This document discusses different types of real numbers including natural numbers, integers, rational numbers, irrational numbers, and real numbers. It defines each type of number and provides examples. It also defines the absolute value or modulus of a real number, listing properties such as the modulus being non-negative and greater than or equal to both the number and its opposite. Formulas are given for the absolute value of products and sums of real numbers.

1.  Real Nos.
 Course: BBA
 Subject: Business Mathematics
 Unit: 1.1
1.

2. Real nos.:
1) Natural nos: The nos. 1,2,3,….. which are used for
counting are called natural nos. The set of natural nos.
denoted by N.
Thus N = {1, 2, 3, ……. }
2) Integers: The integers are whole numbers positive,
negative or zero. The set of integers is denoted by Z.
Thus Z = {…..-3, -2, -1, 0, 1,2,3,……}
3) Rational nos: The nos. which can be expressed in the
form of p/q where p and q are positive integer and q ≠
0, is called a rational no. The set of rational no. is
denoted by Q.
Thus Q = {p/q such that p Є Z , q Є N }

3. 4) Irrational Nos: If nth root of a rational no. a is not a
rational no., then a is called an irrational no.
An irrational no. cannot be expressed as p/q.
5) Real no.: The set of all rational and irrational number is
called the set R of real no.

4. Examples:
1) Prove that √2 is an irrational number.
2) Express 0.0272727……. Into a quotient form.
3) Express 2.3333……. Into a quotient form.

5. Absolute or Modulus of a real nos.:
The modulus or absolute value of a real no. a is denoted by │a│
and it is defined as follows:
if a > 0
if a < 0
if a = 0.
Properties (without proof):
1) The modulus of a real no. is non negative.
2) For any a Є R, │a│ = │-a│
3) For every a Є R, │a│≥ a and │a│≥ -a.
4) │a│ denotes the greater of the two numbers a and –a
i.e. │a│= max (a, -a).
0
a
a a


 



6. 5) If a and b are any two real numbers.
 │ab│= │a│. │b│
 │a +b│ ≤ │a│+ │b│
 │a - b│ ≥ │a│- │b│
,(b 0)
aa
b b
 
1 1
,(a 0)
a a
 

7. References:
1. http://mathforum.org/mathimages/imgUpload/Venn
_Diagram.jpg
2. Business mathematics by G.C. Patel & A.G. Patel (For
content)- Atul prakashan.