4'4 Eulers Form Appendix B does not include this work' See notes in Study Book 4'4

1
4.4 Eulers Form Appendix B does not include
this work. See notes in Study Book 4.4 ( some
of 4.3).

• Objectives Know
• how to convert any z a ib to Euler form r
  e i?
• vice versa
• how to use Euler form to simplify calculation of
  powers, multiplication division
• how to find nth roots using Euler form
• the algebraic geometric properties of nth roots.

2

• When complex nos are multiplied their angles
  add.
• When they are divided, their angles subtract.
  
• Hence in the polar form r (cos? i sin? )
• the argument ? behaves like an exponent
• but the modulus r does not.
• To remind us of this, we write ? as an exponent
• We define e i? to be cos? i sin?
• Then r (cos ? i sin? ) r e i? .
• And de Moivres Theorem,
• ( cos ? i sin? ) n cos n? i sin n?
  
• becomes much more intuitive
• (e i? ) n e i n? and z n r n e i
  n?

3

• There are many reasons why we use base e, not
  another base.
• (See Study Book and Taylor Series, Alg Calc
  II.)
• Convert the following to Euler Form (Plot
  first easier!)
• 5 5 e i0
  - 2 2 e ip
• 3i 3 e ip/2
  -1 - i sqrt(2) e i5p/4
• In reverse
• e ip cos(p) i sin(p) -1
  i 0 -1 .
• Confirm by plotting e ip which is 1 e
  ip
• That is the point distance 1 on angle p
• So it gives the -1 on the x-axis.)
• Similarly 2e -ip/2
• is the point 2 units down the y axis,
• ie 0 2i or simply -2i.
• Also see Examples 4.1, 4.2.

4

• Note multiplying any complex number by r e i?
• causes
• an increase of ? in the angle, ie a rotation,
• and distance to change by the factor r.
• Eg Multiplying any z by i (which is 1 e
  i?/2 )
• causes anticlockwise rotation through
  angle ?/2 .
• Multiplying z by 2i (which is 2 e ip/2
  ) iz
• causes rotation through pi/2,
  z
• and doubling of distance (modulus).
• Also see Ex 4.3.
• Positive angles cause an anticlockwise rotation
  through angle ?. Negative angles cause a
  clockwise rotation.

5

• Finding nth roots of z.
• First write z in Euler form r e i?
  . Then
• generalise its angle by adding revolutions 2k?.
• take the (1/n)th power r 1/n e (i? 2k? )
  /n .
• find n different roots using n successive values
  of k, eg k 0 , 1, 2,
• Geometrically, the nth roots
  of a e ib
• are evenly spaced on a circle of radius a 1/n
  .
• Examples Find plot
• 1) the cube roots of 4 ? 3 4 i
  8 p/6
• 2) the 4th roots of - 8 .
• Note If we know one nth root, we
  can plot it
• and deduce the positions of the others.
• Eg One cube root of -8 is -2. Plot it
  deduce the others.

6
Homework

• Re-visit Section 8.3, App B, p 438
• Working in Euler Form, write solutions to
• Q 27, 29, 31, 33, 39, 43, 45, 47, 51, 53.