Number theory

Mathematics is the ‘Queen of Science’ and Number theory is the ‘Queen of Mathematics’.

Introduction

Number theory as a fundamental body of knowledge has played a pivotal role in the development of Mathematics.

The Greek Mathematician Pythagoras and his disciples believed that “everything is number” .

HistoryThe system of writing numerals was developed some 10,000 years ago. India was the main centre for the development of the number system which we use today. It took about 5000 years for the complete development of the number system.

HistoryThe Whole numbers are fountain head of all Mathematics. The present system of writing numerals is known as Hindu-Arabic numeral system.

In this system, we use the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is also called the decimal system with base 10. The word ‘decimal’ comes from Latin word ‘Decem’ which means ‘Ten’.

Number System

Natural numbers N = {1, 2, 3, g}, Whole numbers W = {0, 1, 2, g}, Integers Z = {3, – 2, – 1, 0, 1, 2,3g} Rational numbers Q

The numbers of the form where p and q are integers and q ≠ 0 are knownas rational numbers. The collection of numbers of the form , where q > 0 is denoted by Q.

q

p

q

p

Rational numbers include natural numbers, whole numbers, integers and all negative and positive fractions.

Rational numbers

Representation of Rational Numbers on the Number Line

To express rational numbers appropriately on the number line, divide each unit length into as many number of equal parts as the denominator of the rational number and then mark the given number on the number line.

Representation of Rational Numbers on the Number Line

Express on the number line.7

4

Representation of Rational Numbers on the Number Line

Express on the number line.5

17

Representation of Rational Numbers on the Number Line

Express on the number line.3

2

To find rational numbers between two rational numbers

To find rational numbers between two rational numbers

To find rational numbers between two rational numbers

To find rational numbers between two rational numbers

To find rational numbers between two rational numbers

To find rational numbers between two rational numbers

So, unlike natural numbers and integers, there are countless rational numbers between any two given rational numbers.

Four Properties of Rational Numbers

Addition

Addition

Addition

Addition