Notes
Ncert Solutions
Assignments
Natural and Whole Number
Counting Numbers/Natural Numbers
The numbers 1,2,3,… are called natural numbers or counting numbers.
Whole Numbers
Let us add one more number i.e. ,zero(0), to the collection of natural numbers.
Now the numbers are 0,1,2,… These numbers are called
whole numbers
We can say that whole nos. consist of zero and the natural numbers. Therefore, except zero all the whole nos. are natural numbers.
So all natural Number are whole number but all whole numbers are not natural numbers
Integers
Z={…7,6,5,4,3,2,1,0,1,2,3,4,5,6…..}
Rational Numbers
A number is called rational if it can be expressed in the form p/q where p and q are integers ( q> 0).
Example : ½ , 4/3 ,5/7 ,1 etc.
Properties of Rational Numbers
Closure Property
The Closure Property states that when you perform an operation (such as addition, multiplication, subtraction etc) on any two numbers in a certain type of number, the result of the computation is another number in the same type of number
Here is the summary for the closure property on various mathematical operations on various numbers
Numbers

Closed Under


addition

subtraction

multiplication

division

Rational numbers

Yes

Yes

Yes

No

Integers

Yes

Yes

Yes

No

Whole Numbers

Yes

No

Yes

No

Natural Numbers

Yes

No

Yes

No

Watch this tutorial for more explanation about closure property
Commutativity Property
The Commutativity Property states that order does not matter so if you swap numbers in an operation (such as addition, multiplication, subtraction etc) on any two numbers in a certain type of number, the result of the computation is same as the result without swap
Here is the summary for the commutative property on various mathematical operations on various numbers
Numbers

Commutative Under


addition

subtraction

multiplication

division

Rational numbers

Yes

No

Yes

No

Integers

Yes

No

Yes

No

Whole Numbers

Yes

No

Yes

No

Natural Numbers

Yes

No

Yes

No

Watch this tutorial for more explanation about commutative property
Associativity Property
The Associativity Property states that it doesn't matter how we group the numbers (i.e. which we calculate first) .We can add parenthesis any where and we get the same answer
Here is the summary for the Associative property on various mathematical operations on various numbers
Numbers

Associative Under


addition

subtraction

multiplication

division

Rational numbers

Yes

No

Yes

No

Integers

Yes

No

Yes

No

Whole Numbers

Yes

No

Yes

No

Natural Numbers

Yes

No

Yes

No

Watch this tutorial for more explanation about Associative property
Additive Identity/Role of Zero
We observe
a + 0 = 0 +
a =
a, where
a is a whole number
b + 0 = 0 +
b =
b, where
b is an integer
c + 0 = 0 +
c =
c, where
c is a rational number
Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well
Multiplicative identity/Role of one
We observe
a × 1 = =
a, where
a is a whole number
b × 1 =
b, where
b is an integer
c × 1 =
c, where
c is a rational number
1
is the multiplicative identity for rational numbers. It is the
multiplicative identity for integers and whole numbers as well
Additive Inverse
We observe
a + (a) =
0, where
a is a whole number
b +(b) =
0, where
b is an integer
(a/b) + (a/b) =
0, where
a/b is a rational number
So we say that (a/b) is the additive inverse of a/b and a/b is the additive inverse for (a/b)
Example
What is the additive inverse of following numbers
a) 1/6
b) 8/9
Solution:
Additive Inverse of any rational number a/b is defined as a/b so that (a/b) +(a/b)=0
So for 1/6, additive inverse will be 1/6
Similarly for 8/9 ,it will be 8/9
Reciprocal or multiplicative inverse
The multiplicative inverse of any rational number a/b is defined as b/a so that (a/b) x (b/a) =1
Zero does not have any reciprocal or multiplicative inverse
Example
Find the multiplicative inverse of the following.
54
11/12
Solution
The multiplicative inverse of any rational number a/b is defined as b/a so that (a/b) x (b/a) =1
So answer would be
54=54/1 multiplicative inverse will 1/54 =1/54
Similarly for 11/12, the multiplicative inverse will be 12/11
Watch this tutorial for learning about additive and multiplicative inverse
Distributivity of Multiplication over Addition and Subtraction.
For all rational numbers a, b and c,
a (b + c) = ab + ac
a (b – c) = ab – ac
How to Represent Rational Number on the Number Line
We already know from last grades on how to represent Natural numbers, whole Numbers and integers on the Number Line
Natural Numbers
Whole Numbers
Integers
Rational numbers can also be represented on the Number line
Lets take an example of 4/9
Step 1) In a rational number, the numeral below the bar, i.e., the denominator, tells the number of equal parts into which the first unit has been divided. The numeral above the bar i.e., the
numerator, tells ‘how many’ of these parts are considered
So in this particular example, we need to divide the first unit into 9 parts and we need to move to the 4
^{th} part
Step 2) Draw the straight line. Divide the first unit into equal parts equal to the denominator of the rational number. And then mark each equal part to the right of the line and reach the desired number
Rational Numbers between Two Rational Numbers
There are countless rational number between two rational number.
Now we will see some example to find the rational number between the rational numbers
Example 1
Find the 3 rational number between 1 and 2
Solution
Method 1 Average method
We know the average of two rational number lies between them and it is a rational number
Step 1: First average =(1+2)/2=3/2
So 3/2 lies between 1 and 2
Step 2: Average of 1 and 3/2 will also lie between 1 and 3/2 and hence 1 and 2,
So =1/2 (1+3/2 )=5/4
Step 3.: Again Average of 3/2 and 2 will also lie between 1 and 2. So 1/2 (3/2+2)=7/4
So 3/2,5/4 and 7.4 are rational Number
In General If a and b are two rational numbers, then (a+b)/2 is a rational number between a and b such that a < (a+b)/2< b.
Method 2
We can write the number like 10/10 , 20/10
So the rational number would 11/10, 12/10, 13/10,14/10,15/10
Example 2
Find the 3 rational number between 1/3 and 7/2
Solution
Method 1 Average method
We know the average of two rational number lies between them and it is a rational number
So step 1: First average =(1/3+7/2)/2=23/24
So 23/24 lies between 1/3 and 7/2
Step 2: Average of 1/3 and 23/24 will also lie between 1/3 and 23/24 and hence 1/3 and 7/2
So = ½(1/3+23/24)=31/48
Step 3.: Again Average of 23/24 and 7/2 will also lie between 1/3 and 7/2. So=1/2(23/24+7/2)=107/48
Method 2: Another simple way would be to convert the denominator into same value
1/3 =2/6
7/2=21/6
Now 2/6 <3/6 < 4/6< 5/6 …..< 20/6 < 21/6
So This way we can easy find the rational numbers
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