Class 6 Mathematics - Chapter 2: Whole Numbers

Quick Revision Notes & Formula Sheet (MP Board)

Natural Numbers (प्राकृतिक संख्याएं): Counting numbers starting from 1.
- Set: $1, 2, 3, 4, 5, \dots$
- Smallest Natural Number: $1$
Whole Numbers (पूर्ण संख्याएं): All natural numbers along with zero ($0$).
- Set: $0, 1, 2, 3, 4, 5, \dots$
- Smallest Whole Number: $0$
- Note: Every natural number is a whole number, but $0$ is a whole number which is not a natural number.
Predecessor (पूर्ववर्ती): The number that comes just before a given number.
- Predecessor = Given Number - 1
- Example: Predecessor of $100 = 100 - 1 = 99$
- Note: Whole number $0$ has no predecessor in whole numbers.
Successor (उत्तरवर्ती / परवर्ती): The number that comes just after a given number.
- Successor = Given Number + 1
- Example: Successor of $99 = 99 + 1 = 100$

A straight line with numbers placed at equal intervals.
Addition on Number Line: Move to the RIGHT.
- Example: $3 + 4 \rightarrow$ Start at $3$, move $4$ steps to the right = $7$
Subtraction on Number Line: Move to the LEFT.
- Example: $7 - 3 \rightarrow$ Start at $7$, move $3$ steps to the left = $4$
Multiplication on Number Line: Move in equal jumps to the RIGHT starting from $0$.
- Example: $2 \times 3 \rightarrow$ Start at $0$, make $3$ jumps of $2$ units each = $6$

Let $a$, $b$, and $c$ be any three whole numbers:

Whole numbers are closed under Addition and Multiplication, but NOT under Subtraction and Division.

Addition: $a + b = \text{Whole Number}$ (e.g., $5 + 4 = 9$)
Multiplication: $a \times b = \text{Whole Number}$ (e.g., $5 \times 4 = 20$)
Subtraction: $a - b \neq \text{Always a Whole Number}$ (e.g., $3 - 5 = -2$, not a whole number)
Division: $a \div b \neq \text{Always a Whole Number}$ (e.g., $7 \div 2 = 3.5$, not a whole number)

Order of numbers does not change the result for Addition and Multiplication.

Grouping of numbers does not change the result for Addition and Multiplication.

Addition: (a + b) + c = a + (b + c)
- Example: $(2 + 3) + 4 = 2 + (3 + 4) = 9$
Multiplication: (a * b) * c = a * (b * c)
- Example: $(2 * 3) * 4 = 2 * (3 * 4) = 24$

Multiplication distributes over Addition and Subtraction.

Over Addition: a * (b + c) = (a * b) + (a * c)
- Example: $2 * (3 + 5) = (2 * 3) + (2 * 5) = 6 + 10 = 16$
Over Subtraction: a * (b - c) = (a * b) - (a * c)
- Example: $4 * (5 - 2) = (4 * 5) - (4 * 2) = 20 - 8 = 12$

Rule / Property	Formula	Description	Example
Additive Identity	`a + 0 = a`	Adding $0$ to any whole number gives the number itself. $0$ is the additive identity.	$7 + 0 = 7$
Multiplicative Identity	`a * 1 = a`	Multiplying any whole number by $1$ gives the number itself. $1$ is the multiplicative identity.	$9 * 1 = 9$
Multiplication by Zero	`a * 0 = 0`	Any whole number multiplied by $0$ becomes $0$.	$15 * 0 = 0$
Division by Zero	`a / 0 = Undefined`	Division of any whole number by zero is not defined.	$5 / 0 = \text{Not Defined}$

Numbers can be arranged in elementary shapes using dots:

Line: Every number can be shown as a line (e.g., • • • for 3).
Rectangle: Numbers like $6, 8, 10, 12$ can form rectangles.
Square: Square numbers like $4, 9, 16, 25$ form squares (e.g., $3 \times 3 = 9$).
Triangle: Triangular numbers like $3, 6, 10, 15$ form triangles.

Rearrangement for Easy Addition:
- Find pairs that make multiples of $10, 100, \dots$
- Example: $14 + 17 + 6 = (14 + 6) + 17 = 20 + 17 = 37$
Rearrangement for Easy Multiplication:
- Group terms to form $10, 100, 1000, \dots$
- Example: $2 \times 37 \times 50 = (2 \times 50) \times 37 = 100 \times 37 = 3700$
Using Distributive Law for Fast Multiplication:
- Example: $12 \times 35 = (10 + 2) \times 35 = (10 \times 35) + (2 \times 35) = 350 + 70 = 420$