Number System : Definitions, Operations & Divisibility

Types of Numbers
Natural Numbers: The numbers 1,2,3,4.... are called natural numbers or positive integers.
Whole Numbers: The numbers 0,1,2,3.... are called whole numbers. Whole numbers include “0”.
Integers: The numbers .... -3, -2, -1, 0, 1, 2, 3,.... are called integers. You will see questions on integers in almost all the exams where you see number system aptitude questions.
Negative Integers: The numbers -1, -2, -3, ... are called negative integers.
Positive Fractions: The numbers(2/3) ,(4/5) ,(7/8) ... are called positive fractions.
Negative Fractions: The numbers -(6/8) ,-(7/19) , -(12/17) ... are called negative fractions.
Rational Numbers: Any number which is a positive or negative integer or fraction, or zero is called a rational number. A rational number is one which can be expressed in the following format ⇒(a/b) , where b ≠ 0 and a & b are positive or negative integers.
Irrational Numbers: An infinite non-recurring decimal number is known as an irrational number. These numbers cannot be expressed in the form of a proper fraction a/b where b ≠ 0. e.g.√2 , √5 , Π, etc.
Surds: Any root of a number, which cannot be exactly found is called a surd. Essentially, all surds are irrational numbers. e.g. √2 , √5 etc.
Even Numbers: The integers which are divisible by 2 are called even numbers e.g. -4, 0, 2, 16 etc.
Odd Numbers: The integers which are not divisible by 2 are odd numbers e.g. -7, -15, 5, 9 etc.
Prime Numbers: Those numbers, which are divisible only by themselves and 1, are called prime numbers. In other words, a number, which has only two factors, 1 and itself, is called a prime number. e.g. 2, 3, 5, 7, etc.
Note: 2 is the only even prime number.
There are 25 prime numbers upto 100. These are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 & 97. These should be learnt by heart.
Co-Prime Numbers: Two numbers are considered to be prime to each other if their HCF is 1. e.g. 5 and 24 are prime to each other. In other words, 5 and 24 are co-prime.
To check whether a number is prime, e.g. 79, we do not need to check all the factors below 79. The square of 8 is 64 & that of 9 is 81. Therefore, check if any of the prime numbers less than 9 is a factor of 79. The prime numbers 2, 3, 5, 7 are not the factors of 79. So, 79 is a prime number.
Composite Number: A number, which has factors other than itself and 1, is called a composite number. e.g. 9, 16, 25....
Note: 1 is neither a composite number nor a prime number.
Illustration: Which of the following is a prime number?
1. 31
2. 91
3. 87
4. 57
Sol: Here 91 is divisible by 7, so 91 is not a prime number. 87 is divisible by 3. So it is also not a prime number. 57 is not a prime number as it is divisible by 3. 31 has only two factors 1 and 31, so it is a prime number.
Consecutive Numbers: Numbers arranged in increasing order and differing by 1 are called consecutive numbers. e.g. 4, 5, 6, 7 etc.
Real Numbers: The natural numbers, integers, whole numbers, rational numbers and irrational numbers constitute the set of real numbers. Every real number can be represented by a point on a number line.
Perfect Numbers: If the sum of all the factors of a number excluding the number itself happens to be equal to the number, then the number is called as perfect number. 6 is the first perfect number. The factors of 6 are 1, 2, 3 & 6. Leaving 6 the sum of other factors of 6 are equal to 6. The next three perfect numbers after 6 are: 28,496 and 8128.
Complex Numbers: Complex numbers have a real and an imaginary component; e.g. (√-2 - 4), (2 +√-3 ), etc. Square root of any negative number is an imaginary number - e.g.√-2 ,√-3 . The square root of a negative number does not exist in the real sense. Such numbers are called imaginary numbers.
Fibonacci Numbers: The numbers, which follow the following series are known as Fibonacci numbers. E.g. 1,1,2,3,5,8,13,21..... The series is obtained by adding the sum of the preceding two numbers. In general for a Fibonacci number X, Xi+2 = Xi+1 + Xi.
Operations on Odd & Even Numbers
  1. Addition or subtraction of any two odd numbers will always result in an even number or zero. E.g. 2 + 5 = 7; 11 - 5 = 6.
  2. Addition or subtraction of any two even numbers will always result in an even number or zero. E.g. 6 + 8 = 14; 8 - 2 = 6.
  3. Addition or subtraction of an odd number from an even number will result in an odd number. E.g. 8 + 5 = 13; 12 - 5 = 7.
  4. Addition or subtraction of an even number from an odd number will result in an odd number.  E.g. 7 + 6 = 13; 11 - 6 = 5.
  5. Multiplication of two odd numbers will result in an odd number. E.g. 5 × 7 = 35.
  6. Multiplication of two even numbers results in an even number. E.g. 4 × 6 = 24.
  7. Multiplication of an odd number and an even number will result in an even number. E.g. 7 × 4 = 28.An odd number raised to an even or an odd power is always odd.
  8. An even number raised to an odd or an even power is always even.
Divisibility Rules
  1. Divisibility rule of 2 - A number is divisible by 2 when its units place is 0 or divisible by 2. e.g. 120, 138.
  2. Divisible by 3 - 19272 is divisible by 3 when the sum of the digits of 19272 i.e. 21 is divisible by 3. Note that if n is odd, then 2n + 1 is divisible by 3 and if n is even, then 2n - 1 is divisible by 3.
  3. Divisibility rule of 4 - A number is divisible by 4 when the last two digits of the number are 0s or are divisible by 4. As 100 is divisible by 4, it is sufficient if the divisibility test is restricted to the last two digits. e.g. 145896, 128, 18400
  4. Divisibility by 5 - A number is divisible by 5, if its unit’s digit is 5 or 0. e.g. 895, 100
  5. Divisibility rule of 7:  How to check whether a number is divisible by 7 or not. Let us check divisibility of 343. Double the last digit of the given number: 3 x 2 = 6, subtract it from the rest of the number: 34 - 6 = 28. Check if the difference is divisible by 7 i.e. 28 is divisible by 7, therefore 343 is divisible by 7.
  6. Divisibility rule of 8 - A number is divisible by 8, if the last three digits of the number are 0s or are divisible by 8. As 1000 is divisible by 8, it is sufficient if the divisibility test is restricted to the last three digits e.g. 135128, 45000.
  7. Divisibility rule of 9 - A number is divisible by 9, if the sum of its digits is divisible by 9. e.g. 810, 92754
  8. Divisibility rule of 11- A number is divisible by 11, if the difference between the sum of the digits at odd places and the digits at even places of the number is either 0 or a multiple of 11. e.g. 121, 65967. In the first case 1 + 1 - 2 = 0. In the second case 6 + 9 + 7 = 22 and 5 + 6 = 11 and the difference is 11. Therefore, both these numbers are divisible by 11.
  9. Divisibility rule of 25- Any number is divisible by 25, if the last 2 digits of the number are 00 or they are multiples of 25.e.g. 2358975 is divisible by 25 as its last two digits are 75.
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Some important points
  1. The greatest number of ‘n’ digits will have ‘n’ 9s straightaway e.g. the greatest five-digit number will have five 9s i.e. it will be 99999.
  2. The smallest number of ‘n’ digits has 1 as the leftmost digit and the rest all the digits are zeroes. E.g. the smallest five digit number is 10000.
  3. The sum of a two-digit number and a number formed by reversing its digits is divisible by 11. E.g. 34 + 43 = 77, which is divisible by 11. At the same time, the difference between those numbers is divisible by 9. e.g. 43 – 34 = 9, which is divisible by 9.
  4. ∑n= (n(n+1)/2) , ∑n is the sum of first n natural numbers.
  5. ∑n2 =(n(n+1)(2n+1)/6) , ∑n2 is the sum of first n perfect squares.
  6. ∑n3 =(n2(n+1)2/4) = (∑n)2, ∑n3 is the sum of first n perfect cubes.
  7. Local Value: The local value or face value of a digit in a number is the actual value of the digit. e.g. The local value of 5 in 7415236 is 5.
  8. Place Value: The place value is the value of the digit at which it is placed. e.g. the place value of 5 in 7415236 is 5000 as 5 is placed at the thousand's place in the number.