Page 20 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Since there lies a rational number between any two rational numbers, it is clear that there lie an
infinite number of rational numbers between two given rationals. This property of rational
numbers make them dense every where. Evidently integral numbers or the natural numbers are
not dense in this sense.

Real Numbers

We know that the equation x = 2 has no solution in Q. Therefore if we have a square of unit length,
2
then there exists no rational number which will give us a measure of the length of its diagonal.
Thus we feel that our system of rational numbers is inadequate and we want to extend it.
The extension of rational numbers into real numbers is done by special methods two of which
are due to Richard Dedekind and George Cantor. We shall not describe these methods here. We
can simply say here that a real number is one which can be expressed in terms of decimals
whether the decimals terminate at some state or we have a system of infinite decimals, repeating
or non-repeating. We know that every repeating infinite decimals is a rational number, also
every terminating decimal is a rational number.

Irrational Number

A real number which cannot be put in the form p/q where p and q are integers is called an
irrational number. The set R of real numbers is the union of the set of rational numbers and the set
of irrational numbers.
If a, b, c are real numbers, then
(i) a + b = b + a, ab = ba (commutative of addition and multiplication)
a (b c ) (a b ) c ,
(ii) Associativity of addition and multiplication
a ( ) ( ) c
ab
bc
(iii) a + 0 = 0 + a = a, i.e., the real number 0 is the additive identity.
(iv) a.1 = 1.a = a, i.e., the real number 1 is the multiplicative identity.
(v) For each a R, these corresponds – a R such that
a + (–a) = – (a) + a = 0
Thus every real number has an additive inverse.
(vi) Each non-zero real number has multiplicative inverse.

(vii) Multiplication composition distributes addition, i.e.,
a (b + c) = ab + ac
(viii) The cancellation law invariably holds good for addition. For multiplication, if a 0, then
ab = ac b = c
(ix) The order relations satisfy the trichotomy law.

Complex Numbers

An ordered pair (a, b) of real numbers is called a complex number. The product set R × R
consisting of the ordered pairs of real numbers is called the set of complex numbers. We shall
denote the set of complex numbers by C.
Thus
C = {z : z = (a, b), a, b R}.

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