The real number system includes natural numbers, integers, rational numbers, irrational numbers, and real numbers. Natural numbers are the non-negative whole numbers starting at 0. Integers add the negative whole numbers. Rational numbers are numbers that can be represented as fractions of integers. Irrational numbers cannot be represented as fractions. Real numbers include all rational and irrational numbers and can be located on the real number line. The real number system is closed under addition and multiplication and follows important properties like commutativity and associativity.

1. Real Number
System
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2. Real number system:

3. The natural number:
 a number that occurs commonly and obviously in
nature
 non-negative number
 The set of natural numbers, denoted N
N = {0, 1, 2, 3, ...}

4. PROPERTIES OF N:
 P1: 1 is natural number; 1€N
 P2: For each n€N, there exists a unique natural
number n'€N, called the successor of n, we can
write it as (n+1) also.
 P3: for each n€N , we have n* or (n+1)≠1.
 P4: If m, n € N and m*=n*, then m=n.
 P5: Any sub set S of N is equal to N if
 1€N
 m€N ⇛ m*€S

5. Addition on N:
 The basic laws of addition composition are:
 A1: Closure law.
 For m, n€N, m+n€N
 A2: Communicative law.
 m+n=n+m V m, n€N
 A3: Associative law:
 m+(n+p)=(m+n)+p
Vm,n,p€N
 A4: Cancellation law:
 m+p=n+p ⇛ m+p
Vm,n,p€N

6. Multiplication on N:
 The operation of multiplication on N is defined as follows:

 M1. Closure law:
 m, n€N; m.n€N ¥ m, n€N
 M2. Communicative law:
 m.n=n.m ¥ m, n€N
 M3. Associative law:
 m. (n. p)=(m. p). n ¥m,n,p€N
 M4: Cancellation law:
 m.p=n.p ⇛m=n ¥ m,n,p€N
 M5: Existence of identity:
 m.1=1.m ¥ m€N

7. Order relation on N:
 The lows governing order relations are:
 Q1. Trichotomy law: Any two natural numbers m and n then one and only one of
the following three possibilities hold.
 m=n
 m>n
 m<n
 Q2:Transitive law:
 m>n and n>p ⇛ m>p ¥ m,n,p€N
 Q3: Anti-symmetric law:
 m>n and n>m ⇛m=n ¥ m,n€N
 Q4: Monotone Property of Addition:
 m>n⇛m+p>n+p ¥ m,n,p€N
 Q5: Monotone Property of Multiplication:
 m>n⇛mp>np ¥ m,n,p€N

8. The integers:
 The natural numbers are whole numbers
positive, negative or zero. We can also defined
them as ratio of two numbers which don’t have a
remainder.
 The numbers -1, -2, -3, -4,……..are negative
integers.
 The numbers +1, +2, +3, +4……are positive
integers
 The number 0 is the only integer that has no sign.

9. Prime numbers:
 An integer other then 0 or 1 is a prime number if
and only if its only divisors are 1 and the number
itself.
 PROPERTIES OF PRIME NUMBERS:
 If p is prime number and if p is a factor of ab
where a, b€I then p is a factor of a or p is a factor
b.
 If p is a prime number and if p is a divisor of the
product of a, b, c,…….r of integers then p is a
divisor of at least one of these.

10. Prime numbers:
 A rational number is any number that can be
expressed as the quotient or fraction a/b of two
integers, with the denominator b not equal to
zero. Since b may be equal to 1, every integer is
a rational number.

11. Properties of Rational
numbers:
 The set of all rational numbers is countable.
 Since the set of all real numbers is uncountable,
we say that almost all real numbers are irrational,
in the sense ofLebesgue measure,
 The set of rational numbers is a null set.

12. Irrational numbers:
 An irrational number is any real number that
cannot be expressed as a fraction a/b, where and
b are integers, with b non-zero, and is therefore
not a a rational number.
 An irrational number cannot be represented as a
simple fraction.

13. The real numbers:
 In mathematics, a real number is a value that
represents a quantity along a continuum, such as
–
 5 (an integer),
 4/3 (a rational number that is not an integer),
 8.6 (a rational number given by a finite decimal
representation),
 √2 (the square root of two, an irrational number)
and
 π (3.1415926535..., a transcendental number).

14. Properties of real numbers:
 a.Addition:
 1: Closure law:If a and b are any two real numbers, their sum (a+b) is also a real
number.
 2. Communicative law: If a and b are two real numbers,then
 a+b=b+a, V a,b∈R
 3. Associative law:(a+b)+c=a+(b+c) V a,b, c∈R
 b.Multiplication:
 1: Closure law:If a and b are any two real numbers, their product ab is also real
number.
 2. Communicative law:
 a, b= b, a V a, b ∈ R
 3. Associative law:
 (a,b) ,c=a, (b,c) V a, b, c ∈ R
 c.Relation between the two Algebric operation:
 a, (b,c)=a, b+a , c
 (b+c) , a= b , a+c ,a

15.  Modulus of real numbers: The absolute value has the
following four fundamental properties:
 Non-negativity
 Positive-definiteness
 Multiplicativeness
 Subadditivity
 Imaginary numbers: An imaginary number is defined as
any number that, when squared, results in a real number
less than zero.

16. Complex numbers:
A complex number is a number consisting of a
real part and an imaginary part. Complex
numbers extend the idea of the one-dimensional
number line to the two-dimensional complex
plane by using the number line for the real part
and adding a vertical axis to plot the imaginary
part.