Complex umbers Summary

1. Cartesian representation and laws:

→+= iyxz xz =)Re( , yz =)Im( , 22|| yxz += , iyxz −=*

(i) )Re(2* zzz =+

(ii) )Im(2* zizz =−

(iii) 2||* zzz =

(iv) ( ) wzwz ±=± **

* , where w is also a complex number

(v) )arg()arg()arg( 2121 zzzz += where )arg( 1z<−π , π≤)arg( 2z

(vi) )arg()arg(arg 21

2

1 zzz

z−=

ote: If a polynomial 0)( =zF has real coefficients throughout its entire

structure, then complex roots (if any) MUST occur in conjugate pairs.

2. Polar representation and laws:

)sin(cos θθ irz += , where 0|| >= zr , θ=)arg(z and πθπ ≤<−

De Moivre’s Theorem:

If )sin(cos θθ irz += , then )sin(cos θθ ninrz nn +=

Useful information:

(i) ;cos)cos( θθ =− θθ sin)sin( −=−

(ii) If )sin(cos θθ irz += , then )sin(cos* θθ irz −=

(iii) ,cos2* θrzz =+ )sin(2* θirzz =−

(Note in particular if 1=r , then z

iz1

sincos* =−= θθ )

3. Euler’s(exponential) representation and laws:

θirez = , where 0|| >= zr , θ=)arg(z and πθπ ≤<−

(i) ( )( )21

21

θθ ii erer = )(

2121 θθ +ierr

(ii) =2

1

2

1

θ

θ

i

i

er

er

)(

2

1 21 θθ −ie

r

r

(iii) ( )θire n = θinre

Useful information:

(i) If θirez = , then θirez −=*

(ii) 2222 cos2)())(( rzzrrerezzrezrez iiii +−=++−=−− −− θθθθθ

4. Solving higher order polynomial equations:

(a) Roots of unity:

)2(1 πkinn ezz =⇒=

∴

= n

ki

ez

π2

, =n 0, 1, 2, 3, …….. 1−n

Note: Ensure that the argument of each single individual complex root is

presented in the standard range required, ie ππ ≤<− )arg(z .

(b) Solving general polynomial equations:

)2()( θπ +=⇒= kinn rezzFz , where )(zF has been transformed into the Euler’s

representation and its argument θ is added to a

mandatory cycling factor πk2

∴

+

= n

ki

ez

θπ2

=n 0, 1, 2, 3, …….. 1−n

Note: Ensure that the argument of each single individual complex root is

presented in the standard range required, ie ππ ≤<− )arg(z .

5. Physical implications of multiplying one complex number by another:

Consider a complex number =z 1

1

θier . If it is multiplied by another complex

number =w 2

2

θier , then the physical effect is that of changing the length

of the cord joining z and the origin (in the Argand diagram)by a factor of 2r ,

and subsequently rotating this cord by an angle of 2θ . The resultant complex

number is therefore )(

2121 θθ +ierr .

Im

zw

z

2θ

1θ

21rr

Re

1r

6. Loci of various complex number constructs

(i) baz =− ||

Interpretation: Set of variable points denoted by z which are always b

units away from a fixed complex number a .

Locus: A circle centered at complex number a with radius b

Im

a Re

b

Note: If baz ≤− || , then the locus shall be the entire interior of the circle

coupled with the boundary of the circle itself. Similar logic must be

applied to varying inequalities for this circular loci construct as well.

(ii) |||| bzaz −=−

Interpretation: Set of variable points denoted by z which are equidistant

from two unique, fixed complex numbers a and b .

Locus: A line that bisects the cord joining complex numbers a and b

in a perpendicular fashion

Im

b

Re

a

(iii) θ=− )arg( az

Interpretation: Set of variable points denoted by z which will form

an argument of θ around a fixed complex number a .

Locus: A line that is pivoted at a and possesses a standard argument θ .

Im

θ

a

Re