Real and
Complex Numbers
The combination of rational and irrational
numbers is called as Real numbers. .
In general, when we talk about a number, we
take it as a real number. The above discussion can be summarised in the
following way. .
Real Numbers
Rational Numbers Irrational
Numbers
Integers Fractions
Negative Positive
integers integers
Factors
When a number is expressed as a product of
two or more numbers, the latter numbers are called factors. Similarly, when an
algebraic expression is expressed as the product of two or more expressions,
each of these later quantities is called a factor of it. The determination of
these is called resolution. .
Example1
(i) 72 = 8 x 9 = 2 x 2 x 2 x 3 x 3 = 23
x 22
(ii) 7a2 - 21ab = 7a(a -
3b)
While resolving algebraic
expression the following may be remembered x2 + (a + b)x + ab = (x +
a) (x + b)
Multiples
If you multiply 5 with 3, the product is 15.
Here, 5 and 3 are called the factors of 15 (as described above) and 15 is a
multiple of 5 and 3. .
Thus if p divides q completely without
leaving any remainder, then p is a factor of q ; and q is a multiple of p. In
the above example, 15 is called a “common multiple” of 3 and 5 because it is a
multiple of both 3 and 5. .
Co-Primes
If two numbers are such that there is no
common factor between them expect one, then they are said to be relatively
prime or Co-Prime to each other. .
The two numbers individually may be prime or
composite. .
Eg: 15 and 23; 13 and 29; 15 and 32 are
co-primes. .
For understanding certain mathematical
concepts, it is necessary to know the classification of numbers. .
Number of factors of a given number
If N is a composite number such that N = ap.bq.cr……where
a,b,c are prime factors of N and p,q,r..... are positive integers, then the
number of factors of N is given by the expression (p + 1) (q + 1)(r + 1)... For
example 144 = 24 x 32. Hence 144 has (4 + 1)(2 + 1),
i.e., 15 factors. This figure includes 1 and the number itself also counted as
factors. .
Number of ways of expressing a given number
as a product of two factors
The given number N (which can be written as
equal to ap.bq.cr ……..where a,b,c are prime
factors of N and p, q, r……. are positive integers) can be expressed as the
product of two factors in different ways. The number of ways in which this can
be done is given by the expression 1/21(p + 1)(q + 1) (r + 1).....} .
If p, q, r etc. are all even, then the
product (p + 1)(q + 1) (r + 1)... becomes odd and the above rule will not be
valid. If p, q, r,... are all even, it means that N is a perfect square. .
So, to find out the number of ways in which a
perfect square can be expressed as a product of 2 factors, we have the
following 2 rules
(1) as
a product of two DIFFERENT factors is ½{(p + 1) (q + 1)(r + 1)... -1} ways
(excluding 
x )
(2) as
a product of two factors (including x ) is ½{(p + 1)(q + 1)(r + 1)... +1} way.
