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 ; i^{3} = i ; i^{4} = 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 a^{2} + b^{2} = 0 then a = 0 = b but in complex numbers,
z_{1}^{2} + z_{2}^{2} = 0 does not imply z_{1} = z_{2} = 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 (x^{2} + y^{2})^{2} = (x^{2} – y^{2})^{2} + 4x^{2}y^{2} = a^{2} + b^{2} or x^{2 }+ y2 = √(a^{2} + b^{2} )...(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)].
209 videos443 docs143 tests

1. What is the algebraic representation of a complex number? 
2. How do you add complex numbers? 
3. How do you multiply complex numbers? 
4. How do you find the conjugate of a complex number? 
5. How do you represent complex numbers in polar form? 
209 videos443 docs143 tests


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