yyO xPrz = x + iy
= r cos q + jr sin qxqFigure 12.1The Argand diagram.
cisθ is shorthand for cosθ+isinθ, using the alternative notationi forj.^ (θ)is the
standard shorthand for cosθ+jsinθthat we will use throughout this book.
Remember thatθ is measured anti-clockwise positive from thex-axis, and, being an
angle, it will represent the same point in the plane at an infinite number of different values,
separated by integer multiples of 2π. To obtain a single valued representation ofz,θmust
be confined to a range of length 2π. The particular choice−π<θ≤π is called the
principal valueofθand is written Argz. If the angleθis not so confined we use a lower
case notation argz. The important point is that the argument of a complex number is not
unique. Thus forz=iwe can takeθ=
π
2,5 π
2,−3 π
2,..., etc. In general we have(^) (θ)= (^) (θ+ 2 kπ)
for any integerk. This fact will play a crucial role in taking roots of a complex number
(➤363). It is also worth noting that while it is not essential to work in radians in complex
numbers, it tends to be the safest policy, particularly for theoretical work.
In terms of Cartesian coordinates we have Argz=tan−^1
(y
x
)
, the sign and value of
the angle being determined byxandy. Also note that|z|=
√
zz ̄.
You may find the above polar representation of complex numbers rather strange – what’s
wrong with the simplex+jyform, you may ask? Well, as always there is method in
mathematics – bear with me, it turns out that the polar form of a complex number ismuch
easier to deal with in certain circumstances.
Exercises on 12.3
- Plot the following complex numbers on the Argand plane and put them into polar form.
(i) 1 (ii) j (iii) − 3 j (iv) 1−j(v) 2+j (vi) − 3 − 2 j (vii) − 3 + 2 j- Convert into Cartesian form
(i) 2^ ( 0 ) (ii) 3^ (π) (iii)^ (π/ 2 )(iv) 3^(
3 π
4)
(v) 1^(
−π
3)
(vi) 2^(
−π
2)