**(i) Naturalnumber system**

1, 2, 3, … are called natural numbers (or counting numbers). The set of all natural numbers is denoted by **N**.

**N** = {1, 2, 3, …}

**(ii)** **Whole number system**

0, 1, 2, 3, … are called whole numbers. The set of all whole numbers is denoted by **W**.

**W** = {0, 1, 2, 3, 4, …}

All natural numbers are whole numbers, but the whole number 0 is not a natural number. Addition of two (or more) natural numbers or two (or more) whole numbers is also a natural number of a whole number respectively.

(e.g.,) 23 + 17 = 40. 40 is a natural number

0 + 20 = 20. 20 is a whole number

What applies to addition, also applies to multiplication.

Subtraction of one natural or whole number from another natural or whole number need not always be a natural number or a whole number.

For example, 20 – 25 = -5, -5 is neither a natural number nor a whole number.

0 – 7 = -7. Again -7 is neither a whole number nor a natural number.

What applies to subtraction also applies to division.

For example, $$\frac{28}{13} = 2\frac{2}{13}$$

is neither a natural number nor a whole number. What about division by zero? It is not permitted.

**(iii) Integers:**

Since zero is not an element in the set of natural numbers N, the set of whole numbers W was invented. 0 is called the **additive identity **for 0 + any whole number = the same whole number. (0 + 8) = 8 (0 + 0 = 0). Both the natural number system and the whole number system do not have negative numbers. To answer this need came the set of integers denoted by **Z**.

**Z** = {… -3, -2, -1, 0, 1, 2, 3, …}

(Zero is neither positive nor negative).

All whole numbers and natural numbers are integers but all integers are not whole numbers or natural numbers.

In **Z**, addition, subtraction and multiplication have got solutions, but not division.

For example, (i) 12 – 20 = -8. -8 belongs to **Z**

(ii) (-10) + (-17) = -27, -27 belongs to **Z**

** **(iii) 40 x 0 = 0, 0 belongs to **Z**

(iv) -8 x -7 = 56, 56 belongs to **Z**

But $$\frac{-10}{3} = -3\frac{1}{3}$$

does not belong to **Z**. $$ -3\frac{1}{3}$$ is not an integer

Hence the necessity of another system to include such non integer numbers arose.

**(iv)Rational numbers**

A number of the form $$\frac{p}{q}$$ where p and q are integers and q ? 0 is called a rational number. The set of all rational numbers is denoted by **Q**. Every integer (every whole number and every natural number) is a rational number. As -17 is an integer. It can be expressed as $$\frac{-17}{1}$$ which is a rational number. But the converse is not true (i.e., Every rational number need not be an integer). For $$\frac{12}{13}$$ is not an integer, but only a rational number.

**(v) Decimal representation of rational numbers**

A rational number can be expressed as a decimal. For example $$\frac{2}{5}=0.4$$ which is a terminating decimal. Again $$\frac{3}{8}=0.375$$ which is also a terminating decimal.

Now consider $$\frac{3}{7}$$ when 3 is divided by 7,

$$\frac{3}{7}=0.428571428571\ldots$$

It is a non terminating repeating (or recurring) decimal.

are examples of non terminating recurring decimals. From this it is observed that all terminating decimals and repeating (recurring) decimals are rational numbers.

A terminating decimal can be given as a rational number (of the form p/q).

For example (i) -0.35 (a terminating decimal)

$$=\frac{-35}{100}=\frac{-7}{20}$$

(ii) $$1.075=\frac{1075}{1000}=\frac{43}{40}$$

A recurring decimal can also be rewritten as a rational number.

For example

Thus every terminating decimal as well as every recurring decimal is a rational number.

**(vi) Non-terminating and non-recurring decimals ? Irrational numbers**

0.10100100100001 … is an example for non-terminating non-recurring decimal.

$$\sqrt{2}=1.4142135 \ldots$$

$$\sqrt{3}=1.7320508 \ldots$$

are also examples for non terminating, non-recurring decimals.

Such non-terminating, non recurring decimals are called irrational numbers.

**A rational number is either a terminating or non-terminating but recurring decimal.**

**An irrational number is a non-terminating and non-recurring decimal.**

It cannot be given in the form

$$\frac{p}{q}$$

where p and q are integers (q ? 0)

**Real numbers:**

Rational numbers and irrational numbers taken together form the real number system. The set ofreal numbersis denoted by the letter **R**.

**Q** is the set of rational numbers and **Q’** is the set of irrational numbers

**R = Q U Q’**

**The set of real numbers includes all natural numbers, whole numbers, integers, rational numbers and irrational numbers.**