1. Real Number System Presentation on

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; 1N P2: For each nN, there exists a unique natural number n'N, called the successor of n, we can write it as (n+1) also. P3: for each nN , 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 1N mN m*S 5. Addition on N: The basic laws of addition composition are: A1: Closure law. For m, nN, m+nN A2: Communicative law. m+n=n+m V m, nN A3: Associative law: m+(n+p)=(m+n)+p Vm,n,pN A4: Cancellation law: m+p=n+p m+p Vm,n,pN 6. Multiplication on N: The operation of multiplication on N is defined as follows: M1. Closure law: m, nN; m.nN m, nN M2. Communicative law: m.n=n.m m, nN M3. Associative law: m. (n. p)=(m. p). n m,n,pN M4: Cancellation law: m.p=n.p m=n m,n,pN M5: Existence of identity: m.1=1.m mN 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 mn and n>p m>p m,n,pN Q3: Anti-symmetric law: m>n and n>m m=n m,nN Q4: Monotone Property of Addition: m>nm+p>n+p m,n,pN Q5: Monotone Property of Multiplication: m>nmp>np m,n,pN 8. The integers: The natural numbers are whole numbers positive, negative or zero. We can also defined them as ratio of two numbers which dont have a remainder. The numbers -1, -2, -3, -4,..are negative integers. The numbers +1, +2, +3, +4are 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, bI 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 (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,bR 3. Associative law:(a+b)+c=a+(b+c) V a,b, cR 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.