TET Syllabus - Mathematics Study Materials in Number System (Part V)

 

 

      Teachers Eligibility Test - Paper 2

 

Class – VI Mathematics

 

                                       NUMBER SYSTEM

   

              WHOLE NUMBERS

  

          The collection of counting numbers {1,2,3...} is called Natural numbers, denoted by N. If this collection includes 0 as well, then the collection              {0, 1,2,3...} is called Whole numbers, denoted by W.

 

Facts on Natural and Whole Numbers

 

v The smallest natural number is 1.

v The smallest whole number is 0.

v Every number has a successor. The number that comes just after the given number is its successor.

v Every number has a predecessor. The number 1 has a predecessor in W namely ‘0’, but it has no predecessor in N. The number ‘0’ has no predecessor in W.

v There is an order to numbers. By comparing the two given numbers the larger of the two can be identified.

v Numbers are endless. By adding 1 to any chosen large number, the next number can be found.

 

                   Properties of Whole Numbers

Commutatively of addition and multiplication

              When two numbers are added (or multiplied), the order of the numbers does not affect the sum (or the product). This is called commutatively of addition (or multiplication).

Example

                        43 + 57 = 57 + 43

                        12 × 15 = 15 × 12

35,784 + 48, 12, 69,841 = 48, 12, 69,841 + 35,784

          39,458 × 84,321 = 84,321 × 39,458

 

 

 

Example

7 – 3 = 4 but 3 – 7 will not give the same answer.

Similarly, the answers of 12 ÷ 6 and 6 ÷ 12 are not equal.

That is, 7 – 3 ≠ 3 – 7 and 12 ÷ 6 ≠ 6 ÷ 12

Hence, subtraction and division are NOT commutative.

 

Associativity of addition and multiplication

                   When several numbers are added, the order in which the numbers are added does not matter. This is called associativity of addition. Similarly, when several numbers are to be multiplied, the order in which the numbers are multiplied does not matter. This is called associativity of multiplication.

Example

                     (43 + 57) + 25 = 43 + (57 + 25)

                       12 × (15 × 7) = (12 × 15) × 7

   35,784 + (48, 12, 69,841 + 3) = (35,784 + 48, 12, 69,841) + 3

           (39,458 × 84,321) × 17 = 39,458 × (84,321 × 17)

It is to be noted that here too, subtraction and division are NOT associative.

 

Distributive of multiplication over addition or subtraction

              An interesting fact relating to addition and multiplication comes from the following patterns:

                  (72 × 13) + (28 × 13) = (72 + 28) × 13

                                     37 × 102 = (37 × 100) + (37 × 2)

                                       37 × 98 = (37 × 100) – (37 × 2)

In the last two cases, we are actually noting down the following equations:

                             37 × (100 + 2) = (37 × 100) + (37 × 2)

                           37 × (100 − 2) = (37 × 100) – (37 × 2)

                     It can be noted that the product of a number and a sum of numbers can be written as the sum of two products.

Similarly, the product of a number and a number got by subtraction can be written as the difference of two products. This property is called as property of distributive of multiplication over addition or subtraction. It is a very useful property to group numbers in a convenient way.

 

Identity for addition and multiplication

              When zero is added to any number, we get the same number.

0	+	9	=	9
0	+	10	=	10
16	+	0	=	16
37	+	0	=	37
50	+	0	=	50

 

 

 

 

 

Similarly, when we multiply any number by 1, we get the same number.

So, zero is called the additive identity and one is called the multiplicative identity for whole numbers.

1	x	55	=	55
25	x	1	=	25
400	x	1	=	400
1	x	12	=	12

Finally, these are some simple observations that are important.

                        When we add any two natural numbers, we get a natural number. Similarly when we multiply any two natural numbers, we get a natural number.

                        When we add any two whole numbers, we get a whole number. Similarly when we multiply any two whole numbers, we get a whole number.

          When we add a natural number to a whole number, we get a natural number.

           When we multiply a natural number by a whole number, we get a whole number.

 

NOTE:

        Any number multiplied by zero gives zero.

        Division by zero is not defined.