Page 21 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Two complex numbers (a, b) and (c, d) are equal if and only if Notes
a = c and b = d.
The sum of two complex numbers (a, b) and (c, d) is defined to be the complex number (a + c, b + d)
and symbolically, we write
(a, b) + (c, d) = (a + c, b + d)
The addition of complex numbers is commutative, associative, admits of identity element and
every complex number possesses additive inverse.
If u and v are two complex numbers, then u – v = u + (–v).
The cancellation law for addition in C is
(a, b) + (c, d) = (a, b) + (e, f) (c, d) = (e, f) (a, b), (c, d), (e, f) C.

The product of the complex numbers (a, b) and (c, d) is defined to be the complex number (ac – bd,
ad + bc) and symbolically we write
(a, b) (c, d) = (ac – bd, ad + bc).

The multiplication of complex numbers is commutative, associative admits of identity element
and every non-zero complex number possesses multiplicative inverse.
Cancellation law for multiplication in C is

[(a, b) (c, d) = (a, b) (e, f) and (a, b) (0, 0)] (c, d) = (e, f)
In C multiplication distributes addition.
A complex number (a, b) is said to be divided by a complex number (c, d) if there exists a complex
number (x, y) such that (x, y) (c, d) = (a, b).
The division, except by (0, 0), is always possible in the set of complex numbers.

Usual Representation of Complex Numbers

Let (a, b) be any complex number.
We have (a, b) = (a, 0) + (0, b)
= (a, 0) + (0, 1) (b, 0)

Also, we have (0, 1) (0, 1) = (–1, 0) =–1. If we denote the complex number (0, 1) by i, we have
2
i = –1. Also we have (a, b) = a + ib, which is the usual notation for a complex number.
In the notation Z = a + ib for a complex number, a is called the real part and b is called the
imaginary parts. A complex number is said to be purely real if its imaginary part is zero, and
purely imaginary if its part is zero but its imaginary part is not zero.

For each complex number z = (a, b), we define the complex number z = (a, –b) to be the conjugate
of z. In our usual notation, if
z = a + ib

then z = a – ib

If z = (a, b) be any complex number, then the non-negative real number (a b 2 ) is called the
modulus of the complex number z and is denoted by |z|.

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