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www.paperadda.com – Online Education and Examination Portal, , Natural Numbers : Numbers which are used for counting the objects are called, natural numbers. They are denoted by N., N = { 1, 2, 3………………..}, All positive integers are natural numbers., , Whole numbers :- When ‘zero’ is included in the natural numbers, they are known as, whole numbers., They are denoted by W., W= { 0, 1, 2, 3……………….}, Integers : All natural numbers, zero and negatives of natural numbers are called as, integers., They are denoted by I., I = { ………………..,-3, -2, -1, 0, 1 , 2, 3………………}, Rational numbers : The numbers which can be expressed in the form of, are integers and, are called rational numbers, , where P and Q, , They are called by Q., , Irrational numbers : The numbers which cannot be written in the form of, Q are integers and, are called irrational numbers., , where P and, , When these numbers are expressed in decimal form, they are neither terminating nor, repeating., , Number System, , Page 1

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www.paperadda.com – Online Education and Examination Portal, , Real numbers : Real numbers include both rational as well as irrational numbers., , Positive or negative, large or small, whole numbers or decimal numbers are all real, numbers., , e.g.= 1, 13.79, -0.01,, , etc., , Imaginary numbers : An imaginary number is a complex number that can be written as a, real number multiplied by the imaginary unit ‘i’ which is defined by its properly, Note : Zero (0) is considered to be both real and imaginary number., Prime number : A prime number is a natural number greater than 1 and is divisible only, by 1 and itself., e.g.2, 3, 5, 7, 11, 13, 17 ,19 ………….etc., Note :- 2 is the only even prime number., Composite Numbers : A number, other than 1, which is not a prime number is called a, composite number ., E.g. 4, 6, 8, 9, 10, 12, 14, 15 ……….etc., Note :1 1 is neither a prime number nor a composite number., 2 there are 25 prime numbers between 1 and 100., To find whether a number is prime or notTo check whether the number is prime or not,, 1 We take an integer larger than the square root of the number. Let the number be ‘k’., 2 Test the divisibility of the given number by every prime number less than ‘k’., 3 If it is not divisible by any of them, then the given number is prime otherwise it is a, composite number., E.g.= Is 881 a prime number ?, Sol- The appropriate square root of 881 is 30., , Number System, , Page 2

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www.paperadda.com – Online Education and Examination Portal, , Prime number less than 30 are 2, 3, 5, 7, 11, 13, 17,19, 23,29., , 881 is not divisible by any of the above numbers, so it is a prime minister., Co-prime numbers : Two numbers are co-prime of their HCF is 1., E.g. (2,3), (3,4) , (5,7), (3,13) etc., Even numbers : The number which is divisible by 2 is called even number., E.g. – 2, 4, 6, 8……………., Odd numbers – The number which is not divisible by 2 is called odd number., e.g.= 3, 5, 7, 9………….., Consecutive numbers : A series of numbers in which the succeeding number is greater, then the preceding number by 1 is called a series of consecutive numbers., i.e., Difference between two consecutive numbers is 1., Some Rules on Counting Numbers, 1. Sum of all the first n natural numbers, , Q. Find the sum of first 20 natural numbers., Ans- Sum of 1 to 20, , Q. Find the sum of numbers from 11 to 20., , Number System, , Page 3

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www.paperadda.com – Online Education and Examination Portal, , 2. Sum of first n old numbers =, , Q. What is the sum of first 10 odd numbers ?, Ans- Sum of first 10 odd numbers =, Q. Find the sum of 9+11+13+………+29, Ans – 1+3+5+…………+29=, (as there are 15 odd numbers from 1 to 29), 1+3+5+7=, 9+11+13+29=225-16=209, 3. Sum of first n even numbers, , Q. What is the sum of even numbers between 1 and 50 ?, , Q. Find the value of 12+14+……….+30., Ans- (2+4+6+…….+30) has 15 even numbers, , Number System, , Page 4

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www.paperadda.com – Online Education and Examination Portal, 2+4+6+………30=15(15+1)=240, , Similarly 2+4+6+8+10= 5(5+1)=30, 12+14………….+30=240-30=210, 4. Sum of squares of first n natural numbers, , Divisibility Rules, Divisibility by 2 : Number Whose last digit is either even or zero is divisible by, 2., Divisibility by 3 : If the sum of the digits of a number is divisible by 3, the, number is also divisible by 3., Divisibility by 4 :If the last two digits of a Number is divisible by 4 or the, number having two or more zeros at the end, the numbers is divisible by 4., , Number System, , Page 5

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www.paperadda.com – Online Education and Examination Portal, ,  Divisibility by 5: If a number is divisible by 5 or 0, the number is divisible by, 5., Divisibility by 6 : If a number is divisible by both 2 and 3 the number is also, divisible by 6., , Divisibility by 8 : If the last three digits of a number is divisible by 8 or the last, three digits of a number are zeros, the number is divisible by 8., Divisibility by 9 : If the sum of all the digits of a number is divisible by, 9, the number is also divisible by 9., Divisibility by 10. The number which ends with zero is divisible by 10.,  Divisibility by 11. If the sums of digits at odd and even places are equal, , or differ by a number divisible by 11, then the number is also divisible by, 11., , Divisibility by 12. The number which is divisible by both 3 and 4 is also, divisible by 12., , Number System, , Page 6