = 21− 22i + 8 = 29 − 22i
You try it
3. Do these complex number multiplications.
a. (–2 + i)(4 – 6i)
b. (3 – 4i)2
c. (a + bi)2
Review: Dividing complex numbers
To divide complex numbers, begin with a step where you multiply top-and-bottom by the
conjugate of the denominator.
Example: Divide
4 − 5i
.
2 + 3i
The conjugate of (2 + 3i) is (2 – 3i), so begin by multiplying top-and-bottom by (2 – 3i).
4 − 5i (4 − 5i)(2 − 3i) 8 − 12i − 10i + 15i 2 8 − 22i − 15 − 7 − 22i
7 22
=
=
=
=
= − − i.
2
2 + 3i (2 + 3i)(2 − 3i)
4+9
13
13 13
4 − 6i + 6i − 9i
Name
May 13, 2016
Math 4
Complex number system page 3
You try it
4. Divide complex numbers. Simplify your answers until in the form a + bi, as done in the
example above.
a.
6+i
2 + 3i
b.
2 + 3i
6+i
d.
1
a + bi
Absolute value
Here we consider the question: What should be the meaning of | a + bi |, the absolute value of a
complex number?
For guidance we can turn to the real numbers and the number line. For real number r, the
absolute value | r | is the distance between 0 and r on the number line.
What would be the equivalent idea in the complex plane? The role of 0 would be played by the
point (0, 0), and a + bi is represented by point (a, b). So the definition of absolute value comes
down to answering the question posed in problem 5 below.
You figure it out
5. In the plane, what is the distance between the points (0, 0) and (a, b)? Hint: Draw a picture
involving a right triangle, then use the Pythagorean Theorem.
Name
May 13, 2016
You should have gotten an answer of
Math 4
Complex number system page 4
a 2 + b 2 for problem 3, which leads to this definition.
Definition of absolute value: | a + bi | =
a2 + b2 .
You try it
6. Calculate these absolute values of complex numbers.
a. | 3 + 4i |
b. | 3 – 4i |
c. | 5 + 2i |
d. | –12 – 5i |
e. | 6i |
7. In general, how are | a + bi | and | a – bi | related? Explain why this is true.