**Notes For All Chapters Maths Class 6**

### Natural Numbers

All the positive counting numbers starting from one are called Natural Numbers.

### Predecessor and Successor

If we add 1 to any natural number, we get the next number, which is called the Successor of that number.

12 + 1 = 13

So 13 is the successor of 12.

If we subtract 1 from any natural number, we get the predecessor of that number.

12 – 1 = 11

So 11 is the predecessor of 12.

Remark: There is no predecessor of 1 in natural numbers.

### Whole Numbers

Whole numbers are the collection of natural numbers including zero. So, the zero is the predecessor of 1 in the whole numbers.

### Number Line

To draw a number line, follow these steps-

(i) Draw a line and mark a point 0 on it.

(ii) Now mark the second point to the right of zero and label it as 1.

(iii) The distance between the 0 and 1 is called the unit distance.

(iv) Now you can mark other points as 2, 3, 4 and so on with the unit distance.

This is the number line for the whole numbers.

**1. The distance between two points**

The distance between 3 and 5 is 2 units. Likewise, the distance between 1 and 6 is 5 units.

**2. The greater number on the number line**

The number on the right is always greater than the number on the left.

As number 5 is on the right of the number 2, Hence 5>2.

**3. A smaller number on the number line**

The number on the left of any number is always smaller than that number.

As number 3 is on the left of 7, so 3 < 7.

### Addition on the Number Line

If we have to add 2 and 5, then start with 2 and make 5 jumps to the right. As our 5th jump is at 7 so the answer is 7.

The sum of 2 and 5 is 2 + 5 = 7

### Subtraction on the Number Line

If we have to subtract 6 from 10, then we have to start from 10 and make 6 jumps to the left. As our 6th jump is at 4, so the answer is 4.

The subtraction of 6 from 10 is 10 – 6 = 4.

### Multiplication on the Number Line

If we have to multiply 4 and 3, then Start from 0, make 4 jumps using 3 units at a time to the right, as you reach to 12. So, we say, 3 × 4 = 12.

Properties of Whole Numbers

**1. Closure Property**

Two whole numbers are said to be closed if their operation is also the whole number.

Operation | Meaning | Example | Closed or not |

Addition | Whole numbers are closed under addition as their sum is also a whole number. | 2 + 5 = 7 | Yes |

Subtraction | Whole numbers are not closed under subtraction as their difference is not always a whole number. | 9 – 2 = 7 2 – 9 = (-7) which is not a whole number. | No |

Multiplication | Whole numbers are closed under multiplication as their product is also a whole number. | 9 × 5 = 45 | Yes |

Division | Whole numbers are not closed under division as their result is not always a whole number. | 5 ÷ 1 = 5 5 ÷ 2 =, not a whole number. | No |

**2. Commutative Property**

Two whole numbers are said to be commutative if their result remains the same even if we swap the positions of the numbers.

Operation | Meaning | Example | Commutative or not |

Addition | The addition is commutative for whole numbers as their sum remains the same even if we interchange the position of the numbers. | 2 + 5 = 7 5 + 2 = 7 | Yes |

Subtraction | Subtraction is not commutative for whole numbers as their difference may be different if we interchange the position of the numbers. | 9 – 2 = 7 2 – 9 = (-7) which is not a whole number. | No |

Multiplication | Multiplication is commutative for whole numbers as their product remains the same even if we interchange the position of the numbers. | 9 × 5 = 45 5 × 9 = 45 | Yes |

Division | The division is not commutative for whole numbers as their result may be different if we interchange the position of the numbers. | 5 ÷ 1 = 5 1 ÷ 5 =, not a whole number. | No |

**3. Associative Property**

The two whole numbers are said to be associative if the result remains the same even if we change the grouping of the numbers.

Operation | Meaning | Example | Associative or not |

Addition | The addition is associative for whole numbers as their sum remains the same even if we change the grouping of the numbers. | 3 + (2 + 5) = (3 + 2) + 5 3 + 7 = 5 + 5 10 = 10 | Yes |

subtraction | Subtraction is not associative for whole numbers as their difference may change if we change the grouping of the numbers. | 8 – (10 – 2) ≠ (8 – 10) – 2 8 – (8) ≠ (-2) – 2 0 ≠ (-4) | No |

Multiplication | Multiplication is associative for whole numbers as their product remains the same even if we change the grouping of the numbers. | 3 × (5 × 2) = (3 × 5) × 2 3 × (10) = (15) × 2 30 = 30 | Yes |

Division | The division is not associative for whole numbers as their result may change if we change the grouping of the numbers. | 24 ÷ 3 ≠ 4 ÷ 2 8 ≠ 2 | No |

**4. Distributivity of Multiplication over Addition**

This property says that if we have three whole numbers x, y and z, then

x(y + z) = xy + xz

Example

Evaluate 15 × 45

Solution

15 × 45 = 15 × (40 + 5)

= 15 × 40 + 15 × 5

= 600 + 75

= 675

**5. Identity for Addition**

If we add zero to any whole number the result will the same number only. So zero is the additive identity of whole numbers.

a + 0 = 0 + a = a

This clearly shows that if we add zero apples to 2 apples we get the two apples only.

**6. Identity for Multiplication**

If we multiply one to any whole number the result will be the same whole number. So one is the multiplicative identity of whole numbers.

### Patterns

Patterns are used for easy verbal calculations and to understand the numbers better.

We can arrange the numbers using dots in elementary shapes like triangle, square, rectangle and line.

**1. We can arrange every number using dots in a line**

**2. We can arrange some numbers using a rectangle.**

**3. We can arrange some numbers using a square.**

**4. We can arrange some numbers using a triangle.**

### Use of Patterns

Patterns can be used to simplify the process.

1. 123 + 9 = 123 + 10 – 1 = 133 -1 = 132

123 + 99 = 123 + 100 – 1 = 223 – 1 = 222

2. 83 × 9 = 83 × (10-1) = 830 – 83 = 747

83 × 99 = 83 × (100-1) = 8300 – 83 = 8217