We have learned about the natural numbers from 1 to 10. Whole numbers are the set natural numbers including with zero. 0 is the smallest whole number. Whole numbers are 0, 1, 2, 3, ……… All-natural numbers are whole numbers, but all whole numbers are not natural numbers
Properties of Whole Numbers
- Addition and multiplication of any 2 whole number give a whole number.
- Subtraction and division of any 2 whole number may or may not give a whole number.
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What is a Number line?
A number line is a picture of a graduated straight and horizontal line in which numbers are written. A number written on the left-hand side of the number line is lesser and number written on the right-hand side of the number line is greater. Lets us look into some solved example problems.
Find 12 × 35 using distributivity.
12 × 35 = 12 × (30 + 5)
= 12 × 30 +12 × 5
= 360 + 60 = 420.
Calculate – (2 + 3) + 4 = ? = 5+ 4 = 9.
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Evolution of Numbers
- Numbers that are used for counting and ordering are called natural Numbers.
- 1,2,3,4,5,6… are natural numbers
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- Natural numbers along with zero form the collection of whole numbers.
- 0,1,2,3,4,5… are called whole numbers.
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Predecessors and Successors
Predecessor and Successor
- A successor of any number is the next number to it, which is obtained by adding 1.
- A predecessor of any number is the previous number to it, which is obtained by subtracting 1.
- For example, predecessor and successor of the number 12 is 12 – 1 and 12 + 1 which is 11 and 13
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Where Do Whole Numbers Live?
- It is the infinitely long line containing all the whole numbers.
- The line starts at zero, and any two consecutive whole numbers have the same distance between them.
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Describing Number Line
Operations on a number line
⇒ Addition on a number line. For example, addition of 1 and 5 (1 + 5 = 6). First, locate 1 on the number line. Then move 5 places to the right will give 6.
⇒ Subtraction on a number line. For example, subtraction of 3 from 7 (7 – 3 = 4). First, locate 7 on the number line. Then move 3 places to the left will give 4.
⇒ Multiplication on a number line. For example product of 3 and 4 (3 × 4 = 12). Start from 0 and skip 3 places to the right 4 times.
⇒ Division on a number line. For example 6 ÷ 3 = 2. Start from 6 and subtract 3 for a number of times till 0 is reached. The number of times 3 is subtracted gives the quotient.
Properties of Operators: Commutative Associative and Distributive
Division by zero
Division of any whole number by 0 is not defined.
Mathematical operations are simplified due to certain properties that every number follows. They are:
Addition and multiplication are commutative for whole numbers. i.e whole numbers can be added or multiplied in any order.
For e.g: 2 + 3 = 5 = 3 + 3 × 4 = 12 = 4 × 3
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Associativity of addition and multiplication
For eg: (5 +6) + 4 = 15 = 5 + (6 + 4)
(2 × 3) × 4 = 24 =2 × (3 × 4)
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With distributivity property, 4 × (5 + 3) can be written as (4 × 5) + (4 × 3)
Here, 4 × (5 + 3) = 4 × 13 = 52
Also, (4 × 5) + (4 × 3) = 20 + 32 = 52
There exists certain numbers, when included in mathematical operations like addition and multiplication, the value of the operation remains unchanged. Such numbers are called as identities.
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Additive identity gives the same whole number when added to another whole number.
Zero is the additive identity as a + 0 = a, (a is any whole number).
Multiplicative identity gives the same whole number when multiplied by another whole number.
1 is the Multiplicative identity as a × 1 = a, (a is any whole number)
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Let’s Play With Whole Numbers
- Every number can be arranged as a line.
- E.g : 5 = •••••
- Some whole numbers can be expressed as squares.
- E.g :
- Some whole numbers can be expressed as rectangles.
- E.g : 6 can be shown as 3 × 2
- Some numbers can also be arranged as triangles.
Numbers between square numbers
- Between 2 successive square numbers there exists 2n non-square numbers. Between, n² and (n + 1)² there are non-square numbers. Here is a whole number.
- For example, between 9 (3)² and 16 (4)², there are 10 , 11, 12, 13, 14, 15 which is 6 = 2 × 3 numbers.
Adding odd numbers
- Sum of the first n natural odd numbers gives n² which is a perfect square.
- For example : Sum of first 5 natural odd numbers ⇒ 1 + 3 + 5 + 7 + 9 = 25 = 5²
Properties of Operators: Closure Properties
Whole numbers are closed under addition and also under multiplication.
|3||+||1||=||4, a whole number|
|5||+||3||=||8, a whole number|
|4||×||4||=||16, a whole number|
|9||×||2||=||18, a whole number|
Whole numbers are not closed under subtraction and division.
|8||–||5||=||3, a whole number|
|5||–||8||=||-3, not a whole number|
|12||÷||4||=||3, a whole number|
|9||÷||2||=||9/2, not a whole number|
Learn more about the whole numbers from the topics given below: