Chapter 2: Whole numbers (Notes)
Posted on : 15-02-2019 Posted by : Admin

Natural numbers

All numbers which we generally use for counting like 1, 2, 3, 4 etc. are known as ‘Natural numbers’, because these numbers come naturally when we start to count.

 

Successor

If number 1 is added to any natural number then we can get the next number. This next number is called its “successor”

For example,

12 is a natural number and when 1 is added to 12 we get 13 (12+1=13). So here 13 is the successor of 12. Similarly the successor of 13 is 14 (13+1=14) and so on.

 

Predecessor

If number 1 is subtracted from any natural number then we get the previous number. This previous number is called its “predecessor”

For example,

12 is a natural number and when 1 is subtracted from 12 we get 11 (12-1=11). So here 11 is the predecessor of 12. Similarly the predecessor of 13 is 12 (13-1=12) and so on.

The population in our village can be counted; even the population in our country can be counted. But, it is difficult to count the stars in the sky. If we are able to count the stars there is a number for that also. If 1 is added to that number, it will be a larger number than original number. Hence we can say that “Every natural number has a successor”. Also “There is no largest natural number”.

 

Whole numbers

There is no predecessor for number 1 in Natural numbers, but we can add ‘0’ as the predecessor for 1. When 0 is added to the set of natural numbers, this becomes set of whole numbers. In other words, the Natural numbers along with ‘0’ forms the collection of ‘Whole Numbers’.

The whole numbers starts from zero and continues like 0, 1, 2, 3…. and so on. Just like natural numbers, there is no largest whole number.

 

The number line

Natural numbers, Whole numbers, Successors, predecessor, number line, addition, subtraction, multiplication, division, number line

Drawing a number line

  1. Draw a line with arrow going right side and mark the initial point as ‘0’.
  2. Mark the second point to the right of zero and label it ‘1’.
  3. The distance between 0 and 1 is a ‘Unit distance’. Again to the right of 1 to a unit distance we mark a point and label it as ‘2’.
  4. The distance between any two consecutive numbers in a number line is 1 unit.
  5. As the number line is starting with 0 we also call it as number line for whole numbers.
  6. If we observe the number line, number 6 is on the right of 2, hence we say that 6>2. Similarly all other numbers.

 

Successor of 10 on number line

Natural numbers, Whole numbers, Successors, predecessor, number line, addition, subtraction, multiplication, division, number line

Predecessor of 9 on number line

Natural numbers, Whole numbers, Successors, predecessor, number line, addition, subtraction, multiplication, division, number line

Let us consider the following example,

Which number is at the farthest left when we mark 15, 6 and 9 on a number line?

Natural numbers, Whole numbers, Successors, predecessor, number line, addition, subtraction, multiplication, division, number line

From the number line, we can see that number 6 is on the farthest left.

Which number is at the farthest right when we mark 1005 and 9756 on a number line?

Natural numbers, Whole numbers, Successors, predecessor, number line, addition, subtraction, multiplication, division, number line

From the number line, we can see that number 9756 is on the farthest right.

 

Addition on number line

Addition of whole numbers is shown in the above image on the number line. Let us consider the addition of 3 and 4 on a number line,

  • First mark four points 0, 1, 2, 3 and 4 on number line.
  • Since we have to perform addition, we have to take the jumps to the right side of the number line.
  • Now as we have to add 4 to 3, we have to take 4 unit steps to the right side of the number 3 (from 3 to 4, 4 to 5, 5 to 6 and 6 to 7).
  • Hence arrive at 7 and we know 3+4=7

Natural numbers, Whole numbers, Successors, predecessor, number line, addition, subtraction, multiplication, division, number line

 

Subtraction on number line

Subtraction of whole numbers is shown in the above image on the number line. Let us consider the subtraction of 6 and 8 on a number line,

  • First mark eight points 0, 1, 2, 3…on number line.
  • Since we have to perform subtraction, we have to take the jumps to the left side of the number line.
  • Now as we have to subtract6from 8, we have to take 6 unit steps to the left side of the number 8 (from 8 to 7, 7 to 6, 6 to5, 5 to 4, 4 to 3 and 3 to 2).
  • Hence arrive at 2 and we know 8-6=2

Natural numbers, Whole numbers, Successors, predecessor, number line, addition, subtraction, multiplication, division, number line

 

Multiplication on the number line

Multiplication of whole numbers is shown in the above image on the number line. Let us consider the multiplication of 2 and 3 on a number line.

  • First mark points 0, 1, 2, 3…etc. on number line.
  • Since we have to perform multiplication, we have to take the long jumps to the right side of the number line.
  • Now as we have to multiply 2 and 3, we have start at 0 and move 3 units at a time to the right, make 2 such moves.
  • Hence we arrive at 6.

Natural numbers, Whole numbers, Successors, predecessor, number line, addition, subtraction, multiplication, division, number line

 

Operations on whole numbers and their properties

We generally perform four basic operations on the whole numbers. They are Addition, Subtraction, Multiplication and Division. Now let us study the properties of these operations on whole numbers.

CLOSURE PROPERTY

Sum of any two whole numbers is a whole number i.e. the collection of whole numbers is closed under addition. This property is known as the closure property for addition of whole numbers.

For Addition and Multiplication

Whole numbers are closed only under ADDITION and MULTIPLICATION.

Consider the following examples,

  • 4+3 is a whole number and also 3+4 is a whole number. Hence we can say that whole numbers are closed under addition.
  • 9 x 4 is a whole number but 4 x 9 is also a whole number. Hence the whole numbers are closed under multiplication.

For Subtraction and Division

Whole numbers are not closed under SUBTRACTION and DIVISION.

Consider the following examples,

  • 7-4 is a whole number but 4-7 is not a whole number. Hence we can say that whole numbers are not closed under subtraction.
  • 9/3 is a whole number but 3/9 is not a whole number. Hence the whole numbers are not closed under division.

*Division by zero: Division of any whole number by zero is not defined.

COMMUTATIVE PROPERTY

For Addition and Multiplication

Let us consider the addition of 4+6 and 6+4 on a number line, we can find that the result is the same in both cases.

So, we can add two whole numbers in any order.

This property is known as commutative property of addition for whole numbers.

Let us consider the multiplication of 2×3 and 3×2, whatever may be the order of multiplication the result will be the same.

This is known as commutative property of multiplication for the whole numbers.

Thus we can say that ADDITION and MULTIPLICATION are commutative for whole numbers.

For subtraction and division:

Let us consider the subtraction 10-7 and 7-10, we can find that the result is not the same in both cases,

Again consider division of 15/5 and 5/15, here also the result is not the same in both cases.

Hence from the above cases we can say SUBTRACTION and DIVISION are not commutative for whole numbers.

ASSOCIATIVE PROPERTY

For addition and multiplication

Let us take the addition (1+2) +3= 3+3 = 6

Now consider let us change the order of addition,

1+ (2+3) = 1 + 5 = 6

In both the cases the result is the same. This is called associative property of addition for the whole numbers.

Let us take the multiplication (1×2) ×3 = 2 × 3 = 6

Now let us change the order of multiplication

1× (2×3) =1×6 =6

Again in both the result is the same. This is called associative property of multiplication for the whole numbers.

Example: Find 10+15+20

This can be written as (10+15) + 20 =45 or 10 + (15 +20) =45

This represents that ADDITION and MULTIPLICATION are associative for whole numbers.

For division

Let us consider the division (16/4)/2=4/2=2

Now let us change the order of division

 16 / (4/2) = 16/2=8

The result is not same in both the cases. Hence we can say that DIVISION is not associative for whole numbers.

DISTRIBUTIVE PROPERTY

For multiplication over addition

Let us consider the example of 4 × (5+8)               

 4 × (5+8) =4× (13) =52

Now take (4×5) + (4×8) = 20+32 =52.

We find in both the cases, the result is the same i.e. 4× (5+8) = (4×5) + (4×8) =52

Hence if we take any three whole numbers   for example, a, b, c we have

a× (b + c) = (a × b) + (a × c).

This is known as distributive property of multiplication over addition.

Example:  A hotel charges Rs.30 for lunch and Rs.10 for milk each day. How much money do you spend in 10 days on these things?

Solution: Add amount for milk for 10 days and amount for lunch for 10 days.

Cost of lunch = 10×30 = Rs.300

Cost of milk = 10×10 = Rs.100

Total cost = Rs. (300+100) = Rs.400

This example shows that 10× (30+10) = (10×30) + (10×10)

 

Identity for Addition and Multiplication

The difference between the collection of whole numbers and natural numbers is the presence of zero.

Let us add zero to whole number,

1+0=1

2+0=2

5+0=0

The same whole number comes again. Hence zero is called the additive identity of whole numbers.

Now let us consider the multiplication of whole numbers with zero. In multiplication any whole number multiplied by zero the result is zero.

1×0=0

2×0=0

5×0=0

Let us take the multiplication of whole numbers with one,

1×1=1

2×1=2

5×1=5

We can observe that whatever may be the whole number when multiplied with one, the result is the same whole number. Hence we can say that one is the multiplicative identity of whole numbers.

 



Hope you have liked this post.

Please share it with your friends through below links.

All the very best from Team Studyandscore

“Study well, Score more…”

  

- Share with your friends! -