**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

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. .

(i) 72 = 8 x 9 = 2 x 2 x 2 x 3 x 3 = 2^{3}
x 2^{2}

(ii) 7a^{2} - 21ab = 7a(a -
3b)

While resolving algebraic
expression the following may be remembered x^{2} + (a + b)x + ab = (x +
a) (x + b)

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. .

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. .

If N is a composite number such that N = a^{p}.b^{q}.c^{r}……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 = 2^{4} x 3^{2}. 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 a^{p}.b^{q}.c^{r} ……..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.