Number System

Let us study different sets of numbers.

W is the set of whole numbers written as

W = {0, 1, 2, 3, 4……}

This familiar set of numbers has the weakness that it cannot supply answers to subtraction problems of the "Smaller take away larger" variety:

For example 2 – 10 = - 8

This problem is overcome by defining the set of integer written as Z.

Z = {…… -4, -3, -2, -1, 0, 1, 2, 3, 4 ……}

Now, we have 2 – 10 = -8 Z

We also know that W$\subset$ Z

A further extension of set of integers is needed, because many division problems with integers do not answer in Z.

Example: 13 ÷ 2 and $\frac{13}{2}$ is not an integer.

Rational number:

To include such numbers we consider set of rational numbers which is defined as follows.

A number is called a rational number if it can be written in the form, where

a, b ? Z, and b ? 0, (also a and b do not have any common factor other than 1)

Example:

$\frac{1}{10}$= 3.33 …. etc.

a rational number is either a terminating decimal or a non – terminating recurring decimal.

Also, W $\subset$ Z $\subset$ Q

Irrational numbers:

There are some numbers such as $\sqrt{2}$, $\sqrt{3}$, $\sqrt{7}$, , etc. which cannot be expressed in the form a/b (where a and b are integers and b?0).

Such numbers are called irrational numbers.

Irrational numbers are non-terminating and non-recurring decimals.

$\sqrt{2}=1.414213……$

$\sqrt{7}= 2.64575……$

$\sqrt{\Pi}=1.7728…...$

Real numbers:

These two distinct set of numbers, i.e. rational numbers and irrational numbers are grouped into one set called the set of " real numbers".

Therefore the set of real numbers 'R' contains all terminating decimals and non- terminating recurring decimals and non-terminating, non-recurring decimals.

This entire set of numbers represented by decimals is called the set of real numbers.

To every point on the number line(real line) there corresponds a real number and to every real number there corresponds a point on the number line.

Thus the collection of all points on the number line can be thought of as the system of real numbers.

Note:

• There is no number that is both rational and irrational, i.e. a real number is either a rational or irrational number.
• The word 'real' is not to be interpreted as meaning anything special. It is a word just to describe the set of numbers