Number System Basics

Natural Numbers : Set of counting numbers is called natural numbers. It is denoted by N. where,

N = {1, 2, 3, ……∞}

Whole Numbers : When zero is included in the set of natural numbers, then it forms set of whole numbers. It is denoted by W. where, W = {0, 1, 2, 3, …..∞}

Integers : When in the set of whole numbers, natural numbers with negative sign are included, then it becomes set of integers. It is denoted by I or Z.

I : [– ∞, …………………………. –4, –3, –2, –1, 0, 1, 2, 3, 4, ……….∞]

Integers can further be classified into negative or positive Integers. Negative Integers are denoted by Z– and positive Integers are denoted by Z+ .

Z – = {– ∞, ………………. –3, – 2, –1} and Z + = {1, 2, 3, ……………. ∞}

Further 0 is neither negative nor positive integer.

Prime Numbers: The natural numbers which have no factors other than 1 and itself are called prime numbers. Note that,

(i) In other words they can be divided only by themselves or 1 only. As, 2, 3, 5, 7, 11 etc.

(ii) All prime numbers other than 2 are odd numbers but all odd numbers are not prime numbers. 2 is the only one even Prime number.

Co-Prime Numbers : Two numbers which have no common factor except 1, are called Co–Prime numbers. Such as, 9 and 16, 4 and 17, 80 and 81 etc. It is not necessary that two co–prime numbers are prime always. They may or may not be prime numbers.

Divisible numbers/composite numbers : The whole numbers which are divisible by numbers other than itself and 1 are called divisible numbers or we can say the numbers which are not prime numbers are composite or divisible numbers. As, 4, 6, 9, 15, …….. Note : 1 is neither Prime number nor composite number. Composite numbers may be even or odd.

Rational Numbers : The numbers which can be expressed in the form of p/q where p and q are integers and co-prime and q ≠ 0 are called rational numbers. It is denoted by Q. These may be positive, or negative. e.g. 4/5, 5/1, -1/2 etc are rational numbers.

Irrational Numbers : The numbers which are not rational numbers, are called irrational numbers. Such as

√2 = 1.414213562……….

π = 3.141592653 ………..

Real Numbers: Set of all rational numbers as well as irrational numbers is called Real numbers. The square of all of them is positive.

Perfect Numbers : If the sum of all divisors of a number N (except N) is equal to the number N itself then the number is called perfect number. Such as, 6, 28, 496. 8128 etc. The factor of 6 are 1, 2 and 3 Since, 6 : 1 + 2 + 3 = 6

28 : 1 + 2 + 4 + 7 + 14 = 28

496 : 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

8128 : 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128. etc

Complex Numbers : Z = a + ib is called complex number, where a and b are real numbers, b ≠ 0 and i = √-1 .

Such as, √-2 , √-3 etc.

So, a + ib or 4 + 5i are complex numbers.

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