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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). It includes all natural, whole number and integers.

Example: 1/2, 4/3, 5/7,1 etc. 

Natural Numbers

All the positive integers from 1, 2, 3,……, ∞.

Whole Numbers

All the natural numbers including zero are called Whole Numbers.

Integers

All negative and positive numbers including zero are called Integers.

Properties of Rational Numbers

1. Closure Property

This shows that the operation of any two same types of numbers is also the same type or not.

a. Whole Numbers

If p and q are two whole numbers then

Operation	Addition	Subtraction	Multiplication	Division
Whole number	p + q will also be the whole number.	p – q will not always be a whole number.	pq will also be the whole number.	p ÷ q will not always be a whole number.
Example	6 + 0 = 6	8 – 10 = – 2	3 × 5 = 15	3 ÷ 5 = 3/5
Closed or Not	Closed	Not closed	Closed	Not closed

b. Integers

If p and q are two integers then

Operation	Addition	Subtraction	Multiplication	Division
Integers	p+q will also be an integer.	p-q will also be an integer.	pq will also be an integer.	p ÷ q will not always be an integer.
Example	– 3 + 2 = – 1	5 – 7 = – 2	– 5 × 8 = – 40	– 5 ÷ 7  = – 5/7
Closed or not	Closed	Closed	Closed	Not  closed

c. Rational Numbers

If p and q are two rational numbers then

Operation	Addition	Subtraction	Multiplication	Division
Rational Numbers	p + q will also be a rational number.	p – q will also be a rational number.	pq will also be a rational number.	p ÷ q will not always be a rational number
Example				p ÷ 0= not defined
Closed or Not	Closed	Closed	Closed	Not closed

2. Commutative Property

This shows that the position of numbers does not matter i.e. if you swap the positions of the numbers then also the result will be the same.

a. Whole Numbers

If p and q are two whole numbers then 

Operation	Addition	Subtraction	Multiplication	Division
Whole number	p + q = q + p	p – q ≠ q – p	p × q = q × p	p ÷ q ≠ q ÷ p
Example	3 + 2 = 2 + 3	8 –10 ≠ 10 – 8 – 2 ≠ 2	3 × 5 = 5 × 3	3 ÷ 5 ≠ 5 ÷ 3
Commutative	yes	No	yes	No

b. Integers

If p and q are two integers then

Operation	Addition	Subtraction	Multiplication	Division
Integers	p + q = q + p	p – q ≠ q – p	p × q = q × p	p ÷ q ≠ q ÷ p
Example	True	5 – 7 = – 7 – (5)	– 5 × 8 = 8 × (–5)	– 5 ÷ 7 ≠ 7 ÷ (-5)
Commutative	yes	No	yes	No

c. Rational Numbers

If p and q are two rational numbers then

Operation	Addition	Subtraction	Multiplication	Division
Rational numbers	p + q = q + p	p –q ≠ q – p	p × q = q × p	p ÷ q ≠ q ÷ p
Example
Commutative	yes	No	yes	No

3. Associative Property

This shows that the grouping of numbers does not matter i.e. we can use operations on any two numbers first and the result will be the same.

a. Whole Numbers

If p, q and r are three whole numbers then

Operation	Addition	Subtraction	Multiplication	Division
Whole number	p + (q + r) = (p + q) + r	p – (q – r) = (p – q) – r	p × (q × r) = (p × q) × r	p ÷ (q ÷ r)  ≠ (p ÷ q) ÷ r
Example	3 + (2 + 5) = (3 + 2) + 5	8 – (10 – 2) ≠ (8 -10) – 2	3 × (5 × 2) = (3 × 5) × 2	10 ÷ (5 ÷ 1) ≠ (10 ÷ 5) ÷ 1
Associative	yes	No	yes	No

b. Integers

If p, q and r are three integers then

Operation	Integers	Example	Associative
Addition	p + (q + r) = (p + q) + r	(– 6) + [(– 4)+(–5)] = [(– 6) +(– 4)] + (–5)	Yes
Subtraction	p – (q – r) = (p – q) – r	5 – (7 – 3) ≠ (5 – 7) – 3	No
Multiplication	p × (q × r) = (p × q) × r	(– 4) × [(– 8) ×(–5)] = [(– 4) × (– 8)] × (–5)	Yes
Division	p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r	[(–10) ÷ 2] ÷ (–5) ≠ (–10) ÷ [2 ÷ (– 5)]	No

c. Rational Numbers

If p, q and r are three rational numbers then

Operation	Integers	Example	Associative
Addition	p + (q + r) = (p + q) + r		yes
Subtraction	p – (q – r) = (p – q) – r		No
Multiplication	p × (q × r) = (p × q) × r		yes
Division	p ÷ (q ÷ r)  ≠ (p ÷ q) ÷ r		No

The Role of Zero in Numbers (Additive Identity)

Zero is the additive identity for whole numbers, integers and rational numbers.

	Identity		Example
Whole number	a + 0 = 0 + a = a	Addition of zero to whole number	2 + 0 = 0 + 2 = 2
Integer	b + 0 = 0 + b = b	Addition of zero to an integer	False
Rational number	c + 0 = 0 + c = c	Addition of zero to a rational number	2/5 + 0 = 0 + 2/5 = 2/5

The Role of one in Numbers (Multiplicative Identity)

One is the multiplicative identity for whole numbers, integers and rational numbers.

	Identity		Example
Whole number	a ×1 = a	Multiplication of one to the whole number	5 × 1 = 5
Integer	b × 1= b	Multiplication of one to an integer	– 5 × 1 = – 5
Rational Number	c × 1= c	Multiplication of one to a rational number

Negative of a Number (Additive Inverse)

	Identity		Example
Whole number	a +(- a) = 0	Where a is a  whole number	5 + (-5) = 0
Integer	b +(- b) = 0	Where b is an integer	True
Rational number	c + (-c) = 0	Where c is a rational number

Reciprocal (Multiplicative Inverse)

The multiplicative inverse of any rational number

Example

The reciprocal of 4/5 is 5/4.

Distributivity of Multiplication over Addition and Subtraction for Rational Numbers

This shows that for all rational numbers p, q and r

1. p(q + r) = pq + pr

2. p(q – r) = pq – pr

Example

Check the distributive property of the three rational numbers 4/7,-( 2)/3 and 1/2.

Solution

Let’s find the value of

This shows that

Representation of Rational Numbers on the Number Line

On the number line, we can represent the Natural numbers, whole numbers and integers as follows

Rational Numbers can be represented as follows

Rational Numbers between Two Rational Numbers

There could be n number of rational numbers between two rational numbers. There are two methods to find rational numbers between two rational numbers.

Method 1

We have to find the equivalent fraction of the given rational numbers and write the rational numbers which come in between these numbers. These numbers are the required rational numbers.

Example

Find the rational number between 1/10 and 2/10.

Solution

As we can see that there are no visible rational numbers between these two numbers. So we need to write the equivalent fraction.

2/10 = 20/100((multiply the numerator and denominator by 10)

Hence, 2/100, 3/100, 4/100……19/100 are all the rational numbers between 1/10 and 2/10.

Method 2

We have to find the mean (average) of the two given rational numbers and the mean is the required rational number.

Example

Find the rational number between 1/10 and 2/10.

Solution

To find mean we have to divide the sum of two rational numbers by 2.

3/20 is the required rational numbers and we can find more by continuing the same process with the old and the new rational number.

Remark: 1. This shows that if p and q are two rational numbers then (p + q)/2 is a rational number between p and q so that

p < (p + q)/2 < q.

2. There are infinite rational numbers between any two rational numbers.

Fractions

• The numbers of the form a/b, where a and bare natural numbers is known as fraction.

• Proper fraction: A fraction whose numerator is less than its denominator.

• Improper fraction: A fraction in which numerator is greater than denominator.

• Mixed fraction: A combination of a natural number and a proper fraction.

Fractions tell about “a part of a whole”.

Here the pizza is divided into 4 equal parts and there are 3 parts left with us.

We will write it in a fraction as 3/4, in which 3 is numerator which tells the number of parts we have and 4 is denominator which tells the total parts in a whole.

The General form of a Fraction

Where, denominator ≠ 0

If numerator = denominator then the fraction becomes a whole i.e. 1. This is called unity of fraction.

Types of Fraction

Type of Fraction	Meaning	Example
Proper fraction	When numerator is less than the denominator. It shows the part of a whole.
Improper fraction	When numerator is more than the denominator. It represents the mixture of whole and a proper fraction.
Mixed Fraction	The improper fraction can be written in the mixed form as it is the mixture of whole number and a fraction.
Like Fraction	The fractions with the same denominator are like fractions.
Unlike Fraction	The fractions with different denominators are unlike fractions.
Equivalent Fraction	The fractions proportional to each other are called equivalent fractions. It represents the same amount with different fractions.

Converting a Mixed Fraction into an Improper Fraction

Converting an Improper Fraction into a Mixed Fraction

Divide the Numerator by the denominators that the quotient will be the whole number and remainder will be the numerator, while denominator will remain the same. 

How to find the equivalent fractions?

To find the equivalent fraction of proper and improper fraction, we have the multiply both the numerator and denominator with the same number.

Example

Reciprocal of a Fraction

If we have two non-zero numbers whose product is one then these numbers must be the reciprocals of each other.

To find the reciprocal of any fraction, we just need to flip the numerator with the denominator.

Multiplication of Fractions

1. How to multiply a fraction with a whole number?

a. If we have to multiply the proper or improper fraction with the whole number then we simply multiply the numerator with that whole number and the denominator will remain the same.

Example

b. If we have to multiply the mixed fraction with the whole number then first convert it in the form of improper fraction then multiply as above.

Example

c. Fraction as an operator “of”.

If it is written that find the 1/2 of 24 then what does ‘of’ means here?

Here ‘of’ represents the multiplication.

2. How to multiply a fraction with another fraction?

If we have to multiply the proper or improper fraction with another fraction then we simply multiply the numerator of both the fractions and the denominator of both the fractions separately and write them as the new fraction.

Example

Value of the products of the fractions

Generally when we multiply two numbers then we got the result which is greater than the numbers.

5 × 6 = 30, where, 30 > 5 and 30 > 6

But in case of a fraction, it is not always like that.

a. The product of two proper fractions

If we multiply two proper fractions then their product will be less than the given fractions.

Example

b. The product of two improper fractions

If we multiply two improper fractions then their product will be greater than the given fractions.

Example

c. The product of one proper and one improper fraction

If we multiply proper fraction with the improper fraction then the product will be less than the improper fraction and greater than the proper fraction.

Example

Division of Fractions

1. How to divide a whole number by a Fraction?

a. If we have to divide the whole number with the proper or improper fraction then we will multiply that whole number with the reciprocal of the given fraction.

Example

b. If we have to divide the whole number with the mixed fraction then we will convert it into improper fraction then multiply it’s reciprocal with the whole number.

Example

2. How to divide a Fraction with a whole number?

To divide the fraction with a whole number, we have to take the reciprocal of the whole number then divide it with the whole number as usual

Example

3. How to divide a fraction with another Fraction?

To divide a fraction with another fraction, we have to multiply the first fraction with the reciprocal of the second fraction.

Example

Decimal Numbers

Fractions which has denominator 10, 100, 1000 etc are called Decimal Fractions.

A decimal number is a number with a decimal point. Numbers left to the decimal are 10 greater and numbers to the right of the decimal are 10 smaller.

Multiplication of Decimal Numbers

1. How to multiply a decimal number with a whole number?

If we have to multiply the whole number with a decimal number then we will multiply them as normal numbers but the decimal place will remain the same as it was in the original decimal number.

Example

35 × 3.45 = 120.75

Here we have multiplied the number 35 with 345 as normal whole numbers and we put the decimal at the same place from the right as it was in 3.45.

2. How to multiply Decimal numbers by 10,100 and 1000?

a. If we have to multiply a decimal number by 10 then we will transfer the decimal point to the right by one place.

Example

5.37 × 10 = 53.7

b. If we have to multiply a decimal number by 100 then we will transfer the decimal point to the right by two places.

Example

5.37 × 100 = 537

c. If we have to multiply a decimal number by 1000 then we will transfer the decimal point to the right by three places.

Example

5.37 × 1000 = 5370

3. How to multiply a decimal number by another decimal number?

To multiply a decimal number with another decimal number we have to multiply them as the normal whole numbers then put the decimal at such place so that the number of decimal place in the product is equal to the sum of the decimal places in the given decimal numbers.

Example

Division of Decimal Numbers

1. How to divide a decimal number with a whole number?

If we have to divide the whole number with a decimal number then we will divide them as whole numbers but the decimal place will remain the same as it was in the original decimal number.

Example

12.96 ÷ 4 = 3.24

Here we divide the number 1296 with 4 as normal whole numbers and we put the decimal at the same place from the right as it was in 12.96.

2. How to divide Decimal numbers by 10,100 and 1000?

a. If we have to divide a decimal number by 10 then we will transfer the decimal point to the left by one place.

Example

5.37 ÷ 10 = 0.537

b. If we have to divide a decimal number by 100 then we will transfer the decimal point to the left by two places.

Example

253.37 × 100 = 2.5337

c. If we have to divide a decimal number by 1000 then we will transfer the decimal point to the left by three places.

Example

255.37 × 1000 = 0.25537

3. How to divide a decimal number by another decimal number?

To divide a decimal number with another decimal number

• First, we have to convert the denominator as the whole number by multiplying both the numerator and denominator by 10, 100 etc
• Now we can divide them as we had done before.

Example

Here we had converted denominator 2.4 in the whole number by multiplying by 10.Then divide it as usual.

Integers

• Integers are the collection of whole numbers and their  negatives. Positive Integers are 1, 2, 3 … . Negative Integers are 1, 2, 3 … .

• Every positive integers is greater than every negative integers.

• Zero is less than every positive integers and greater than every negative integers.

Representation of integers on the number line.

Integers are closed under addition. In general, for any two integers a and b, a + b is an integer.

Integers are closed under subtraction. Thus, if a and b are two integers then a – b is also an integer.

Addition is commutative for integers. In general, for any two integers a and b, we can say a + b = b + a

Subtraction is not commutative for integers.

Addition is associative for integers.

In general for any integers a, b and c, we can say a + (b + c) = (a + b) + c

Zero is an additive identity for integers. In general, for any integer a
a + 0 = a = 0 + a

While multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (-) before the product. We thus get a negative integer. In general, for any two positive integers a and b we can say a × (-b) = (-a) × b = -(a × b)

Product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put positive sign before the product. In general, for any two positive integers a and b, (-a) × (-b) = a × b

Integers are closed under multiplication. a × b is an integer, for all integers a and b,

Multiplication is commutative for integers. In general, for any two integers a and b, a × b = b × a

The product of a negative integer and zero is zero a × 0 = 0 × a=0

1 is the multiplicative identity for integers.
a × 1 = 1 × a = a

Multiplication is associative for integers, (a × b) × c = a × (b × c)

The distributivity of multiplication over addition is true for integers.
a × (b + c) = a × b + a × c

The distributivity of multiplication over subtraction is true for integers.
a × (b – c) = a × b – a × c

When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (-) before the quotient.
a ÷ (-b) = (-a) ÷ b where b ≠ 0

When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+).
(-a) ÷ (-b) = a ÷ b where b ≠ 0

Any integer divided by 1 gives the same number.
a ÷ 1 = a

For any integer a, we have a ÷ 0 is not defined.

Natural numbers, whole numbers and integers: The numbers 1, 2, 3,……… which we use for counting are known as natural numbers. The natural numbers along with zero forms the collection of whole numbers.
The numbers……., -3, -2, -1, 0, 1, 2, 3, form the collection of integers.

Integers	Whole numbers
1. The integers form a bigger group which contains whole numbers and negative numbers.	1. The whole numbers do not form a group as big as integers because they do not contain negative numbers.
2. The group of integers includes all the whole numbers.	2. The group of whole numbers does not include all the integers.
3. There is no smallest integer.	3. 0 is the smallest whole number.
4. Integers are closed under subtraction.	4. Whole numbers are not closed under subtraction.

In this chapter, we shall learn more about integers, their properties and operations.

Properties of Addition and Subtraction of Integers
Closure Under Addition
We know that the addition of two whole numbers is again a whole number. For example, 17 + 24 = 41 which is a whole number. This property is known as the closure property for the addition of whole numbers.

This property is true for integers also, i.e., the sum of two integers is always an integer. We cannot find a pair of integers whose addition is not an integer. Since additions of integers give integers, we can say integers are closed under’addition just like whole numbers. In general, for any two integers a and b, a + b is also an integer.

Closure Under Subtraction
If we subtract two integers, then their difference is also an integer. We cannot find any pair of integers whose difference is not an integer. Since subtraction of integers gives integers, we can say integers are closed under subtraction. In general, for any two integers a and b, a – b is also an integer.

Note: The whole numbers do not satisfy this property.
For example: 5 – 7 = -2 which is not a whole number.

Commutative Property
Commutativity of Addition: We know that 3+ 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In other words, addition is commutative for whole numbers. Similarly, the addition is commutative for integers.
We cannot find any pair of integers for which the sum is different when the order is changed. So, we conclude that addition is commutative for integers also. In general, for any two integers a and b, we can say that a + b = b + a.

Commutativity of Subtraction: We know that the subtraction is not commutative for whole numbers.
For example, 10 – 20 = -10 and 20 – 10 = 10
So, 10 – 20 ≠ 20 – 10
Similarly, the subtraction is not commutative for integers.

Associative Property
We cannot find any example for which sum is different when the order of addition is changed. This shows that addition is associative for integers.
In general, for any integers a, b and c, we can say that a + (b + c) = (a + b) + c

Additive Identity
When we add zero to any whole number {i.e., zero and positive integer), we get the same whole number. So, zero is an additive identity for whole numbers. In particular, we can say that zero is an additive identity for positive integers.
Consider the following examples:
(-8) + 0 = -8
(-23) + 0 = -23
0 + (-37) = -37
0 + (-59) = -59
0 + (-43) = -43
-61 + 0 = -61
-50 + 0 = -50
These examples show that zero is an additive identity for negative integers also. Thus, we can say that zero is an additive identity for integers. In general, for any integer a, a + 0 = a = 0 + a

Product of Three or More Negative Integers
We find that if the number of negative integers in a product is even, the product is a positive integer; if the number of negative integers in a product is odd, the product is a negative integer.

Properties of Multiplication of Integers
Closure Under Multiplication
Closure: Let us observe the following table:

We observe that the product of two integers is an integer. We cannot find a pair of integers whose product is not an integer. This gives an idea that the product of two integers is again an integer. So, we say that integers are closed under multiplication. In general, a × b is an integer, for all integers a and b.

Commutativity of Multiplication
We know that multiplication is commutative for whole numbers (i.e., zero and positive
integers). Now, let us observe the following table:

We observe that two integers can be multiplied in any order. The above examples suggest commutativity of multiplication of integers. So, in general, we can say that for any two integers a and b, a × b = b × a.

Multiplication by Zero
We know that any whole number [i.e., zero and positive integers] multiplied by zero gives zero. Let us observe the following table showing the product of a negative integer and zero.
(-3) × 0 = 0
0 × (-4) = 0
(-5) × 0 = 0
0 × (-6) = 0
This table shows that the product of a negative integer and zero is again zero.
In general, for any integer a, a × 0 = 0 × a = 0

Multiplicative Identity
We know that 1 is the multiplicative identity for whole numbers (i.e., zero and positive integers). Let us observe the following table showing the product of a negative integer and 1.
(-3) × 1 = -3
(-4) × 1 = -4
1 × (-5) = -5
1 × (-6) = -6
This table shows that 1 is the multiplicative identity for negative integers also. In general, for any integer a, we have,
a × 1 = 1 × a = a

Multiplication with (-1): Let us observe the following table showing the product of an integer and (-1).
(-3) × (-1) = 3
3 × (-1) = – 3
(-6) × (-1) = 6
(-1) × 13 = -13
(-1) × (-25) = 25
18 × (-1) = -18.
This table shows that (-1) is not the multiplicative identity for integers because when we multiply an integer with (-1) or (-1) with an integer, the result is the integer with the sign changed, i.e., we do not get the same integer. Therefore, for any integer a, we have, a × (-1) = (-1) × a = -a ≠ a

Note: 0 is the additive identity whereas 1 is the multiplicative identity for integers. We get additive inverse of an integer a when we multiply (-1) to a,
i.e., a × (-1) = (-1) × a = -a.

Associativity for Multiplication
Take the integer (- 3). Multiply it with (- 2) to get 6, i.e., (-3) × (-2) = 6.
Then, multiply the product 6 with 5 to get 30, i.e., [(-3) × (-2)] × 5 = 6 × 5 = 30.
Also, (-2) × 5 = (-10).
Multiply integer (-3) with (-10) to get 30.
i.e., (-3) × [(-2) × 5] = (-3) × (-10) = 30.
So, we get the same answer in both the processes, i.e., we get [(-3) × (-2)] × 5 = (-3) × [(-2) × 5]
We observe that the arrangement of integers does not affect the product of integers.
In general, for any three integers a, b and c, (a × b) × c = a × (b × c)
Thus, like whole numbers, the product of three integers does not depend upon the arrangement of integers and this is called associative property for multiplication of integers.

Distributive Property
(i) Distributivity of Multiplication Over Addition: We know that the property of distributivity of multiplication over addition is true for whole numbers.
For example: 16 × (10 + 2) = (16 × 10) + (16 × 2).

(ii) Distributivity of Multiplication Over Subtraction: We know that the property of distributivity of multiplication over subtraction is true for whole numbers (i.e. zero and positive integers).
For example: 4 × (3 – 8) = 4 × 3 – 4 × 8
This property is also true for integers.
For example:
(-9) × [10-(-3)] = (-9) × 13 = -117
and, -9 × 10 – (-9) × (-3) = -90 – 27 = -117
So, (-9) × [10-(-3)]=(-9) × 10 – (-9) × (-3).
We find that these are also equal.
In general, for any three integers a, b and c, a × (b – c) = a × b – a × c.

Division of Integers
1. The division is the inverse operation of multiplication.

Observing the entries in the above table, we find that

• When we divide a negative integer by a positive integer, we get a negative integer.
• When we divide a positive integer by a negative integer, we get a negative integer.
• When we divide a negative integer by a negative integer, we get a positive integer.

2. Division of a negative integer by a positive integer
We observe that
(-12) ÷ 6 = -2 = -(12 ÷ 6)
(-32) ÷ 4 = -8 = -(32 ÷ 4)
(-45) ÷ 5 = -9 = -(45 ÷ 5)
(-12) ÷ 2 = -6 = -(12 ÷ 2)
(- 20) ÷ 5 = -4 = -(20 ÷ 5)
So, we find that while dividing a negative integer by a positive integer, we divide them as whole numbers and put a minus sign (-) before the quotient (i.e. we get a negative integer).

3. Division of a positive integer by a negative integer
We also observe that
72 ÷ (- 8) = -9 = – (72 ÷ 8)
21 ÷ 7 = -3 = -(27 ÷ 7)
This shows that while dividing a positive integer by a negative integer, we divide them as whole numbers and put a minus sign (-) before the quotient (i.e., we get a negative integer).

4. If the dividend and divisor are of opposite sign, then the quotient is negative integer.
Wehave, (—48) ÷ 8= -(48 ÷ 8) = -6
(48) ÷ (-8) = -(48÷8) = -6
So, (-48) ÷ 8 = -6 = 48 ÷ (-8)

5. Division of a negative integer by a negative integer
Lastly we observe that
(-20) ÷ (-4) = 5 = 20 ÷ 4
(-12) ÷ (-6) = 2 = 12 ÷ 6
(-32) ÷ (-8) = 4 = 32 ÷ 8
(-45) ÷ (-9) = 5 = 45 ÷ 9
Here, we notice that while dividing a negative integer by a negative integer, we divide them as whole numbers and put a positive sign i.e. we get a positive integer. We can say that if dividend and divisor are of same signs, then the quotient is a positive integer.

Properties of Division of Integers
(i) Closure: We know that integers are closed under addition, subtraction and multiplication. However, the integers are not closed under division. It can be observed from the following table:

(ii) Commutativity: We know that division is not commutative for whole numbers. For example 16 ÷ 4 ≠ 4 ÷ 16.
Similarly, the division is not commutative for integers.
Note: The division is commutative for integers when the dividend and divisor are equal.

(iii) Like whole numbers, any integer divided by zero is meaningless and zero divided by any integer (other than zero) is equal to zero, i.e., for any integer a, a + 0 is not defined but 0 ÷ a (≠0) = 0.

(iv) When we divide a whole number (i.e., zero and positive integers) by 1, it gives the same whole number.
It is true for negative integers also. For example:
(-8) ÷ 1 = -8
(-11) ÷ 1 = -11
These examples show that negative integer divided by one gives the same negative integer. So, any integer divided by 1 gives the same integer. In general, we can say that for any integer a, a ÷ 1 = a.

The iron pillar on pillar

The iron pillar at Mehrauli, Delhi, shows the outstanding skill of Indian crafts persons. It is made of iron, 7.2. m high, and weighs over 3 tonnes. It was made about 1500 years ago. The date has been inscribed on the pillar mentioning a ruler named Chandra, who probably belonged to the Gupta dynasty. One interesting fact of the pillar is that it has not rusted in all these years.

Buildings in brick and stone

Stupas also reflect the skills of crafts persons that have survived. The word stupa means a mound. Stupas can be of different kinds round and tall, big and small, which will have certain common features. At the centre of the stupa, a small box will be placed, which contained bodily remains (such as teeth, bone or ashes) of the Buddha or his followers, or things they used, as well as precious stones, and coins.

This box, known as a relic casket, was covered with earth. Later, a layer of mud brick or baked brick was added on top. And then, the dome-like structure was sometimes covered with carved stone slabs.

A path, known as the pradakshina patha, was laid around the stupa surrounded with railings. The entrance of the path was through gateways. Devotees walked around the stupa, in a clockwise direction, as a mark of devotion. Both railings and gateways were often decorated with sculpture.

During this period, some of the earliest Hindu temples were also built. Deities such as Vishnu, Shiva, and Durga were worshipped in these shrines. The most important part of the temple was the room known as the garbhagriha, where the image of the chief deity was placed. It was here that priests performed religious rituals, and devotees offered worship to the deity.

Bhitargaon, a tower, known as the shikhara, was built on top of the garbhagriha, to mark this out as a sacred place. Most temples also had a space known as the mandapa a hall where people could assemble.

How were stupas and temples built?

Building stupa and temple went through several stages. Kings or queens wanted to builds these stupas or temples, which was an expensive affair. To start building these sculptures, good quality stone had to be found quarried, and transported to the place chosen for the new building. Here, these rough blocks of stone had to be shaped and carved into pillars, and panels for walls, floors and ceilings. After these, it had to be placed in precisely the right position.

To build these splendid structures, kings and queens spent money from their treasury to pay the crafts persons. Besides, when devotees came to visit the temple or the stupa, they often brought gifts, which were used to decorate the buildings.

Others who paid for these decorations were merchants, farmers, garland makers, perfumers, smiths, and hundreds of men and women who are known only by their names, which were inscribed on pillars, railings and walls.

Painting

Ajanta is a place where several caves were hollowed out of the hills over centuries. Most of these were monasteries for Buddhist monks, and some of them were decorated with paintings.

The world of books

During this period, some of the best-known epics were written. Epics are grand, long compositions, about heroic men and women, and include stories about gods.

A famous Tamil epic, the Silappadikaram, was composed by a poet named Ilango, around 1800 years ago. It is the story of a merchant named Kovalan, who lived in Puhar and fell in love with a courtesan named Madhavi, neglecting his wife Kannagi. Later, he and Kannagi left Puhar and went to Madurai, where he was wrongly accused of theft by the court jeweller of the Pandya king. The king sentenced Kovalan to death. Kannagi, who still loved him, was full of grief and anger at this injustice and destroyed the entire city of Madurai.

Another Tamil epic, the Manimekalai was composed by Sattanar around 1400 years ago. This describes the story of the daughter of Kovalan and Madhavi.

Recording and preserving old stories

A number of Hindu religious stories were written down around the same time. These include the Puranas, meaning old. The Puranas contain stories about gods and goddesses, such as Vishnu, Shiva, Durga or Parvati. They also contain details on how they were to be worshipped.

The Puranas were written in simple Sanskrit verse and were meant to be heard by everybody, including women and Shudras, who were not allowed to study the Vedas. They were probably recited in temples by priests, and people came to listen to them.

Two Sanskrit epics, the Mahabharata and Ramayana had been popular for a very long time. The Mahabharata is about a war fought between the Kauravas and Pandavas, who were cousins. The Ramayana is about Rama, a prince of Kosala, who was sent into exile. His wife Sita was abducted by the king of Lanka, named Ravana, and Rama had to fight a battle to get her back. Valmiki is the author of the Sanskrit Ramayana.

Stories told by ordinary people

Ordinary people also told stories, composed poems and songs, sang, danced, and performed plays. Some of these are preserved in collections of stories such as the Jatakas and the Panchatantra, which were written down around this time. Stories from the Jatakas were often shown on the railings of stupas and in paintings in places such as Ajanta.

Writing books on science

Aryabhata, a mathematician and astronomer, wrote a book in Sanskrit known as the Aryabhatiyam. He stated that day and night were caused by the rotation of the earth on its axis, even though it seems as if the sun is rising and setting everyday. He developed a scientific explanation for eclipses as well. He also found a way of calculating the circumference of a circle.

Prashastis and what they tell us

We all know about Samudragupta, a famous ruler of a dynasty known as the Guptas from a long inscription, inscribed on the Ashokan pillar at Allahabad. It was composed as a Kavya by Harishena, a poet and a minister at the court of Samudragupta. This inscription is of a special kind known as a prashasti, a Sanskrit word, meaning ‘in praise of’.

Samudragupta’s prashasti

In Samudragupta’s prashasti, the poet praised the king in glowing terms such as a warrior, as a king who won victories in battle, who was learned and the best of poets. He is also described as equal to the gods. The prashasti was composed in very long sentences.

Harishena described four different kinds of rulers and told us about Samudragupta’s policies towards them.

1. The rulers of Aryavarta, where nine rulers were uprooted, and their kingdoms were made a part of Samudragupta’s empire.
2. The rulers of Dakshinapatha where twelve rulers surrendered to Samudragupta after being defeated and later he allowed them to rule again.
3. The inner circle of neighbouring states, including Assam, coastal Bengal, Nepal, and a number of gana sanghas in the northwest. They brought tribute, followed his orders, and attended his court.
4. The rulers of the outlying areas, perhaps the descendants of the Kushanas and Shakas, and the ruler of Sri Lanka submitted to him and offered daughters in marriage.

Genealogies

Samudragupta’s prashastis mentioned the ancestors’ names such as Samudragupta’s great grandfather, grandfather, father and mother. His mother, Kumara Devi, belonged to the Lichchhavi gana, while his father, Chandragupta, was the first ruler of the Gupta dynasty who adopted the grand title of maharaj-adhiraja, a title that Samudragupta also used.

Samudragupta figures in the genealogies of later rulers of the dynasty, such as his son, Chandragupta II. He led an expedition to western India, where he overcame the last of the Shakas. According to later belief, his court was full of learned people, including Kalidasa the poet, and Aryabhata the astronomer.

Harshavardhana and the Harshacharita

Harshavardhana, who ruled nearly 1400 years ago, and his biography was written by his court poet, Banabhatta in Sanskrit. He was not the eldest son of his father but became king of Thanesar after both his father and elder brother died. His brother-in-law ruled Kanauj who was killed by the ruler of Bengal. Harsha took over the kingdom of Kanauj and then led an army against the ruler of Bengal.

Harsha was successful in the east and conquered both Magadha and Bengal. He tried to cross the Narmada to march into the Deccan but was stopped by a ruler belonging to the Chalukya dynasty, Pulakeshin II.

The Pallavas, Chalukyas and Pulakeshin’s prashasti

During this period, the Pallavas and Chalukyas were the most important ruling dynasties in south India. The kingdom of the Pallavas spread from the region around their capital, Kanchipuram, to the Kaveri delta, while that of the Chalukyas was centred around the Raichur Doab, between the rivers Krishna and Tungabhadra.

Aihole, the capital of the Chalukyas, was an important trading centre. It developed as a religious centre, with a number of temples. Pulakeshin II was the best-known Chalukya ruler. His prashasti was composed by his court poet Ravikirti, which talks about his ancestors, who are traced back through four generations from father to son.

According to Ravikirti, he led expeditions along both the west and the east coasts. Besides, he checked the advance of Harsha. There is an interesting play of words in the poem. Harsha means happiness. The poet says that after this defeat, Harsha was no longer Harsha!

How were these kingdoms administered?

Land revenue remained important and the village remained the basic unit of administration. But, new developments were also introduced. Kings adopted a number of steps to win the support of men who were powerful, either economically, or socially, or because of their political and military strength. For instance:

• Some important administrative posts were hereditary.
• Sometimes, one person held many offices.
• Besides, important men probably had a say in local administration.

A new kind of army

Kings maintained a well-organised army, with elephants, chariots, cavalry and foot soldiers. Military leaders provided kings with troops whenever he needed them but they were not paid regular salaries. Instead, of salary, some of them received grants of land. They collected revenue from the land and used this to maintain soldiers and horses, and provide equipment for warfare. These men were known as samantas.

Assemblies in the southern kingdoms

The inscriptions of the Pallavas mentioned a number of local assemblies, which included the sabha, an assembly of brahmin landowners. This assembly functioned through subcommittees, which looked after irrigation, agricultural operations, making roads, local temples, etc. There was a village assembly found in areas where the landowners were not brahmins. And the nagaram was an organisation of merchants.

Ordinary people in the kingdoms

Kalidasa was known for his plays depicting life in the king’s court. An interesting feature about these plays is that the king and most brahmins are shown as speaking Sanskrit, while women and men other than the king and brahmins use Prakrit. His most famous play, Abhijnana Shakuntalam, is the story of the love between a king named Dushyanta and a young woman named Shakuntala.