# NCERT Class 6 Maths Chapter 2: Whole Numbers: Natural Numbers

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## Questions?

- Is Division Closed on Whole Numbers?
- What is multiplicative identity?
- What is distributive property of multiplication over addition?
- Can addition distribute over multiplication?

## Natural Numbers: Predecessor and Successor

- All-natural numbers except 1 have a natural numbers as predecessor and successor.
- To get successor we add 1. We can keep adding 1 to get the next successor and then the next. Thus, natural numbers can become very, very large- there is no end to how large they can get!
- To get predecessor we subtract 1 from the natural number. For 1 we have a successor but no predecessor (in natural numbers) .
- This set of numbers starting from 1 and going on by adding 1 is called set of natural numbers

## Use of Number 0

On number line each number is unit distance apart- 1 inch 1 cm anything, but it has to be same.

## Operations on Number Line

- Drawing number line
- Distances on number line (difference)
- Greater than relation on number line
- Addition on number line
- Subtraction on number line
- Multiplication on number line – multiplication is just repeated addition

## The Number Zero

- Aryabhatta invented zero that means he thought that some number like zero exists and one can represent Ten as Symbol of one as ten digit and Symbol of zero as unit digit. This was firstly added in Bakhshali Manuscript and then it was added in other Lipis.
- Brahmagupta also deserves some credit for invention of zero. A Hindu astronomer and mathematician named Brahmagupta developed a symbol for zero — a dot underneath numbers. He also developed mathematical operations using zero, wrote rules for reaching zero through addition and subtraction, and the results of using zero in equations.

## Sets/Collections of Numbers

Whole Numbers = Natural Numbers + 0

- Drawing number line
- Distances on number line (difference)
- Greater than relation on number line
- Addition on number line
- Subtraction on number line
- Multiplication on number line – multiplication is just repeated addition

## Problems

- Zero is the smallest natural number.
- 400 is the predecessor of 399.
- Zero is the smallest whole number.
- 600 is the successor of 599.
- All-natural numbers are whole numbers.
- All whole numbers are natural numbers.
- The predecessor of a two-digit number is never a single digit number.
- 1 is the smallest whole number.
- The natural number 1 has no predecessor.
- The whole number 1 has no predecessor.
- The whole number 13 lies between 11 and 12.
- The whole number 0 has no predecessor.
- The successor of a two-digit number is always a two-digit number.

## Operations on Number Line

- When you draw a number line distance between 2 number is unit distance- 1 inch, 1 cm etc.
- On the number line distance between 2 numbers is equal to their difference.

## Closure Property of Addition over Whole Numbers

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.

## Closure Property of Multiplication over Whole Numbers

Since multiplication is repeated addition therefore system of whole numbers is also closed under multiplication.

## Closure Property of Multiplication and Addition over Whole Numbers

- 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.
- Since multiplication is repeated addition therefore system of whole numbers is also closed under multiplication.

## Is Subtraction Closed over Whole Numbers?

## Division as Repeated Subtraction

Assume that we want to divide 12 by 3. Thus, we start with 12 and start subtracting as many threes as we can.

## Is Division Closed over Whole Numbers?

- Division of whole number can produce fractions
- Division by zero is not defined.

## Additive Identity

## Multiplicative Identity

## Multiplication Property of Zero

Sridhar Acharya invented the operations of zero in India in 8^{th} century. He clearly mentioned the properties of Zero. “If zero is added to any number, the sum is the same number; if zero is subtracted from any number, the number remains unchanged; if zero is multiplied by any number, the product is zero” .

## Problems?

- If the product of two whole numbers is zero, can we say that one or both of them will be zero?
- If the product of two whole numbers is 1, can we say that one or both of them will be 1?

## Commutativity of Addition of Whole Numbers

- When we do 2 + 3 it is same as 3 + 2. Similarly, we can say 15 + (7 + 5) or 15 + (5 + 7) are same. This is known as commutativity.
- Now associativity is different- if I do (15 + 7) + 5 or 15 + (7 + 5) - that is I associate the 7 differently now. Now instead of changing the order I have changed the actual additions- numbers (operands) which are on either side of + sign.

## Associativity of Addition of Whole Numbers

- When we do 2 + 3 it is same as 3 + 2. Similarly, we can say 15 + (7 + 5) or 15 + (5 + 7) are same. This is known as commutativity.
- Now associativity is different- if I do (15 + 7) + 5 or 15 + (7 + 5) - that is I associate the 7 differently now. Now instead of changing the order I have changed the actual additions- numbers (operands) which are on either side of + sign.
- There are operations which can be commutative but not associative and there are operations which are associative but not commutative (matrix multiplication)

## Multiplication as Row × Column and Tape

- When we do 2 + 3 it is same as 3 + 2. Similarly, we can say 15 + (7 + 5) or 15 + (5 + 7) are same. This is known as commutativity.
- Now associativity is different- if I do (15 + 7) + 5 or 15 + (7 + 5) - that is I associate the 7 differently now. Now instead of changing the order I have changed the actual additions- numbers (operands) which are on either side of + sign.

## Commutativity of Multiplication of Whole Numbers

When we do 3 x 2, it is same as 2 x 3.

## Associativity of Multiplication of Whole Numbers

When we do 2 x (3 x 4) , or we do (2 x 3) x 4

## Is Subtraction of Whole Numbers Commutative?

## Is Subtraction of Whole Numbers Associative?

let՚s do 4- (2 - 1) = 3, and (4 - 2) -1 = 1

## Division of Whole Numbers is Not Commutative

4/2 is not the same as 2/4

## Division of Whole Numbers is Not Associative

12/ (4/2) = 6 and (12/4) /2 = 3/2

## Application of Associativity & Commutativity of Addition & Multiplication

- 14 + 17 + 6
- 12 × 35
- 6 × 3 × 5
- 8 × 1769 × 125
- 37
- 420
- 90
- 1769000

## Distributivity of Multiplication over Addition

## Distributivity of Multiplication over Addition

- 12 + 6 + 10 = 28
- 2 (3 + 5 (2 + 4) ) = 66

## Does Multiplication Distribute over Subtraction?

## Distributive Law in Normal Multiplication

435 (100 + 30 + 6) = 43500 + 13050 + 2610 = 59160

59160

## Distributive Law Questions

1. A family spends ₹ 100 for lunch and 4 for milk for each day. How much money will they spend in a week?

Answer: 728

## Find the Values

3845 × 5 × 782 + 769 × 25 × 218

824 × 21

769 ⚹ 5 = 3845

3845 ⚹ 5 (782 + 218) = 3845 ⚹ 5 ⚹ 1000 = 19225 ⚹ 1000 = 19225000

824 × 21 = 824 ⚹ 20 + 824 = 17304

## Patterns in Numbers: Triangular Numbers

- The triangles are equilateral
- They are a type of figurate number- other examples being square and cube numbers

## Patterns in Numbers: Square Numbers

769 ⚹ 5 = 3845

3845 ⚹ 5 (782 + 218) = 3845 ⚹ 5 ⚹ 1000 = 19225 ⚹ 1000 = 19225000

## Relation between Triangular and Square Numbers

1, 3, 6, 10,15, 21

## Patterns

- 1 × 9 + 1 = 10
- 12 × 9 + 2 = 110
- 123 × 9 + 3 = 1110
- 1234 × 9 + 4 = 11110
- 12345 × 9 + 5 = 111110
- Patterns when we multiply by 9 - 99, 108,117, 126,
- Pattern in table of 11 = 121,132, 143.
- Pattern of 11 x 11 = 121,111x111 = 12321,1111 x 1111 = 1234321

## Patterns

- 1 × 9 + 1 = 10
- 12 × 9 + 2 = 110
- 123 × 9 + 3 = 1110
- 1234 × 9 + 4 = 11110
- 12345 × 9 + 5 = 111110
- 1 × 8 + 1 = 9
- 12 × 8 + 2 = 98
- 123 × 8 + 3 = 987
- 1234 × 8 + 4 = 9876
- 12345 × 8 + 5 = 98765

824 × 25

## Addition by Completing 10՚s

23 + 64 + 27 = 64 + 50 = 114

25 + 66 + 28 = 25 + 66 + 24 + 4 = 25 + 90 + 4 = 29 + 90 = 119

## Multiplication by Producing 0՚s

824 × 25 = 206 ⚹ 4 ⚹ 25 20600

84 × 99 = 84 (100 - 1) = 8400 – 84 = 8316

84 × 98 = 84 (100 - 2) = 8400 – 168 = 8232

96 × 125 = 12 x 8 x 125 = 12 x 1000 = 12000

64 × 35 = 32 x 2 x 5 x 7 = 224 x 10 = 2240

## More Advanced Topics

- Square triangular numbers
- Beyond 0- Negative Integers
- Concept of infinity
- Real numbers

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✍ Mayank