Complex Numbers - 2

(2) Second Question : a < 0 , b < 0 then the principal value is arg (z)= is an obtuse angle & positive.



(3) a < 0, b < 0 then the principal value is argz =

is an obtuse angle & negative

(4) fourth Question : a > 0, b < 0 then the principal value is arg (z) is an acute angle & negative

Dumb Question:- Find the modulus & amplitude of
(a) - 4, 3i (b) 4, 3i (c) + 4, 0 (d) 4e^-3,

Ans: At first sight it looks like , but modulus can’t be negative & also can’t be imaginary so to make modulus positive we write

Now comparing with we get

ILLVATRATION - 2. What does arg (z) =



arg(z) = shoues that Z is in first Quadrant & join of Zwith origin makes an angle of with positive direction of x axis. Hence of Z is an open ray as shown in figure

CONJUGHTE OF A COMPLEX NUMBER

The complex number of z=a+ib & are called complex conjugate of each other. The complex conjugate is obtained by changing the sign of imaginary part.


In polar from complex conjvgates are having same modulus & modulus of angle is same but the Angles differ in sing,

PROPERTIES OF CONJUGATE
(1)
Why ?

(ii) z is purely real (both sided equality)
Why ?

ie purely real

(iii) is purely imaginary
Why ? if z = ki
then = - ki Hence z = -

(iv)
Why ? Let z₁ - a₁ + ib₂
then
&
clearly (1) = (2)

(v)

(vi)
Why ? Let


(vii)

(viii)
Why ? Let


PROPERTIES OF MODULUS

(i) & if (iff) z = 0
Why? |2| = r = distance of any point from orgin & distance can’t be negative.

(ii)
Why ? Re(z) = r cos Im(z) = r sin
So, which is true.

(iii) (most important & frequently used property)
Why ?

(iv)


(v)

(vi)

(viii)

When point z is joined kto origin O then the vector can be used to represent complex
number z. We know that in a triangle sum of two sides is always greater thann third side. By using this concept the above property will be proved.



In figure 1:


Dumb Question:- What happens when

Ans:- By same way we can also prove that

combining both (1) & (2)we can write

In figure2:


BY (3) & (4)


(viii)
Why ? Look at figure 1 of last property
we can writ
because sum of two sides of triangle is greater than third side.
Modulus is always +ve. hence we can write.

Then
&
Hence

Dumb Question:- How can you kget eqn(7)from eqn(5)& 6 ?
Ans: suppose

Hence 10 > 4 & nalso 10 > -4
If 10 > 4 it is obviously reater than - 4 but when we can use eqn. (7) to make sure that we compare + ve quantitics.

(ix)
Why? Let
then L.H.S.

R.H.S.

Dumb Question: Why ?
Ans:- We know that


why ? By one of the previous properties
using this



Dumb Question: Why
we know that
here

PROPERTIES OF ARGUMENT

(i) Arg
Why ? Let
then L.H.S>

R.H.S.

s (ii) Arg

(iii) Arg
Let