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Types of Numbers-Important short notes on Number System for IBPS PO/ SSC exams

In this article, we will look at some important short notes on Number System which is a very important topic asked in IBPS, SSC and other banking exams. So read them carefully and keep practising at GeekVis. We start with the types of Numbers used in Number system.

Review of fundamentals on Number system:

Numbers are divided into the following categories:

  1. Natural numbers: Counting numbers 1,2,3,4,5.. are known as natural numbers. The set of all natural numbers is represented as; N = {1,2,3,4,5,....}.
  2. Whole numbers: If 0 is included in the set of natural numbers then the collection of numbers, 0,1,2,3,4,5...are called whole numbers. The set of all whole numbers is represented as; W = {0,1,2,3,4,5,....}.
  3. Integers: All counting numbers, including negative numbers and zero are called integers. The set of all integers is represented as; Z = {....-4,-3,-2,-1,0,1,2,3,4..}. 
  4. Positive integers: The set represented as I+={1,2,3,4,5...}, which include all positive numbers, is called set of positive integers. The set of natural numbers and positive integers is same. 
  5. Negative Integers: The set represented by I-={-1,-2,-3,...}, which includes all negative numbers, is called set of negative integers.
  6. Non negative Integers: The set represented by {0,1,2,3....} is called set of non negative integers. 
  7. Rational numbers: These are numbers of the form 'p/q', where p and q are both integers and . Some examples of rational numbers are  etc. To represent the set of all rational numbers, we use the letter 'Q' and the set representation is 
  8. Irrational numbers: These are numbers which are expressed in a decimal form and are neither terminating nor repeating decimals. For example, 
  • The number 0 is neither negative nor positive.
  • Every natural number 'a' can be written as , which means every natural number is a rational number.
  • The number 0 can also be expressed as  and every non-zero integer 'a' can be written as , so every integer is a rational number.
  • Every rational number has a property that when they are expressed in decimal form, they can be expressed either in terminating decimals or in non-terminating repeating decimals. For example: , etc. The recurring decimals are denoted as 
  • The exact value of  is not 22/7. The number 22/7 is a rational number, but the actual value of  is an irrational number. So the value 22/7 is an approximate value of .

In addition to the above numbers, there are some other type of numbers, as mentioned below:

Real numbers: The rational and irrational numbers combined together are called real numbers, e.g. . To denote the set of real number we use the letter R. Remember this, the sum, the difference or product of a rational and irrational number is irrational.

Even numbers: These are numbers which are exactly divisible by 2, e.g. 2,4,6,8,10 etc. In general all even numbers are expressed as '2n', where 'n' is an integer. So -2, -4, -6 etc. are also even numbers.

Odd numbers: These are numbers which are not exactly divisble by 2 and are generally expressed as '2n+1', e.g. 1,3,5,7... etc. In the general expression, like in even numbers, 'n' is an odd number so -1, -3, -5 are also odd numbers.

Prime numbers: All natural numbers other than 1, is a prime number if it is divisible by 1 and itself only. For example 2,3,5,7...etc.

Composite numbers: Natural numbers other than the prime numbers and greater than 1 are knwon as composite numbers. For example, 4,6,8,9,12 etc. 

Perfect numbers: These are defined as those numbers, whose sum of the divisors excluding the number itself is equal to the number itself, them the number is called perfect number. For example, 6 = 1+2+3, where 1,2 and 3 are divisors of 6 so 6 is a perfect number.

Trick useful in solving questions:

If 'n' is odd, then

  • n(n2-1) is divisble by 24
  • 2n+1 is divisible by 3
  • 22n+1 is divisible by 5
  • 52n+1 is divisble by 13

If 'n' is even then

  • 2n-1 is divisble by 3
  • 22n-1 is divisble by 5
  • 52n-1 is divisible by 13

Even number raised to even or odd number is even.

Odd number raised to even or odd number is oddd.

We hope the above notes have given you a good insight into the world of numbers and number system. In the next article, we move ahead in the chapter and look at some problem solving tricks on Numbers. 

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