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