Number System

Introduction

A number is a quantity that is used in counting and measuring.  In mathematics, a 'number system' is a set of numbers together with one or more operations, such as addition or multiplication.

Examples of number systems include: natural numbers, integers, rational numbers, algebraic numbers, real numbers, complex numbers, surreal numbers and hyperreal numbers.

Classification of Numbers

  • Natural Numbers

Natural numbers can be used for counting objects (one orange, two orange, three orange ...).

  • Whole Numbers

All Natural numbers together with zero form the set of whole numbers. Thus, 0 is the only whole number which is not a natural number but every natural number is a whole number.

  • Integers

The natural numbers and their negatives together with the number 0 are called Integers. They are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 65, 7, and −756 are integers; 1.6 and 1½ are not integers. In other words, integers are the numbers one can count with items such as apples or fingers, and their negatives, as well as 0.

The integers are the only integral domain whose positive elements are well-ordered, and in which order is preserved by addition. Like the natural numbers, the integers form a countable infinite set. Integers can be thought of as points on an infinitely long number line.

  • Rational Numbers

A rational number is a number that can be expressed as the quotient a/b, where a and b are two integers, and the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number.

The decimal expansion of a rational number always either terminates after finite number of digits or begins to repeat the same sequence of digits over and over. Thus, any repeating or terminating decimal represents a rational number

  • Irrational Numbers

An Irrational Number is a number that cannot be written as a simple fraction - the decimal goes on forever without repeating such as, pi and the square root of two. The value of Pi is 3.1415926535897932384626433832795 (continues). Similarly the value of square root of 2 is 1.41421356237309504880 (continues without repeating any pattern)

  • Real Numbers

The real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an infinitely long number line.

 

Classification of Natural Numbers

  • Even and Odd Numbers

A number divisible by 2 is called an even number. Examples of even numbers are: 2, 4, 6, 8, 10, etc. A number not divisible by 2 is called an odd number. Examples: 1, 3, 5, 7, 9 etc.

  • Prime Numbers

A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself. e.g., 1, 3, etc. 2 is the only number which is a prime as well as an even number.

  • Composite Numbers

A natural number, other than 1, is a composite number when it is divisible by 1, the number itself and at least one other number. For example, 9 is a composite number because it is divisible by 1, 3, and 9.

  • Perfect Number

If the sum of the divisors of a number, excluding the number itself,  is equal to the number, then it is said to be a perfect number. For example, 6 is a perfect number because sum of all divisors = 1 + 2 + 3 + 6 = 12= 2(6).

 

POINTS TO REMEMBER

  • The numbers written by the symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are known as Digits.
  • The numbers which are divisible by 2 such as 2, 4, 6, etc are known as Even Numbers.
  • The numbers which are not divisible by 2 such as 1, 3, 5 etc are known as Odd Numbers.
  • The numbers 1, 2, 3,……..which are used for counting are known as the natural numbers. Thus 528 is a natural number but 0, − 3, 14∙6, and 3∕4 etc. are not Natural Numbers.
  • If zero is included in natural numbers, the set so obtained is known as whole numbers. For example 0, 1, 2, 3, ....etc are whole numbers but  − 1, −2, 3∙2, 1/9 are not Whole Numbers.
  • The whole numbers positive as well as negative are known as Integer. For example −7, −6, −5,……… 0, 1, 2, 3, 4 etc are integers but √5, 0∙32, −1∙2, and 1/3 etc are not integers.
  • The numbers, which can be expressed in the form of p/q, where p and q are integers but q ≠ 0, are known as Rational Numbers. For example 25, −62, -2/3 etc. are rational numbers.
  • A number, which can not be expressed in terminating decimal, is known as an Irrational Number. For example √5, 3 + √2, etc are irrational numbers.
  • The numbers, which are either rational or irrational, are known as Real Numbers. For example 3, −2, √7 etc are real numbers.
  • A natural number other than 1, is a Prime Number if and only if it is divisible by 1 and the number itself. For example 2 is a prime number but 6 it not a prime number.
  • A natural number other than 1 is a Composite Number if it is divisible by 1, the number itself and at least one other number. For example 6 is a composite number because it is divisible by 1, 2, 3, and 6.

      Note­:

  • 1 is neither prime number nor composite.
  • 2 is the only number which is both prime as well as even number.

            Properties of Rational and Irrational Numbers

  • The sum or product of two rational numbers is always a rational number.
  • The sum or product of two irrational numbers is sometimes a rational number and sometimes an irrational number.
  • The sum or product of a rational number and an irrational number is always an irrational number.
  • There are infinite rational numbers and infinite irrational numbers between two rational numbers or two irrational numbers.

           Some Important Results on Numbers

  • Sum of natural numbers = {1 + 2 + 3 +4 +5 + …+ n} 
  • Sum of first n odd numbers = {1 + 3 + 5 + 7 + …..} =  $n^2$n2 
  • Sum of first n even numbers = {2 + 4 + 6 + 8 + …} = n ( n + 1)
  •  
  • If n is odd n  $\left(^{n^2-1}\right)$(n21)   is divisible by 24 and if n is prime greater than 3, then  $\left(^{n^2-1}\right)$(n21)  is divisible by 24.
  • If n is odd  $\left(2^n+1\right)$(2n+1) is divisible by 3 and if n is even then  $\left(2^n+1\right)$(2n+1) is divisible by 3.
  •  $n\left(n^4-1\right)$n(n41) is always divisible by 30 for a natural number n > 1.
  •  If n is odd  $\left(2^{2n}+1\right)$(22n+1)   is divisible by 5 and if n is even then  $\left(2^{2n}-1\right)$(22n1) is divisible by 5.
  • If n is odd,  $5^{2n}+1$52n+1 is divisible by 13 and if n is even,  $5^{2n}-1$52n1 is divisible by 13 and 12.
  • The fifth power of any single digit number has the same unit digit as the number itself.
  • If p is prime number, 1 + (p - 1)! is divisible by p.
  • If p is prime number, and N is a prime to p, then   is a multiple of p.
  • If p is prime greater than 3, then  $\left(p^2-1\right)$(p21)  is divisible by 12.
  • The product of r consecutive integers is divisible by r!
  • If p is a prime number, then  $\left(a+b+c\right)^p=a^p+b^p+c^p+.....M\left(p\right)$(a+b+c)p=ap+bp+cp+.....M(p)  Where, M (p) is a multiple of p.
  • If  $n=p_1^{\alpha_1}.p_2^{\alpha_2}.......p_{\gamma}^{\alpha_{\gamma}}$n=p1α1.p2α2.......pγαγ where  $p_{1,}p_{2,}........,p_{\gamma,}$p1,p2,........,pγ,   are distinct prime numbers, then
  1. The number of divisors of n

       

  1. The sum of divisors of n

             

  1. Product of all the division

             

Tests of Divisibility

  • A number is divisible by 2 if it ends in zero or in a digit which is a multiple of 2.
  • A number is divisible by 3 if the sum of its digits is divisible by 3.
  • A number is divisible by 4 if the number formed by the last two digits i.e., tens and units, is divisible by 4.
  • A number is divisible by 5 if it ends in zero or 5.
  • A number is divisible by 6 if it is divisible by 2 as well as 3.
  • A number is divisible by 8 if the number formed by it’s last three digits i.e., hundreds, tens and units, is divisible by 8.
  • A number is divisible by 9 if the sum of its digits is divisible by 9.
  • A number is divisible by 10 if it ends in zero.
  • A number is divisible by 11 if, the difference between the sums of the digits in the even and odd places is zero or a multiple of 11.
  • A number is divisible by 12, if it is divisible by both 4 and 3.
  • A number is divisible by 14, if it is divisible by both 2 and 7.
  • A number is divisible by 15, if it is divisible by both 3 and 5.
  • A number is divisible by 16, if the number formed by the last 4 digits is divisible by 16.
  • A given number is divisible by 24, if it is divisible by both 3 and 8.
  • A given number is divisible by 40, if it is divisible by both 5 and 8.
  • A given number is divisible by 80, if it divisible by both 5 and 16.

            Modulus of a Real Number

 

         Modulus of a real number x is defined as

 

           Thus, |7| = 7 and |-7| = - (-7) = 7.