Complex Numbers

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Complex Numbers
The more straight forward questions on Complex
Numbers have been found in Question 2 of the HSC
since 1991. Longer (and usually more difficult)
problems will appear in the latter half of the
paper and are often combined with concepts from
other topics such as Polynomials, Induction and
Binomial Expansions.
(i) Basic Operations
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Example
The complex number 2 i is a root of the
quadratic equation
(i) Find the other root
(ii) Determine the value of m
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(ii) Mod-Arg Form
z x iy where
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(iii) Conjugate Properties
(iv) De Moivres Theorem
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Examples
1. Express the complex number
in modulus argument form.
Hence express in the form a ib where a
and b are real.
2. Express in the terms of
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(v) Roots of Complex Numbers
To find the square root of a complex number we
can
a) Let the solution be a ib and solve
simultaneous equations in a and b
b) Convert into mod arg form and use De Moivres
Theorem
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c) Use the alternative formula
Example Find the square root of 8 6i, giving
your answer in the form x iy
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For cube roots (and higher) we use De Moivres
Theorem (unless of course the polynomial in the
equation is easily factorised)
Equations of the form have n
roots which are equally spaced on the circle of
radius r, where r is the modulus of any root.
Example
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The equation has n roots which
lie on the unit circle, centre (0,0), and if
is a complex root of this equation, then the
roots are
and we can show that
Examples
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3. If is a complex cube root of unity, use
the fact that to
c) Form the cubic equation with roots
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Some important results
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Examples 1. Resolve into real
linear and quadratic factors. Hence prove
sum of roots 0
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3. Solve
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(vi) Geometrical Representation of Complex Numbers
Addition
If a point A represents and point B
represents then point C representing is
such that the points OACB form a parallelogram.
Subtraction
If a point D represents and point E
represents then the points ODEB form a
parallelogram. Note
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Multiplication
If we multiply by the vector OA
will rotate by an angle of in an
anti-clockwise direction. If we multiply by
it will also multiply the length of OA by a
factor of r
Note
will rotate OA anticlockwise 90 degrees.
REMEMBER A vector is HEAD minus TAIL
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Examples
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(vii) Locus
Locus problems can be done by an intuitive method
or by letting z x iy and using methods in
Coordinate Geometry
Examples
1. Sketch on an Argand Diagram, the region
satisfying
x
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2. Sketch on an Argand Diagram, the region
satisfying
perpendicular bisector of the line joining i and
3i
3. Describe the locus described by
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4. Describe the locus described by
locus is ellipse
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2
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r
(2,y)