Complex Numbers


   
 
Complex number
Imaginary Number
Square root of a negative number is known as an imaginary number.  a > 0 is an imaginary number.
 
or
 
A number whose square is negative is known as an imaginary number.
 
. . .
 
The symbol i,
 
 
We write,
 
 
 
 
Powers of i,
 
 
 
 
 
 
 
Let p > 0 be a positive integer such that p > 4. Let q be the quotient and r be the remainder, when p is divided by 4.
 
 
 
Complex Numbers
If x and y are real numbers, then x + iy is called a complex number.
 
x is called the real part and y is called the imaginary part.
 
The complex number x + iy is also written as an ordered pair (x, y) and is denoted by z.
 
i.e., z = x + iy
 
The positive valueis called the modulus of Z and is denoted by |Z|.
 
 
In Z = x + iy, if y = 0, then Z is purely real and if x = 0, then Z is imaginary.
 
Example:
 
 
Note:
 
 
Set of real numbers is a proper subset of the set of Complex numbers.
 
Equality of Complex numbers
 
Two complex numbers are equal iff their corresponding real parts and imaginary parts are separately equal.
 
 
Sum of two Complex numbers
 
If Z1 = a + ib and Z2 = x + iy, then we define their sum as
 
Z1 + Z2 = (a + ib) + (x + iy)
 
= (a + x) + i(b + y) which is a complex number.
 
Negative of a Complex number
 
If Z = a + ib, then Z is called the negative of Z.
 
 
-Z = -(a + ib)
 
= -a - ib
 
 
Additive identity of the Complex number
 
The complex number 0 + i0 is the additive identity for the set of complex numbers.
 
0 + i0 is called the additive identity for the complex number.
 
Z + (0 + i 0 ) = a + i b + (0 + i 0)
 
= (a + 0) + i (b + 0)
 
= a + i b
 
Additive inverse of a Complex number
 
Let Z = a + i b and Z' = x + iy be the additive inverse of Z, then
 
 
 
a + x = 0 and b + iy = 0
 
 
\ Additive inverse of a + ib is - a - ib.
 
Product of two Complex numbers
 
Let Z1 = a + ib and Z2 = c + id, then
 
 
 
 
 
Multiplicative identity of Complex numbers
 
Let Z = a + i b and Z' = x + iy, then
 
 
 
ax - by = a ..... (i)
 
and ay + bx = b ..... (ii)
 
Solving (i) and (ii), we have
 
 
x = 1, y =0
 
Multiplicative identity is 1 + i0.
 
Conjugate complex numbers
 
If Z = a + ib is a complex number, then a - ib is called the complex conjugate of a + ib and the conjugate is denoted by
 
Remark:
 
The sum and product of two conjugate numbers is always real.
 
 
 
 
2a + i(0)
 
2a
 
 
 
 
 
 
 
Quotient of two non-zero Complex numbers
 

If Z1 = a + ib and Z2 = c + id are two complex numbers, then quotient is defined as

 
 
 
 
 
 
Reciprocal of a non-zero complex number or multiplicative inverse of a non-zero complex number
 
 
 
 
 
  
     
   
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