Algebra of Complex Numbers | Mathematics (Maths) for JEE Main & Advanced PDF Download

Complex Numbers

A. Definition
Complex numbers are defined as expressions of the form a + ib where a , b ∈ R &   It is denoted by z i.e. z = a + ib . ‘a’ is called as real part of z (Re z) and ‘b’ is called as imaginary part of z (Im z).

Every Complex Number Can Be Regarded As

Purely real  -  Purely imaginary - Imaginary
if b = 0    -   if a = 0  -  if b ≠ 0

Remark :
(a) The set R of real numbers is a proper subset of the complex numbers . Hence the complete number system is
(b) Zero is both purely real as well as purely imaginary but not imaginary .
(c)  is called the imaginary unit . Also i² = - l ; i³ = -i ; i⁴ = 1 etc.
(d)  only if atleast one of either a or b is non - negative.

B. Algebraic Operations
The algebraic operations on complex numbers are similar to those on real numbers treating ‘i’ as a polynomial . Inequalities in complex numbers are not defined . There is no validity if we say that complex number is positive or negative.

e.g. z > 0 , 4 + 2i < 2 + 4 i are meaningless.

However in real numbers if a² + b² = 0 then a = 0 = b but in complex numbers,

z₁² + z₂² = 0 does not imply z₁ = z₂ = 0.

Equality In Complex Number :

Two complex numbers  are equal if and only if their real & imaginary parts coincide.

C.Conjugate Complex

If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by

Remark :
(i)

(ii)

(iii)  which is real
(iv)If z lies in the 1^st quadrant then  lies in the 4th quadrant and -  lies in the 2nd quadrant.

Ex.1 Express (1 + 2i)²/(2 + i)² in the form x + iy.

Sol. 

Ex.2 Show that a real value of x will satisfy the equation 

Sol.

We have  

or   [by componendo and dividendo],

Therefore, x will be real, if

Ex.3 Find the square root of a + ib

Sol.

Let  = x + iy, where x and y are real. Squaring, a + ib =
Equating real and imaginary parts,

Now (x² + y²)² = (x² – y²)² + 4x²y² = a² + b²  or  x² + y2 = √(a² + b² )...(iii)

[ x and y are real, the sum of their squares must be positive]
From (i) and (iii),

If b is positive, both x and y have the same signs and in opposite case, contrary signs. [by (ii)].