Chapter 2 - Whole Numbers

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Introduction to Whole Numbers

The numbers that we use for counting are 1, 2, 3, 4...and so on. These numbers are known as counting numbers or natural numbers.

Note that 0 is not a part of the natural number series. Natural numbers start from 1.

Predecessor of a Number

Consider the numbers 1, 2, 3, 4, and 5.

The number that comes immediately before a number is called a predecessor of that number.

The number that comes immediately before 4 is 3.

So, 3 is the predecessor of 4.

We subtract 1 from a number to get its predecessor.

Successor of a Number

The number that comes immediately after a number is called a successor of that number.

The number that comes immediately after 4 is 5.

So, 5 is the successor of 4.

Similarly, the predecessor of 3 is 2 and successor of 3 is 4.

We add 1 to a number to get its successor.

Whole Numbers

The series of natural numbers that include 0 is called whole numbers.

So the numbers 0,1,2,3,4... are whole numbers. Here 0 is the predecessor of 1 and 2 is its successor.

Natural numbers form a part of whole numbers.

Operations on Number Line

The Number Line

The number line has a 0 in the middle. At a small distance to the right of 0, 1 is plotted. Similarly, 2, 3, 4, and so on are plotted to the right of 1.

The fixed distance between any two consecutive whole numbers is called the unit distance.

Addition is moving right and subtraction is moving left on the number line.

Addition on Number line

Addition is moving towards right on the number line.

The numbers to the right of any particular number on the number line are always greater than the numbers to the left of that number.

For example, since 3 lies to the right of 2, 3 is greater than 2.

Example: Add 2 + 3 on the number line.

Start at 2 on the number line and jump 3 steps to the right.

That gives us,

2 + 3 = 5

Example: Add 5 + 3 on the number line.

Start at 5 and jump 3 steps to the right of 5. You will land on 8.

5 + 3 = 8

Subtraction on Number line

Subtraction is moving towards left on the number line.

Example: Subtract 2 from 8 on the number line.

A start from 8 is done. Move 2 steps to the left to land on 6 So moving left on the number line is subtraction. So, 8 − 2 = 6.

Example: Simplify 8 − 2 − 3.

Start from 8. Jump 2 steps to left to land on 6. Jump further 3 steps to the left and land on 3. So 8 − 2 − 3 = 3

Multiplication on Number line

Multiplication is making jumps of equal sizes, several times on the number line, starting from zero.

Example: Find the product of 3 × 2.

To find the product of 3 × 2, we need to take a jump of 3 steps twice, starting from 0.

On the number line, the multiplication always starts from 0.

Start at 0 and jump 3 steps to the right to land on 3.

Then, take a second jump of 3 steps to the right to reach 6.

So, the product of 3 × 2 = 6.

Properties of Whole Numbers

Closure Property of Whole Numbers

If any two whole numbers are added, a whole number is obtained as the sum. This property is called the closure property of whole numbers under addition.

Addition and multiplication of whole numbers obey the closure property, while subtraction and division do not.

Example

0 + 10 = 10

3 + 8 = 11

9 + 3 = 12

If any two whole numbers are multiplied, a whole number is obtained. This property is called the closure property of whole numbers under multiplication.

Example

0 × 10 = 0

3 × 8 = 24

3 × 9 = 27

If any two whole numbers are subtracted, one may or may not get a whole number as a result. Whole numbers are not closed under subtraction.

Example

0 − 10 = Not a whole number

3 − 8 = Not a whole number

9 − 3 = 6

In the first two instances, the results are not whole numbers. Only in the third case, the result of subtraction is a whole number.

Similarly, if a whole number is divided by another whole number, one may or may not get a whole number as a result. So, whole numbers are not closed under division.

Example

${0}/{10}$ = 0

${3}/{8}$ = ${3}/{8}$, not a whole number

${9}/{3}$ = 3

In the first and third case, the division of any two whole numbers results in a whole number. But in the second case, the result is a fraction and NOT a whole number.

Commutative Property of Whole Numbers

If any two whole numbers are added or multiplied in any order, the result remains the same. For example,

3 + 8 = 11

8 + 3 = 11

Similarly,

3 × 8 = 24

8 × 3 = 24

Addition and multiplication of whole numbers are commutative, but subtraction and division of whole numbers are not commutative. For example,

3 − 8 = Not a whole number

8 − 3 = 5

So subtraction is not commutative.

Similarly, division is not commutative either. For example,

${4}/{8}$ = ${1}/{2}$

${8}/{4}$ = 2

Associative and Distributive Properties

Associative Property

When three or more whole numbers are added or multiplied, the results remain the same regardless of the grouping of the numbers. This is called associative property of whole numbers.

Addition and multiplication of whole numbers are associative.

Example

8 + (4 + 2) = 8 + 6 = 14

(8 + 4) + 2 = 12 + 2 = 14

Similarly,

8 × (4 × 2) = 8 × 8 = 64

(8 × 4) × 2 = 32 × 2 = 64

If the same numbers are subtracted by grouping them differently, we will get different results.

8 − (4 − 2) = 8 − 2 = 6

(8 − 4) − 2 = 4 − 2 = 2

Similarly, division doesn't follow associative property.

8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4

(8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1

Subtraction and division of whole numbers is not associative.

Distributive Property

Let's take the expression:

8 × (4 + 2)

First, the numbers inside the brackets are evaluated. So, we get,

8 × (4 + 2) = 8 × 6 = 48

The same expression can also be calculated as,

8 × (4 + 2)

= (8 × 4) + (8 × 2)

= 32 + 16 = 48

We get the same result in both the cases.

In the second method, the number outside the bracket (8) is multiplied with each of the numbers inside the brackets (4 and 2) separately and then these products are added to get the final answer.

This property of distributing multiplication over all the numbers inside the bracket is called the distributive property.

Identity Numbers

Consider the following:

2 + 0 = 2

5 + 0 = 5

Any number + 0 = Same number

This is called additive identity of whole numbers. So, 0 is the identity number for addition of whole numbers.

Identity numbers can be different for different operations like addition, subtraction, multiplication, and division.

Any number + 0 = Same number

Any number − 0 = Same number

Any number × 1 = Same number

Any number ÷ 1 = Same number

1 is the identity number for multiplication and division of whole numbers.

Patterns Using Whole Numbers

We can use whole numbers to make different patterns like straight line, triangle, square, and rectangle.

Let's suppose one point represents the number 1 and two points represent the number 2, and so on. We can create different patterns with a given set of points.

With 1 point, we cannot create any patterns. It always remains a point.

Straight Line

If two points are given, then the two can be joined together in the shape of a straight line.

With one more point, we can still form a straight line.

We can form a straight line with any number of points, except 1.

Triangular numbers

Numbers that form a triangle are called triangular numbers. For example, 3, 6, 10, 15, and 21 are triangular numbers.

Square Numbers

We get a square number by multiplying a whole number with itself. 4, 9, 16... are the numbers that form square patterns.

If we add any two consecutive triangular numbers, we end up getting a square number. For example,

3 + 6 = 9

6 + 10 = 16

10 + 15 = 25

6 and 10 are consecutive triangular numbers, whereas 16 is a square number.

Rectangular Numbers

Whole numbers that can be arranged in a rectangular pattern are called rectangular numbers

A whole number is a rectangular number if it can be written as a product of two different whole numbers (one of the whole numbers should not be number 1).

Example: 12 is a rectangular number because it can be expressed as,

2 × 6 = 12

3 × 4 = 12

All square numbers are rectangular numbers, but the opposite is not true.

Did You Know About BODMAS?

What is BODMAS?

To evaluate an expression that contains multiple operations, we need to carry out the operations as per their defined levels of priority.

• (B)rackets have the highest priority
• (O)rder has the next priority
• (D)ivision takes the next priority followed by
• (M)ultiplication, then
• (A)ddition and lastly
• (S)ubtraction.

The first letters of these words are taken to form BODMAS.

Example

Question: Evaluate the following numerical expression:
       3 + 9 × 6 − 22 ÷ 2 + (100 ÷ 2)
Solution: This expression has multiple arithmetic operations like addition, subtraction, multiplication, and division.
First priority; evaluate the numbers inside the brackets.
      100 ÷ 2 = 50
The next priority is division, which gives us,
      22 ÷ 2 = 11
Our expression now reduces to,
      3 + 9 × 6 − 11 + 50
Performing multiplication next, we get,
      9 × 6 = 54
The expression now becomes,
      3 + 54 − 11 + 50
Next priority is addition. So, we add,
      3 + 54 + 50 = 107
Last priority is subtraction. So,
      107 − 11 = 96

Example

Question: Evaluate the expression (16 + 3 × 5) − 7 × 3 + 6.
Solution: First priority: Brackets,
      16 + 3 × 5 = 16 + 15 = 31
Next priority: Division: No operation
Next priority: Multiplication,
      7 × 3 = 21
The expression has now become,
      31 − 21 + 6
Next priority: Addition,
      31 + 6 = 37
Last priority: Subtraction,
      37 − 21 = 16
To conclude,
      (16 + 3 × 5) − 7 × 3 + 6 = 16