# Rational Numbers

A number which can be written in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0 is called a **rational number**.

For e.g.

\(\frac{-2}{5}\), \(\frac{6}{7}\)

A number which can be written in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0 is called a **rational number**.

For e.g.

\(\frac{-2}{5}\), \(\frac{6}{7}\)

**The symbols used in Roman numerals are:**

I ---> 1; V ---> 5; X ---> 10; L ---> 50; C ---> 100; D ---> 500; M ---> 1000

**Rules for the system are :**

**a)** If a symbol is repeated, its value is added as many times as it occurs:

i.e. II is equal to 2, XXX is equal to 30, CC is equal to 200

**b)** A symbol is not repeated more than three time. But the symbols **V, L** and **D** are never repeated.

**c)** If a symbol of smaller value is written to the left of a symbol of greater value, its value get added to the value of the greater symbol.

i.e. VI is equal to 5 + 1 = 6; CXX is equal to 100 + 10 + 10 = 120

**d)** If a symbol of smaller value is written to the left of a symbol of greater value, its value is subtracted from the value of the greater symbol.

i.e. IV is equal to 5 - 1 = 4; XC is equal to 100 - 10 = 90

**e)** The symbols **V,** **L** and **D** are never written to the left of a symbol of greater value, i.e. **V, L** and **D** are never subtracted. The symbol **I** can be subtracted from **V** and **X** only. The symbol **X** can be subtracted from **L, M** and **C** only.

Consider the following:

**3 ^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}**

The collection of numbers 3, 4 and 5 is known as **Pythagorean triplet**. Similarly 6, 8 and 10 are also pythagorean triplets because

**6 ^{2} + 8^{2} = 100 = 10^{2}**

For any natural number **m > 1**, we have **(2m) ^{2} + (m^{2} - 1)^{2} = (m^{2} + 1)^{2}**. So,

**Definition: **The natural numbers (i.e. 1, 2, 3, 4, and so no) along with zero form the collection of whole numbers.

Therefore:-

- All natural numbers are whole numbers.

**Properties of Whole Numbers:**

**Closure property:**Whole numbers are closed under addition and also under multiplication.

It means that when you add two or more whole numbers, the result also be a whole number.

Similarly, if you multiply two or more whole numbers, the result will be a whole number.**Commutativity of addition and multipication:**Two whole numbers can be added in any order. Also two whole numbers can be multiplied in any order.**Associativity of addition and multiplication:**If you solve 2 + ( 3 + 4 ) and ( 2 + 3 ) + 4, you will see that the result is same. Similary, if you solve 2 x ( 3 x 4 ) and ( 2 x 3 ) x 4, you will see that the result is same. This property is known as associativity of addition and multiplication.**Distributivity of multiplication over addition:**You can solve 2 x ( 3 + 5 ) in two ways:

2 x ( 3 + 5 ) = 2 x 8 = 16

or

2 x ( 3 + 5 ) = 2 x 3 + 2 x 5 = 6 + 10 = 16

The second way of solving the above equation is known as distributivity of multiplication over addition.