Revise notes ICSE 6 natural numbers and whole numbers (including number line and patterns)

Revise notes
ICSE 6 Maths
Natural Numbers and Whole Numbers (Including Number Line and Patterns)

Topics

1. Natural Numbers
2. Whole numbers
3. Successor and Predecessor of a whole number
4. Properties of Whole numbers
5. Number lines for Natural numbers and whole numbers

The main properties are as follows:

1. Closure Property

2. Commutative Property

3. Associative Property

4. Distributive Property

5. Identity

For Addition:

1. Closure Property

Whole numbers are said to be closed for addition if on adding any two whole

numbers, we get a whole number only.

If x and y are two whole numbers, then x + y is also a whole number.

Example: 1^st whole number is 5 and other whole number is 8 then the sum of both the whole numbers is 5 + 8 = 13, a whole number.

2. Commutative Property (Commutativity)

The sum of two whole numbers is said to be commutative for addition if their sum remains the same even if the order of addition is changed.
if x and y are two whole numbers, then x + y = y + x.
Example: 1^st number = 4 and 2^nd whole number = 3, sum = 4+3 = 7 and 3+4 = 7.

3. Associative Property (Associativity):

For any three whole numbers if the sum of any two whole numbers is added to the third whole number, then whatever be their order, the sum will come to be same.
For any three whole numbers x, y and z,

x + (y + z) = (x + y) + z.

Example: for whole numbers 3,5,6: 3 + (5+6) = 3 +11 = 14

and (3 + 5) + 6 = 8 + 6 = 14

4. Existence of Identity (Additive Identity):

The addition (sum) of any whole number and whole number 0 (zero) is the number itself.
The whole number 0 is called additive identity or identity element for addition.

Example: for any whole number x.

x+0 = x, 0+x = x

For subtraction:

1. Closure property:

If x and y are two different whole numbers, then x - y is not necessarily-a whole number.
Example: 8 and 3 are whole numbers and 8 -3 = 5 is also a whole number.

2. Commutative Property:

If x and y are two whole numbers, then x – y ≠ y - x.
Example: two whole numbers 20 and 32, 20-32 is not a whole number, but 32-20 = 12 is a whole number.

3. Associative Property:

For any three whole numbers x, y and z where z 0
x - (y - z) ≠ (x - y) - z

Example: Consider 3 numbers 15, 10 and 7.

Here, 15 – (10-7) = 15 -3 = 12

And (15-10) – 7 = 5 -7 = -2

Note:

If z = 0, x - (y - z) = x - (y - 0) = x - y

(x - y) - z = (x - y) - 0 = x – y

4. Existence of identity:

For any whole number x, x - 0 = x but 0 - x x.
Here, it can be stated that for subtraction of whole numbers, no identity element exists

Example: 12-0 = 12 but 0 – 12 12.

For Multiplication:

1. Closure property

If x and y are two whole numbers, then x × y is also a whole number.
Example: one whole number = 5 and other whole number = 4 then the product of both the whole numbers = 5 × 4 = 20.

2. Commutative Property:

If x and y are two whole numbers, then x × y = y × x.
Example: for any two whole numbers 5 and 4 , 5 × 4 = 4 × 5 = 20.

3. Associative Property:

For any three whole number x, y and z.
x × (y × z) = (x × y) ×z
Example: For any whole numbers 4,8, 10: 4 × (8×10) = 4 × 80 = 320

and (4 × 8) × 10 = 320

4. Distributive Property:

For any three whole numbers x, y and z,
x × (y+z) = x × y + x × z
The multiplication of whole numbers is distributive over their addition.
Example: For any whole numbers 5, 3, and 4.

5 × (3 + 4) = 5 × 3 + 5 × 4

35 = 35

The multiplication of whole numbers is distributive over their subtraction

i.e., (i) x × (y - z) = x × y – x × z.

(ii) (y -z) × x = y × x – z × x.

5. Existence of Identity:

Identity element for multiplication is 1.
For any whole number x, their exists whole number 1 for which x × 1 = 1 × x = x.
The whole number 1 is called the multiplicative identity or identity element for multiplication.
Example: For whole number 9: 9 × 1 = 1 × 9 = 9

For Division:

1. Closure Property:

If x and y are two whole numbers, then x y is not necessarily a whole number.
Examples: 5 and 8 are whole numbers, but 5 8 is not a whole number.
For whole numbers 24 and 12; 24 12 = 2, which is a whole number, whereas 12 24 is not a whole number.

2. Commutative Property:

If x and y are two whole numbers, then x y y x.
Example: 3 5 5 3
Division of whole numbers is not commutative.

3. Associative Property:

If x, y and z are any three whole numbers where z 1, then
x (y z) (x ) z

4. Existence of Identity:

The identity element for division of whole numbers does not exist.

Note:

If a is any non-zero whole number, then

(i) a a = 1, i.e., 5 5 = 1

(ii) a 1 = a, i.e., 5 1 = 5

(iii) 0 a = 0, i.e., 0 8 = 0

(iv) a 0 is not defined, i.e., 8 0 is not defined.

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