Complex Numbers - Complex Analysis
Card 0 of 104
Evaluate: 
Evaluate:
The general formula to figure out the modulus is
.
We apply this notion to get.


The general formula to figure out the modulus is
.
We apply this notion to get.
Compare your answer with the correct one above
Evaluate: 
Evaluate:
The general formula to figure out the modulus is

We apply this to get


The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate:

Evaluate:
The general formula to figure out the modulus is

We apply this to get



The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate:

Evaluate:
The general formula to figure out the modulus is

We apply this to get



The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate:

Evaluate:
The general formula to figure out the modulus is

We apply this to get



The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate:

Evaluate:
The general formula to figure out the modulus is

We apply this to get



The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate:

Evaluate:
The general formula to figure out the modulus is

We apply this to get



The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate:

Evaluate:
The general formula to figure out the modulus is

We apply this to get



The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate:

Evaluate:
The general formula to figure out the modulus is

We apply this to get



The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate:

Evaluate:
The general formula to figure out the modulus is

We apply this to get



The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate:

Evaluate:
The general formula to figure out the modulus is

We apply this to get



The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate:

Evaluate:
The general formula to figure out the modulus is

We apply this to get



The general formula to figure out the modulus is
We apply this to get
Compare your answer with the correct one above
Evaluate 
Evaluate
Converting from rectangular to polar coordinates gives us


So

Converting from rectangular to polar coordinates gives us
So
Compare your answer with the correct one above
Compute 
Compute
Converting from Rectangular to Polar Coordinates

Evaluating for 
we get that

Converting from Rectangular to Polar Coordinates
Evaluating for
we get that
Compare your answer with the correct one above
What is the magnitude of the following complex number?

What is the magnitude of the following complex number?
The magnitude of a complex number
is defined as

So the modulus of
is
.
The magnitude of a complex number is defined as
So the modulus of is
.
Compare your answer with the correct one above
What is the magnitude of the following complex number?

What is the magnitude of the following complex number?
The magnitude of a complex number
is defined as

So the modulus of
is
.
The magnitude of a complex number is defined as
So the modulus of is
.
Compare your answer with the correct one above
What is the magnitude of the following complex number?

What is the magnitude of the following complex number?
The magnitude of a complex number
is defined as

Because the complex number
has no imaginary part, we can write it in the form
. Then the modulus of
is
.
The magnitude of a complex number is defined as
Because the complex number has no imaginary part, we can write it in the form
. Then the modulus of
is
.
Compare your answer with the correct one above
What is the argument of the following complex number?

What is the argument of the following complex number?
Note that the complex number
lies in the first quadrant of the complex plane.
For a complex number
, the argument of
is defined as the real number
such that
,
where
is in radians.
Then the argument of
is

.
The angle
lies in the third quadrant of the complex plane, but the angle
lies in the first quadrant, as does
. So
.
Note that the complex number lies in the first quadrant of the complex plane.
For a complex number , the argument of
is defined as the real number
such that
,
where is in radians.
Then the argument of is
.
The angle lies in the third quadrant of the complex plane, but the angle
lies in the first quadrant, as does
. So
.
Compare your answer with the correct one above
What is the argument of the following complex number in radians, rounded to the nearest hundredth?

What is the argument of the following complex number in radians, rounded to the nearest hundredth?
Note that the complex number
lies in the fourth quadrant of the complex plane.
For a complex number
, the argument of
is defined as the real number
such that
,
where
is in radians.
Then the argument of
is

.
The angle
lies in the second quadrant of the complex plane, but the angle
lies in the fourth quadrant, as does
. So
.
Note that the complex number lies in the fourth quadrant of the complex plane.
For a complex number , the argument of
is defined as the real number
such that
,
where is in radians.
Then the argument of is
.
The angle lies in the second quadrant of the complex plane, but the angle
lies in the fourth quadrant, as does
. So
.
Compare your answer with the correct one above
Which of the following is equivalent to this expression?

Which of the following is equivalent to this expression?
Note that
lies in the first quadrant of the complex plane.
Any nonzero complex number
can be written in the form
, where
and
.
(We stipulate that
is in radians.)
Conversely, a nonzero complex number
can be written in the form
, where
and
.
We can convert
by using the formulas above:

,
and


Since
lies in the first quadrant of the complex plane, as does
,
.
So
.
We now substitute this into our original expression and expand.
.
Finally, we convert this number back to the form
.


So our final answer is
.
Note that lies in the first quadrant of the complex plane.
Any nonzero complex number can be written in the form
, where
and
.
(We stipulate that is in radians.)
Conversely, a nonzero complex number can be written in the form
, where
and
.
We can convert by using the formulas above:
,
and
Since lies in the first quadrant of the complex plane, as does
,
.
So .
We now substitute this into our original expression and expand.
.
Finally, we convert this number back to the form .
So our final answer is .
Compare your answer with the correct one above