1 Number System Introduction In this chapter, we will study about the number system and number line. We will also learn about the four fundamental operations on whole numbers and their properties. Natural Numbers (N) The counting numbers 1, 2, 3, 4, are called natural numbers. The set of natural numbers is given by N = {1, 2, 3, 4 }. Even natural numbers (E) The collection of natural numbers, divisible by 2, forms the set of even natural numbers. It is given by E = {2, 4, 6, 8, }. An even natural number can be represented by 2n where n N. Odd natural numbers (O) The collection of natural numbers, not divisible by 2, forms the set of odd natural numbers. It is given by O = {1, 3, 5, 7, }. An odd natural number can be represented by 2n 1, where n N. Whole Numbers (W) All natural numbers together with zero form the collection of whole numbers. The set of whole numbers is given by W = {0, 1, 2, 3, }. Integers (Z) The collection of negatives of natural numbers, zero and natural numbers form the set of integers. It is denoted by Z or I. So, Z = {, 3, 2, 1, 0, 1, 2, 3, }. The set of positive integers is given by Z + = {1, 2, 3, 4, } and the set of negative integers is given by Z = { 1, 2, 3, 4, }. Prime Numbers (P) A natural number greater than 1 which is divisible only by 1 and itself is known as a prime number. The set of prime numbers is given by P = {2, 3, 5, 7, }. Natural numbers which are not prime numbers are known as composite numbers. For example, 4, 6, 8, 9, etc. Face Value The face value or true value of a digit at any place in a numeral is the value of the digit itself. For example, the face value of 5 in 659 is 5. Place Value The place value or local value of a digit in a numeral is the product of the face value and the value of the place of the digit in the given numeral. For example, the place value of 7 in 37,580 is 7,000. 1
Number Line A number line is a straight line on which points are marked to divide it into equal parts. The middle point of this line is marked as zero. To the left of zero, equally spaced points are marked as negative numbers and to the right of zero, they are marked as positive numbers. Thus, the number line is obtained as given below. 4 3 2 1 0 1 2 3 4 For any two numbers on the number line, the number which is on the right is greater than the number on the left. Example 1: How many 2-digit whole numbers are there? Solution: 10, 11, 12,, 99 are 2-digit whole numbers. \ Number of 2-digit numbers = 99 10 + 1 = 90 Example 2: How many 4-digit whole numbers are there? Solution: 1,000, 1,001, 1,002,, 9,999 are 4-digit whole numbers. \ Number of 4-digit numbers = 9,999 1,000 + 1 = 9,000 Example 3: Write all the 2-digit numbers formed using the digits 3, 6 and 4 when: (a) repetition of digits is allowed (b) repetition of digits is not allowed Solution: (a) The 2-digit numbers are 36, 63, 64, 46, 34, 43, 33, 66 and 44. (b) The 2-digit numbers are 36, 63, 64, 46, 34 and 43. Example 4: Write all the 3-digit numbers formed using the digits 9, 7 and 5 when repetition of digits is not allowed. Solution: The 3-digit numbers are 975, 957, 579, 597, 759 and 795. Example 5: What is the smallest 6-digit number formed when: (a) repetition of digits is allowed (b) repetition of digits is not allowed Solution: (a) The 6-digit number is 1,00,000. (b) The 6-digit number is 1,02,345. Example 6: Find the difference between the two place values of 8 in the number 5,86,890. Solution: In 5,86,890, one 8 is at hundred s place. So, its place value is 800. The other 8 is at ten thousand s place. So, its place value is 80,000. Difference between the two place values = 80,000 800 = 79,200 Exercise 1.1 1. Write True or False. (a) All natural numbers are whole numbers. (b) Zero is the only number whose face value and place value is same. Maths Info Largest number of the series Smallest number of the series + 1 = Total numbers in the series 2
(c) There are infinite even prime numbers. (d) The set of negative integers is finite. (e) If x > y, then x > y. 2. Using a number line, fill in the boxes with < or >. (a) 0 16 (b) 24 24 (c) 4 5 (d) 19 9 (e) 5 9 (f) 13 16 3. From the numbers 16, 19, 21, 23, 31, 37, 39 and 51, write which are: (a) prime (b) composite 4. Write all the 2-digit numbers formed using the digits 2, 0 and 7 when: (a) repetition of digits is allowed (b) repetition of digits is not allowed 5. Write all the 3-digit numbers formed using the digits 4, 9, 7 and 5 when repetition of digits is not allowed. 6. How many 3-digit whole numbers are there? 7. How many 5-digit whole numbers are there? 8. Find the difference between the smallest 4-digit natural number and the smallest 4-digit whole number formed when repetition of digits is allowed. 9. Write the greatest 5-digit number using distinct natural numbers. Fundamental Operations We have already studied the operations of addition, subtraction, multiplication and division on whole numbers. Now, let s study some properties of these fundamental operations. Properties of addition Closure The sum of whole numbers is always a whole number. Mathematically, if a and b are whole numbers, then a + b is also a whole number. For example, (a) 32 + 0 = 32 (b) 206 + 19 = 225 Commutative The sum of two whole numbers remains the same irrespective of the order in which they are added. Mathematically, a + b = b + a, where a and b are whole numbers. For example, (a) 25 + 9 = 34 = 9 + 25 (b) 130 + 21 = 151 = 21 + 130 Associative The sum of any three whole numbers remains the same even if their grouping is changed. Mathematically, (a + b) + c = a + (b + c) = a + b + c, where a, b and c are whole numbers. For example, (a) (34 + 61) + 75 = 95 + 75 = 170 Also, 34 + (61 + 75) = 34 + 136 = 170 \ (34 + 61) + 75 = 34 + (61 + 75) = 170 3
(b) (104 + 88) + 112 = 192 + 112 = 304 Also, 104 + (88 + 112) = 104 + 200 = 304 \ (104 + 88) + 112 = 104 + (88 + 112) = 304 Additive identity The sum of any whole number and 0 is always the number itself. So, 0 is called the additive identity for whole numbers. Mathematically, a + 0 = 0 + a = a, where a is any whole number. For example, (a) 39 + 0 = 0 + 39 = 39 (b) 210 + 0 = 0 + 210 = 210 Cancellation law Cancellation law states that if a, b and c are whole numbers, then a+ b = c+ b a = c For example, a + 14 = 5 + 14 a = 5 Properties of subtraction Closure Whole numbers are not closed under subtraction as the difference of two whole numbers need not be a whole number. For example, 5 10 = 5, which is not a whole number. Commutative Whole numbers do not obey commutative law under subtraction as a b need not be equal to b a, where a and b are whole numbers. For example, 12 6 6 12. Associative Whole numbers do not obey associative law under subtraction as (a b) c need not be equal to a (b c), where a, b and c are whole numbers. For example, (5 2) 1 5 (2 1) as (5 2) 1 = 2 and 5 (2 1) = 4 Cancellation law Cancellation law states that if a, b and c are whole numbers, then a b = c b a = c. For example, a 19 = 26 19 a = 26 Properties of multiplication Closure The product of whole numbers is always a whole number. Mathematically, if a and b are whole numbers, then a b is also a whole number. For example, (a) 12 9 = 108 (b) 125 8 = 1,000 Commutative The product of two whole numbers remains the same irrespective of the order in which they are multiplied. Mathematically, a b = b a, where a and b are whole numbers. For example, (a) 95 6 = 570 = 6 95 (b) 107 7 = 749 = 7 107 4
Associative The product of any three numbers remains the same even if their grouping is changed. Mathematically, (a b) c = a (b c) = a b c where a, b and c are whole numbers. For example, (a) (5 9) 3 = 45 3 = 135 Also, 5 (9 3) = 5 27 = 135 \ (5 9) 3 = 5 (9 3) (b) (3 7) 6 = 21 6 = 126 Also, 3 (7 6) = 3 42 = 126 \ (3 7) 6 = 3 (7 6) Property of zero The product of any whole number and 0 is always 0. For example, (a) 4 0 = 0 (b) 947 0 = 0 Multiplicative identity The product of any whole number and 1 is the number itself. So, number 1 is called the multiplicative identity for whole numbers. Mathematically, a 1 = 1 a = a, where a is any whole number. For example, (a) 34 1 = 1 34 = 34 (b) 104 1 = 1 104 = 104 Cancellation law Cancellation law states that if a, b and c are whole numbers, then a b = c b a = c. For example, 19 6 = a 6 a = 19 Distributive According to distributive property of multiplication over addition, if a, b and c are whole numbers, then a (b + c) = (a b) + (a c) Similarly, by distributive property of multiplication over subtraction, we have a (b c) = (a b) (a c) For example, (a) 65 (5 + 3) = (65 5) + (65 3) = 325 + 195 = 520 Also, 65 (5 + 3) = 65 8 = 520 \ 65 (5 + 3) = (65 5) + (65 3) (b) 3 (45 8) = (3 45) (3 8) = 135 24 = 111 Also, 3 (45 8) = 3 37 = 111 \ 3 (45 8) = (3 45) (3 8) Example 7: Find the following products using distributive property. (a) 105 33 (b) 999 55 Solution: (a) 105 33 = (100 + 5) 33 5
Properties of division Closure 6 = (100 33) + (5 33) [Using a (b + c) = (a b) + (a c)] = 3,300 + 165 = 3,465 (b) 999 55 = (1,000 1) 55 = (1,000 55) (1 55) [Using a (b c) = (a b) (a c)] = 55,000 55 = 54,945 The quotient obtained on dividing two whole numbers need not be a whole number. For example, 10 4 = 10 4 = 5, which is not a whole number. So, whole numbers do not obey the closure property 2 under division. Commutative Whole numbers do not obey commutative law under division as a b need not be equal to b a, where a and b are whole numbers. For example, 25 5 5 25 as 25 5 = 5 and 5 25 = 1 5 Associative Whole numbers do not obey associative law under division as (a b) c need not be equal to a (b c), where a, b and c are whole numbers. For example, (45 9) 3 45 (9 3) as (45 9) 3 = 5 3 and 45 (9 3) = 45 3 = 9 Property of zero If 0 is divided by any non-zero whole number, the quotient is always 0. For example, (a) 0 9 = 0 (b) 0 124 = 0 Identity If a is any whole number then a 1 = 1. For example, (a) 21 1 = 21 (b) 105 1 = 105 Division algorithm If a and b are whole numbers and a > b (b 0), then there exist two other whole numbers q and r such that a = bq + r, where r = 0 or r < b. This relation is known as division algorithm or rule of division. We can also say that, Maths Info Dividend = Divisor Quotient + Remainder Division by 0 is not defined. Example 8: Divide 1,509 by 27 and verify the division algorithm. Solution: Dividend = 1,509, divisor = 27, quotient = 55, remainder = 24 55 Divisor Quotient + Remainder = 27 55 + 24 27 1509 135 = 1,485 + 24 = 1,509 159 = Dividend 135 \ Division algorithm is verified. 24
Example 9: Find the number which when divided by 28 gives the quotient 13 and remainder 4. Solution: Divisor = 28, quotient = 13 and remainder = 4 Required number = Dividend = Divisor Quotient + Remainder Dividend = 28 13 + 4 = 364 + 4 = 368 Example 10: Find the number which on dividing 2,558 gives 150 as quotient and 8 as remainder. Solution: Dividend = 2,558, quotient = 150 and remainder = 8 Dividend = Divisor Quotient + Remainder 2,558 = Divisor 150 + 8 2,558 8 = Divisor 150 2,550 150 = Divisor Divisor = 17 \ Required number = 17 Example 11: Find the smallest 4-digit number which is exactly divisible by 24. Solution: The smallest 4-digit number is 1,000. 1,000 divided by 24 gives 16 as remainder. So, if we add 24 16 = 8 in 1,000, then the sum will be exactly divisible by 24. Hence, the required number is 1,008. Note: If we subtract 16 from 1,000, then also the result will be exactly divisible by 24, but in that case the number will be a 3-digit number. Example 12: Find the largest 5-digit number which is exactly divisible by 52. 1923 Solution: The greatest 5-digit number is 99,999. 52 99999 52 99,999 divided by 52 gives 3 as remainder. 479 So, the required number is 99,999 3 = 99,996 468 119 104 159 156 3 Exercise 1.2 1. Fill in the blanks. (a) 657 + = 757 + 657 (b) 35 + (49 + ) = 35 + 49 + 97 (c) 74 ( 201) = 74 225 74 201 (d) (109 + 35) = 24 109 + 24 35 24 41 1000 96 40 24 16 7
2. Find the sum of the following numbers using the most convenient grouping. (a) 3,526, 516, 474 (b) 1,486, 285, 1,014, 215 (c) 2,547, 108, 242, 1,953 (d) 500, 516, 358, 484, 442 3. Find the following products using distributive property. (a) 456 102 (b) 999 54 (c) 13 955 (d) 1,004 36 4. Find the following products using the most convenient grouping. (a) 25 595 4 (b) 5 123 20 (c) 16 5,456 125 5. Divide and verify the division algorithm. (a) 2,542 44 (b) 8,924 58 (c) 17,654 251 6. Find the number which when divided by 37 gives the quotient 16 and remainder 8. 7. Find the largest 6-digit number which is exactly divisible by 41. 8. Find the smallest 6-digit number which is exactly divisible by 180. 9. On dividing 3,487 by 112, the remainder is found to be 15. Find the quotient. 10. Find the number which on dividing 5,498 gives 43 as quotient and 123 as remainder. 8 SUMMARY The counting numbers 1, 2, 3, 4, are called natural numbers. The collection of natural numbers, divisible by 2, forms the set of even natural numbers. The collection of natural numbers, not divisible by 2, forms the set of odd natural numbers. All natural numbers together with zero form the collection of whole numbers. The collection of negatives of natural numbers, zero and natural numbers forms the set of integers. A natural number greater than 1 which is divisible only by 1 and itself is known as a prime number. The face value of a digit at any place in a numeral is the value of the digit itself. The place value of a digit in a numeral is the product of the face value and the value of the place of the digit in the given numeral. A number line is a straight line on which points are marked to divide it into equal parts. The set of whole numbers is closed, commutative and associative under addition and cancellation law also holds true. 0 is the additive identity of whole numbers. The set of whole numbers is not closed, not commutative and not associative under subtraction but cancellation law holds true. The set of whole numbers is closed, commutative and associative under multiplication and cancellation law also holds true. 1 is the multiplicative identity of whole numbers. For any three whole numbers a, b and c, a (b + c) = (a b) + (a c) and a (b c) = (a b) (a c) The set of whole numbers is not closed, not commutative and not associative under division. Division algorithm states that if a and b are whole numbers (a > b, b 0), then there exist two other whole numbers q and r such that a = bq + r, where r = 0 or r < b.
REVIEW EXERCISE Mental Maths 1. Write True or False. (a) 2 is the only even prime number. (b) Whole numbers are closed under addition and subtraction. (c) Whole numbers are associative under multiplication. (d) 1 is the additive identity of whole numbers. (e) For whole numbers a, b and c, a (b + c) = (a b) + (b c). (f) If a is a whole number which is divisible by b (b 0) and a = bq + r, then r = 0. 2. Fill in the blanks. (a) { 1, 2, 3, 4, } is the set of integers. (b) 86 ( + ) = (86 63) + (86 95) (c) is the multiplicative identity of whole numbers. (d) Dividend = Divisor + Solve and Answer 1. What is the difference in the place values of two odd digits in each of the following? (a) 2,756 (b) 56,289 (c) 24,305 (d) 48,770 2. Write all the 4-digit numbers using the digits 1, 0, 9 and 7 without repetition. 3. Write all possible 2-digit numbers using the digits 5, 4 and 8 if repetition of digits is not allowed. 4. How many natural numbers are there between 9 and 29, both inclusive? 5. How many natural numbers are there between 80 and 125, both inclusive? 6. What is the product of the smallest 4-digit number and the greatest 3-digit number? 7. Find the sum of the smallest and the largest 4-digit number formed by using the digits 0, 3, 5 and 7 without repetition. 8. Find the sum of the following numbers and verify the commutative law of addition. (a) 549, 6,134 (b) 954, 24,055 (c) 44,096, 55,013 9. Find the following products using distributive property or the most convenient grouping. (a) 106 643 (b) 25 84 8 (c) 474 96 (d) 744 102 (e) 50 8 45 (f) 40 552 50 10. Find the greatest 5-digit number which is exactly divisible by 225. 11. Find the smallest 4-digit number which is exactly divisible by 23. 12. Find the number which on dividing 66,495 gives 554 as quotient and 15 as remainder. 13. Find the remainder when 54,978 is divided by 113 and the quotient is 486. 14. Find the least number which is to be added to 1,193 so that the sum is exactly divisible by 214. 15. Find the least number which is to be subtracted from 21,325 so that the difference is exactly divisible by 126. 9