Complex number arithmetic
Since complex numbers are legitimate
mathematical entities, just like scalar numbers, they can be added,
subtracted, multiplied, divided, squared, inverted, and such, just like
any other kind of number. Some scientific calculators are programmed to
directly perform these operations on two or more complex numbers, but
these operations can also be done "by hand." This section will show you
how the basic operations are performed. It is highly recommended
that you equip yourself with a scientific calculator capable of
performing arithmetic functions easily on complex numbers. It will make
your study of AC circuit much more pleasant than if you're forced to do
all calculations the longer way.
Addition and subtraction with complex
numbers in rectangular form is easy. For addition, simply add up the
real components of the complex numbers to determine the real component
of the sum, and add up the imaginary components of the complex numbers
to determine the imaginary component of the sum:
When subtracting complex numbers in
rectangular form, simply subtract the real component of the second
complex number from the real component of the first to arrive at the
real component of the difference, and subtract the imaginary component
of the second complex number from the imaginary component of the first
to arrive the imaginary component of the difference:
For longhand multiplication and division,
polar is the favored notation to work with. When multiplying complex
numbers in polar form, simply multiply the polar magnitudes of
the complex numbers to determine the polar magnitude of the product, and
add the angles of the complex numbers to determine the angle of
the product:
Division of polar-form complex numbers is
also easy: simply divide the polar magnitude of the first complex number
by the polar magnitude of the second complex number to arrive at the
polar magnitude of the quotient, and subtract the angle of the second
complex number from the angle of the first complex number to arrive at
the angle of the quotient:
To obtain the reciprocal, or "invert"
(1/x), a complex number, simply divide the number (in polar form) into a
scalar value of 1, which is nothing more than a complex number with no
imaginary component (angle = 0):
These are the basic operations you will
need to know in order to manipulate complex numbers in the analysis of
AC circuits. Operations with complex numbers are by no means limited
just to addition, subtraction, multiplication, division, and inversion,
however. Virtually any arithmetic operation that can be done with scalar
numbers can be done with complex numbers, including powers, roots,
solving simultaneous equations with complex coefficients, and even
trigonometric functions (although this involves a whole new perspective
in trigonometry called hyperbolic functions which is well beyond
the scope of this discussion). Be sure that you're familiar with the
basic arithmetic operations of addition, subtraction, multiplication,
division, and inversion, and you'll have little trouble with AC circuit
analysis.
- REVIEW:
- To add complex numbers in rectangular
form, add the real components and add the imaginary components.
Subtraction is similar.
- To multiply complex numbers in polar
form, multiply the magnitudes and add the angles. To divide, divide
the magnitudes and subtract one angle from the other.
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