Complex Numbers

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Complex Numbers

• GE 111

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COMPLEX NUMBERS

• Because x2 must be greater than 0 for every real
  number, x, the equation
• x2 -1
• Has no real solutions. To deal with this problem,
  Mathematicians of the eighteenth century
  introduced the imaginary number
• Which they assumed had the property
• But otherwise could be treated as a real number.

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Expressions of the form where a and b are real
numbers, were called complex numbers, and these
were manipulated according to the standard rules
of arithmetic with the added property that By
the beginning of the nineteenth century it was
recognized that a complex number, Could be
regarded as an alternate symbol for the ordered
pair (a,b), of real numbers and that operations
of addition, subtraction, multiplication, and
division could be defined on these ordered pairs
so that the familiar laws of arithmetic hold and
i2 -1. Thus A complex number is an ordered
pair of real numbers, denoted either by (a,b) or
abi.
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Examples of complex numbers in both notations

• For simplicity, the last three complex numbers
  would usually be abbreviated as
• Geometrically, a complex number can be viewed
  either as a point or a vector in an xy plane.

Figure 2
Figure 1
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Some complex numbers are shown as points (Figure
1) and as vectors (Figure 2). It may also be
convenient to use a single letter, such as z, to
denote a complex number. Thus we might write The
real number, a, is called the real part of z and
the real number, b, is called the imaginary part
of z. These numbers are sometimes denoted Re(z)
and Im(z), respectively. Therefore When
complex numbers are represented geometrically, in
an xy-coordinate system, the x-axis (horizontal)
is called the real axis, the y-axis the imaginary
axis, and the plane is called the complex
plane. Recall, that we say that 2 vectors are
defined to be equal if they have the same
components, so we define two complex numbers to
be equal if both their real and imaginary parts
are equal.
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Two complex numbers, abi and cdi, are defined
to be equal (i.e. abicdi) if ac and
bd. Note that all real numbers are special
cases of complex numbers.... the imaginary
component is zero. Geometrically, the real
numbers correspond to points on the real
axis. If the real part of a complex number is
zero and the imaginary portion is non zero, these
points lie on the imaginary axis and these
numbers are considered purely imaginary (called
pure imaginary numbers). Addition of complex
numbers Complex numbers are added by adding
their real parts and adding their imaginary
parts (abi)(cdi)(ac)(bd)i Subtraction of
real numbers Similar to addition (but in the
opposite sense) subtraction is performed by
subtracting like parts (abi)-(cdi)(a-c)(b-d)i
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Multiplication by a scalar

• Both components of the complex number are
  multiplied by the scalar
• k(abi)(ka)(kb)i (if k is real)
• k(abi)(-kb)(ka)i (if k is purely imaginary)
• Graphically

Figure 3
Figure 4
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Recall that when adding vectors graphically, the
vectors are placed head to tail, while
subtraction is performed by placing consecutive
vectors head to head. Note that the vector
z1-z2 followed by the (tail of) vector z2
results in z1. This makes sense algebraically,
as well (z1-z2z2z1).
K gt 0
K lt 0
Figure 6
Figure 5

• Multiplying a complex number (vector) by a scalar
  simply changes the amplitude of the (vector)
  complex number, as long as the scalar is greater
  than zero (Figure 5). If the scalar is less than
  zero, the (vector) complex number is positioned
  on the opposite side of the origin. This is often
  referred to as a 180 degree phase shift.
• To this point, there have been parallels between
  complex numbers and vectors in 2-Dimensional
  space.

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However, let's now consider multiplication of
complex numbers, an operation without a vector
analog in 2-D space. When calculating products
of complex numbers, follow the usual rules of
algebra, but treat i2 as -1. Other
properties of complex arithmetic
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Modulus, Complex Conjugate, Division
If zabi is any complex number, then the
conjugate of z, denoted by is defined by In
words, is obtained by reversing the sign of
the imaginary part of z. Geometrically, z is the
reflection of about the real axis
Figure 7. A complex number and its conjugate.
It is interesting to note that if and
only if z is a real number.
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If a complex number is viewed as a vector in 2-D
space, then the norm or length of the vector is
called the modulus (or absolute value) of z. The
modulus of a complex number zabi, denoted by
z, is defined by
Note that if b0, then za is a real number, and
so the modulus of a real number is simply its
absolute value. It is for this reason, that the
modulus of z is called the absolute value of z.
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Example Prove that
Division of complex numbers is typically
considered the opposite of multiplication. Thus,
if
then the definition of should be
such that
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Thus
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Let
equating real and imaginary parts gives two
simultaneous equations
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Using Cramers Rule
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So for
Although this may seem complex, this is merely
the original quotient multiplied by the complex
conjugate of the denominator
in the form
Example Express
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Properties of Complex Conjugates
Polar Form
If zxiy is a nonzero complex number, r z and
? indicates the angle from the positive real axis
to the vector z, then as suggested in the figure
on the next slide
The projection of the vector on the X axis is
and the projection of the vector on the Y axis is
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Such that zxiy can be written as
or
r is the amplitude (modulus) of the complex
number and ? is the angle between the vector and
the "x" axis, (arg(z) or phase angle)
Note that the angle, ?, can be determined using
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However, care must be taken in the calculation of
? as it will depend on the quadrant location of
the complex number as illustrated in the graph
below
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Both quadrants II and IV produce negative numbers
in the calculation of the tan-1 function but the
calculated angle is for quadrant IV. Similarly,
quadrant I and III produce positive numbers for
the tan-1 calculations but the result applies
only to quadrant I. Hence the need to add 180
deg (p rad) to the angle values of the complex
numbers when located in quadrants II or
III. Calculate r and ? for z -2 3i, -3 4i
and 1 3i
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Complex Exponentials
Euler's Formula can also be used as another form
for expressing the phase angle of a complex
number and is given by
Other relationships with exponentials and complex
numbers
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Euler's formula allows us to envision the
geometrical implications of complex
multiplication more easily.
For example, the complex number with amplitude r1
and phase ?1 multiplied by
a second complex number with amplitude r2 and
phase ?2 can be calculated by
Similarly
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This means that the result of these operations is
another complex number, whose amplitude is the
product of the 2 amplitudes and the phase angle
is the sum of the 2 phase angles. Consider the
example problem on slide 16 done using polar
coordinates C A/B A 3 4i B 1
2i
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• Note that on slide 16 our answer for A/B was
• A/B -1 2i
• The modulus and phase angle for this complex
  number are
• Which agrees with previous slide
• Compute using polar method
• D AB/C where A -7i B 2 3i C 4 5i