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L7 1 1Review Complex Numbers

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    >> We're going to continue on with our
    introduction to Electrical and Computer
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    Engineering by moving now into
    the Sinusoidal Steady-state,
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    we're analyzing circuits
    that involve resistors,
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    capacitors and inductors, that
    are driven by sinusoidal sources.
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    My name's Lee Brinton, I'm
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    an electrical engineering instructor
    in Salt Lake Community College.
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    We're going to start this discussion
    with a review of complex numbers.
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    We find that electrical engineers
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    use complex numbers just
    a little bit differently
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    than are typically taught in
    college algebra and trig classes.
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    So we're going to start
    out with just going back
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    to remembering where
    complex numbers come from,
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    we'll review rectangular and polar form
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    of complex numbers, we'll introduce to you,
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    I think for many of you
    will be the first time,
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    the complex exponential
    form of a complex number,
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    and then we'll go through
    some example calculations to show how
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    these different forms make
    certain types of calculations
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    easier than doing those same
    calculations in other forms.
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    First of all, where do
    complex numbers come from?
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    But probably, the easiest or
    at least an easy way to see it is to look
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    at the graph of the function
    y equals x squared minus 1,
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    it's simply a parabola
    dropped down one unit.
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    You can say, "Well, what are
    the x-intercepts there?"
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    To find the x-intercepts,
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    you of course set the equation or
    the function y equal to 0 and solve for x.
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    So we have x squared minus 1
    equals 0, add one to both sides,
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    you get x squared equals one,
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    and take the square root of both sides,
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    and you get x equals plus or minus 1.
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    So sure enough, the way x-intercepts
    are at plus or minus one.
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    Well, now let's change
    this function to simply
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    changing this minus sign to a plus sign,
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    and we can then do the same calculations.
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    Now, obviously, there's
    no x-intercepts here.
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    Nonetheless, we can still set y equal
    to zero and try and solve for x.
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    We have x squared plus 1 equals 0.
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    Subtract one from both sides gives
    us x squared equals minus one.
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    Now take the square root of
    both sides and we go x equals,
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    we're looking for a number that when
    multiplied by itself equals negative one,
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    and in the real number system we
    all know there is no such number.
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    There's no number that
    when multiplied by itself
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    equals negative one in
    the real number system.
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    Well, that's not very satisfactory,
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    there's got to be
    a solution just because we
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    change the sign from minus to plus,
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    and all of a sudden we
    don't have solutions,
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    now there's got to be a solution.
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    So we define or we invented
    the solution to that equation.
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    We called it i, so we have x then
    is equal to plus or minus i,
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    where i then equals is defined as,
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    so let's make a third line there,
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    i is defined as the square root
    of negative one.
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    Now, in electrical engineering,
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    i is typically used to represent current,
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    and so electrical engineers say, "Well,
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    let's let j equal i or j then is equal
    to the square root of negative one."
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    Thus we have this new type of number,
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    it's called an imaginary number,
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    rather than unfortunate choice
    of term from my perspective,
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    because it's no more imaginary
    than the real numbers,
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    but you get my point,
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    they're numbers that exist just
    not in the real number system.
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    In fact, we're going to show that
    we need to introduce a new domain.
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    To do so, we already
    introduced a new domain,
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    is known as the imaginary number line,
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    and we can plot imaginary
    numbers on a number line
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    just like we can plot real numbers
    on the real number line.
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    Now, it becomes a little bit more
    interesting when we look at this equation.
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    Now what we've done is
    we've shifted the axis of
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    symmetry of the parabola off of the y-axis
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    over a distance of two units to the right.
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    Let's take this and set that equal to zero.
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    We have x minus minus 2 quantity
    squared plus 1 equals 0.
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    Let's read my one go, plus 1 equals 0.
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    Subtract one from both sides,
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    we get x minus 2 squared equals negative 1.
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    Take the square root of both sides,
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    we get x minus 2 equals plus or minus i,
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    x then equals two plus i,
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    and x equals 2 minus i.
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    Now we have x representing
    not a pure real number,
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    not a pure imaginary number,
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    but a combination of the two
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    and when you've got a real number
    plus an imaginary number,
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    we call that a complex number.
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    As we know, when solving equations that
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    involve real numbers, real coefficients,
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    that lead to imaginary solutions,
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    those solutions come as complex conjugates,
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    2 plus i and 2 minus i are said to be
    conjugates of each other because they
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    differ only by the sign on
    the imaginary part of the number.
Title:
L7 1 1Review Complex Numbers
Description:

Review of complex numbers as used in electrical engineering.

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Video Language:
English
Duration:
05:32
CDStunes edited English subtitles for L7 1 1Review Complex Numbers Mar 28, 2019, 2:46 PM

English subtitles

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