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.
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REVIEW:
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To add complex numbers in rectangular
form, add the real components and add the imaginary
components. Subtraction is similar.
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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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