**Introduction**

A number system is a way to express the numbers of a given set using digits or symbols in a consistent manner. Let us recall some of them that we studied earlier.

**Natural Numbers**

The numbers 1, 2, 3, 4,... are called natural numbers. They are usually represented by the symbol *N*.

**Whole Numbers **

The numbers 0, 1, 2, 3, 4,... are called whole numbers. They are usually represented by the symbol W.

**Integers **

The numbers −3,−2,−1, 0, 1, 2, 3,... are called integers. They are usually represented by the symbol Z, which is the abbreviation of the German word 'Zahlen', which means ‘to count’.

Co-prime integers: Two integers are said to be co-prime if they have no common factors except 1.

**Rational Numbers**

A rational number is a number of the form $\frac{p}{q}$, where *p *and *q *are integers and *q ≠ 0. *The word 'rational' is derived from the word 'ratio'. The set of all rational numbers is represented by the symbol *Q*, which comes from the word 'Quotient'.

-2, 0, 3, $\frac{2}{3},\frac{9}{4},\frac{-2}{3}$ etc. are rational numbers. In other words, all the integers, whole numbers, and natural numbers are included in the collection of rational numbers.

Notes:

* Between any two given rational numbers, there are infinitely many rational numbers.

* Equivalent rational numbers: Equivalent rational numbers are obtained by either multiplying or dividing both the numerator and the denominator by the same value.

For example, to find the equivalent rational numbers of the number $\frac{3}{5}$, multiply both the numerator and the denominator by a number, say 2.

$\frac{6}{12},\frac{12}{20},\frac{24}{40}$ etc. are rational numbers equivalent to the number $\frac{3}{5}$.

* Finding a rational number between two rational numbers:

1. If $\frac{\mathrm{a}}{\mathrm{b}}$ and $\frac{\mathrm{c}}{\mathrm{d}}$ are two rational numbers, then $\frac{\mathrm{a}+\mathrm{c}}{\mathrm{b}+\mathrm{d}}$ is a rational number that lies between $\frac{\mathrm{a}}{\mathrm{b}}$ and $\frac{\mathrm{c}}{\mathrm{d}}$.

2. If $\frac{\mathrm{a}}{\mathrm{b}}$ and $\frac{\mathrm{c}}{\mathrm{b}}$ are two rational numbers, then $\frac{\mathrm{a}+\mathrm{c}}{2\mathrm{b}}$ is a rational number that lies between $\frac{\mathrm{a}}{\mathrm{b}}$ and $\frac{\mathrm{c}}{\mathrm{b}}$.

**Irrational Numbers**

Numbers that are not rational are called irrational numbers, i.e., numbers that cannot be written as simple fractions.

$\sqrt{2},2.4747..,\mathrm{\pi}$ etc, are irrational numbers.

If a and b are two distinct positive irrational numbers, then $\sqrt{\mathrm{ab}}$ is an irrational number lying between a and b.

**Real Numbers**

Every real number is either a rational number or an irrational number.

**Q1: **Find a rational number between $\frac{1}{2}$ and $\frac{1}{3}$.

**Solution** : Add the numerators: 1 + 1 = 2

Add the denominators: 2 + 3 = 5

$\frac{2}{5}$ is a rational number lies between $\frac{1}{2}$ and $\frac{1}{3}$

Alternative method: Make the denominators of both the fractions common

$\frac{1}{2}=\frac{1\times 3}{2\times 3}=\frac{3}{6}$

$\frac{1}{3}=\frac{1\times 2}{3\times 2}=\frac{2}{6}$

It is difficult to find rational numbers between $\frac{2}{6}$ and $\frac{3}{6}$; hence, multiply the numerator and denominator by 10.

$\frac{2}{6}=\frac{20}{60}$

$\frac{3}{6}=\frac{30}{60}$

Hence, the rational numbers between $\frac{20}{60}$ and $\frac{30}{60}$ are $\frac{21}{60},\frac{22}{60},..........\frac{29}{60}$.

∴ $\frac{24}{60}=\frac{2}{5}$ is a rational number between $\frac{1}{2}$ and $\frac{1}{3}$.

**Q2: **Find a rational number between $\frac{3}{5}$ and $\frac{7}{8}$.

Solution: Add the numerators: 3 + 7 = 10

Add the denominators: 5 + 8 = 13

$\frac{10}{13}$ is a rational number between $\frac{3}{5}$ and $\frac{7}{8}$.

**Note:**

There are infinitely many rational numbers between any two rational numbers. We can split two numbers into smaller parts in order to find the numbers between them. It is not possible to pick up each and every number in the number line because of the infinite numbers present in it.

### Irrational Numbers

A number that cannot be expressed in the form $\frac{p}{q}$where '*p' *and '*q'* are integers and *q ≠ 0 *is called an irrational number. The Pythagoreans (followers of the famous mathematician Pythagoras) proved that $\sqrt{2}$ is irrational. In 425 BC, Theodorus of Cyrene demonstrated that $\sqrt{3},\sqrt{5},\sqrt{6},\sqrt{7},\sqrt{10},\sqrt{11},\sqrt{12},\sqrt{13},\sqrt{14},\sqrt{15}\sqrt{17}$ are also irrationals.

In the 1700s, Lamberte and Legendre proved that $\mathrm{\pi}$ is irrational.

Examples:

(a). $\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11}.......$

(b). 1.101001000100001......

(c). $\mathrm{\pi}$ = 3.14159...

The set of all rational and irrational numbers together forms the collection of **real numbers,** denoted by *R*. So, every real number is represented by a point on the number line. Also, every point on the number line represents a unique real number. This had been proved by two German mathematicians, G. Cantor and R. Dedekind. Hence, the number line is called a **real number line**.

**Q1:** Locate $\sqrt{2}$ on the number line.

Here pythagoras theorem is taken for locating $\sqrt{2}$.

By pythagoras theorem,

*Hypotenuse ^{2 }= Base^{2} + Perpendicular^{2}*

^{${\sqrt{2}}^{2}={1}^{2}+{1}^{2}$}

Transfer the figure onto the number line such that the vertex* **C** *coincides with 0 on the number line.

Using a compass, with* **C** *as the center and* **CA** *as the radius, draw an arc touching the number line. The point* **P** *where the arc touches the number line denotes $\sqrt{2}$ on the line.

**Q2:** Locate $\sqrt{5}$ on the number line.

Pythagoras theorem is taken for locating $\sqrt{5}$.

By Pythagoras theorem,

*Hypotenuse ^{2 }= Base^{2} + Perpendicular^{2}*

${\sqrt{5}}^{2}={2}^{2}+{1}^{2}$

Transfer the figure onto the number line such that the vertex* **C** *coincides with 0 on the number line.

Using a compass, with* **O** *as the center and* **OB** *as the radius, draw an arc touching the number line. The point* **P** *where the arc touches the number line denotes $\sqrt{5}$ on the line.