Further Pure 1

1
Further Pure 1

• Lesson 5 Complex Numbers

2
Numbers

• What types of numbers do we already know?
• Real numbers All numbers ( 2, 3.15, p ,v2)
• Rational Any number that can be expressed as a
  fraction
• (4, 2.5, 1/3)
• Irrational Any number that cant be expressed
  as a fraction. ( p ,v2, v3 1)
• Natural numbers The counting numbers (1, 2, 3,
  .)
• Integers All whole numbers ( -5, -1, 6)
• Complex Numbers (Imaginary numbers)

3
Complex Numbers

• What is the v(-1)?
• We define the v(-1) to be the imaginary number j.
  (Hence j2 -1)
• Note that lots of other courses use the letter i,
  but we are going to use j.
• We can now use this to calculate a whole new
  range of square roots.
• What is v(-144)?
• Answer is 12j and -12j, or 12j.

4
Complex Numbers

• Now we can define a complex number (z) to be a
  number that is made up of real and imaginary
  parts.
• z x y j
• Here x and y are real numbers.
• x is said to be the real part of z, or Re(z).
• y is said to be the imaginary part of z, or
  Im(z).

5
Solving Quadratics

• Use the knowledge you have gained in the last few
  slides to solve the quadratic equation
• z2 6z 25 0
• Remember
• Solution

6
Solving Quadratics
7
Addition and Subtraction

• To Add and subtract complex numbers all you have
  to do is add/subtract the real and imaginary
  parts of the number.
• (x1y1j) (x2y2j) x1 x2 y1j y2j
• (x1 x2) (y1 y2)j
• (x1y1j) (x2y2j) x1 - x2 y1j - y2j
• (x1 - x2) (y1 - y2)j

8
Multiplying

• Multiplying two complex numbers is just like
  multiplying out two brackets.
• You can use the FOIL method.
• First Outside Inside Last.
• Remember j2 -1
• (x1y1j)(x2y2j) x1x2 x1y2j x2y1j y1y2j2
• x1x2 x1y2j x2y1j - y1y2
• x1x2 - y1y2 x1y2j x2y1j
• (x1x2 - y1y2) (x1y2 x2y1)j
• What is j3, j4, j5?

9
Multiplying

• Alternatively you could use the box method.
• (x1y1j)(x2y2j) (x1x2 - y1y2) (x1y2 x2y1)j

10
Questions

• If z1 5 4j z2 3 j z3 7 2j
• Find
• a) z1 z3 12 2j
• b) z1 - z2 3 3j
• c) z1 z3 -2 6j
• d) z1 z2 11 17j
• e) z1 z3 43 18j

11
Complex Conjugates

• The complex conjugate of
• z (x yj) is z (x yj)
• If you remember the two solutions to the
  quadratic from a few slides back then they where
  complex conjugates.
• z -3 8j z -3 8j
• In fact all complex solutions to quadratics will
  be complex conjugates.
• If z 5 4j
• What is z z
• What is z z

12
Activity

• Prove that for any complex number z x yj,
  that z z and z z are real numbers.
• First z z (x yj) (x yj)
• x x yj yj
• 2x Real
• Now z z (x yj)(x yj)
• x2 xyj xyj y2j2
• x2 y2(-1)
• x2 y2 Real
• Now complete Ex 2A pg 50

13
Division

• There are two ways two solve problems involving
  division with complex numbers.
• First you need to know that if two complex
  numbers are equal then the real parts are
  identical and so are there imaginary parts.
• If we want to solve a question like 1 (2 4j)
  we first write it equal to a complex number p
  qj.
• Now we re-arrange the equation to find p and q.
• (p qj)(2 4j) 1

14
Division

• Expanding the equation gives
• 2p 4q 2qj 4pj 1
• The number 1 can be written as 1 0j
• So
• (2p 4q) (2q 4p)j 1 0j
• Now we can equate real and imaginary parts.
• 2p 4q 1
• 4p 2q 0
• Solve these equations
• p 1/10 q -1/5
• Therefore 1 (2 4j) 0.1 0.2j

15
Division

• The second method is similar to rationalising the
  denominator in C1.
• The 20 on the bottom comes from the algebra we
  proved a few slides back. (x yj)(x yj) x2
  y2
• Now see if you can find (3 - 5j) (29j)
• Now complete Ex 2B pg 53

16
Argand Diagrams

• Complex numbers can be shown Geometrically on an
  Argand diagram
• The real part of the number is represented on the
  x-axis and the imaginary part on the y.
• -3
• -4j
• 3 2j
• 2 2j

Im
Re
17
Modulus of a complex number

• A complex number can be represented by the
  position vector.
• The Modulus of a complex number is the distance
  from the origin to the point.
• z v(x2y2)
• Note x x

Im
y
x
Re
18
Modulus of a complex number

• Find
• a) 3 4j 5
• b) 5 - 12j 13
• c) 6 - 8j 10
• d) -24 - 10j 26

19
Sum of complex numbers

• z1 z 2

Im
Re
20
Difference of complex numbers

• z2 - z1
• Now complete
• Ex 2C pg 57

Im
Re
21
Sets of points in Argand diagram

• What does z2 z1 represent?
• If z1 x1 y1j
• z2 x2 y2j
• Then z2 z1
• (x2 x1) (y2 y1)j
• So z2 z1
• v((x2 x1)2 (y2 y1)2)
• This represents the distance between to complex
  numbers
• z1 z2.

Im
(x2,y2)
y2- y1
(x1,y1)
x2- x1
Re
22
Examples

• Draw an argand diagram showing the set of points
  for which z 3 4j 5
• Solution
• First re-arrange the question
• z (3 4j) 5
• From the previous slide this represents a
  constant distance of 5 between the point (3,4)
  and z.
• This will give a circle centre (3,4)
• Now do Ex 2D pg 60

23
Examples

• How would you show the sets of points for which
• i) z 3 4j) 5
• ii) z 3 4j) lt 5
• iii) z 3 4j) 5