Use the definition of addition to show that the real and
imaginary parts can be expressed as a sum/differenceof a complex number and its conjugate.
and
.
. Similarly,
Got questions? Get instant answers now!
Complex numbers can also be expressed in
an alternate form,
polar form , which we will find
quite useful. Polar form arises arises from the geometric interpretation of complex numbers.The Cartesian form of a complex number can be re-written as
By forming a right triangle having sides
and
, we see that the real and imaginary parts correspond to the cosine and sine of the triangle's base angle.
We thus obtain the
polar form for complex numbers.
The quantity
is known as the
magnitude of the complex number
, and is frequently written as
.
The quantity
is the
complex number's
angle .
In using the arc-tangent formula to find the angle, we must take into account the quadrant in which the complex number lies.
Convert
to polar form.
To convert
to polar form, we first locate the number in the complex
plane in the fourth quadrant. The distance from the originto the complex number is the magnitude
, which equals
. The angle equals
or
radians
(
degrees).
The final answer is
degrees.
Got questions? Get instant answers now!
Surprisingly, the polar form of a complex number
can be expressed mathematically as
To show this result, we use
Euler's relations that express exponentials with imaginary arguments in terms of trigonometric functions.
The first of these is easily derived from the Taylor's series
for the exponential.
Substituting
for
, we find that
because
,
, and
. Grouping separately
the real-valued terms and the imaginary-valued ones,
The real-valued terms correspond to the Taylor's series for
, the imaginary ones to
,
and Euler's first relation results. The remaining relationsare easily derived from the first.
We see that multiplying the exponential in
[link] by a real constant corresponds to setting the radius of the complex number to the constant.
Calculating with complex numbers
Adding and subtracting complex numbers expressed in Cartesian
form is quite easy: You add (subtract) the real parts andimaginary parts separately.
To multiply two complex numbers in Cartesian form is not quite
as easy, but follows directly from following the usual rules of arithmetic.
Note that we are, in a sense, multiplying two vectors to obtain another vector.
Complex arithmetic provides a unique wayof defining vector multiplication.
Division requires mathematical manipulation. We convert the
division problem into a multiplication problem by multiplyingboth the numerator and denominator by the conjugate of the
denominator.
Because the final result is so complicated, it's best to
remember
how to perform
division—multiplying numerator and denominator by thecomplex conjugate of the denominator—than trying to
remember the final result.
The properties of the exponential make calculating the product and ratio of two complex numbers much simpler when the numbers are expressed in polar form.
To multiply, the radius equals the product of the radii and
the angle the sum of the angles. To divide, the radius equalsthe ratio of the radii and the angle the difference of the
angles. When the original complex numbers are in Cartesianform, it's usually worth translating into polar form, then
performing the multiplication or division (especially in thecase of the latter). Addition and subtraction of polar forms
amounts to converting to Cartesian form, performing thearithmetic operation, and converting back to polar form.
When we solve circuit problems, the
crucial quantity, known as a transfer function, will always beexpressed as the ratio of polynomials in the variable
. What we'll need to understand the circuit's effect
is the transfer function in polar form. For instance, supposethe transfer function equals
Performing the required division is most easily accomplished
by first expressing the numerator and denominator each inpolar form, then calculating the ratio. Thus,
Got questions? Get instant answers now!
