NUMBER SYSTEM – Part I – Theory

Natural Numbers: Set of counting numbers is called natural numbers. It is denoted by N. where, N = {1, 2, 3 …}

Even Numbers: The set of all natural numbers which are divisible by 2 are called even numbers. It is denoted by E. Where, E = {2, 4, 6, 8, 10 …}

 Odd Numbers: The set of all natural numbers which are not divisible by 2 are called odd numbers. In other words, the natural numbers which are not even numbers are odd numbers. i.e., O = {1, 3, 5, 7 …}

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  

(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 is not equal to 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……….

∏ = 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.

Cyclic Numbers : Cyclic numbers are those numbers of n digits which when multiplied by any other number upto n gives same digits in a different order. They are in the same line.

As 142857

2 × 142857 = 285714 : 3 × 142857 = 428571 4 × 142857 = 571428 : 5 × 142857 = 714285

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.

Note : In a perfect number, the sum of inverse of all of its factors including itself is 2 always.

e.g. Factors of 28 are 1,2,4,7,14 are

 = 1/ 1+ 1/ 2 + 1 /4 + 1 /7 + 1 /14 + 1 /28 =  56 /28 = 2

Complex Numbers : Z = a + ib is called complex number, where a and b are real numbers, b is not equal to 0 and i = -1 . Such as, √-2 , √-3 etc.

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

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