NUMBER SYSTEMS

Maths Project

REAL NUMBERS

Real Number

s

Rational

Irrational

INTRODUCTION

In mathematics, a real number is a

value that represents a quantity along

a continuous line. The real numbers

include all the rational numbers, such

as the integer −5 and the fraction 4/3,

and all the irrational numbers such as

√2 (1.41421356… the square root of

two, an irrational algebraic number)

and π (3.14159265…, a

transcendental number).

Real numbers can be thought of as

points on an infinitely long line called

the number line or real line, where the

points corresponding to integers are

equally spaced. The reals

are uncountable, that is, while both the

set of all natural numbers and the set

of all real numbers are infinite sets.

BASIC PROPERTIES

A real number may be

either rational or irrational;

either algebraic or transcendental;

and either positive, negative, or zero.

Real numbers are used to

measure continuous quantities. They

may be expressed by decimal

representations that have an infinite

sequence of digits to the right of the

decimal point; these are often

represented in the same form as

324.823122147…

More formally, real numbers have the

two basic properties :-

The first says that real numbers

comprise a field, with addition and

multiplication as well as division by

nonzero numbers, which can be totally

ordered on a number line in a way

compatible with addition and

multiplication.

The second says that if a nonempty

set of real numbers has an upper

bound, then it has a real least upper

bound. The second condition

distinguishes the real numbers from

the rational numbers: for example,

the set of rational numbers whose

square is less than 2 is a set with an

upper bound (e.g. 1.5) but no

(rational) least upper bound: hence

the rational numbers do not satisfy

the least upper bound property.

It is divided into

two parts :-

Rational And

Irrational

Real Numbers

RATIONAL NUMBERS

INTRODUCTION

In mathematics, a rational

number is any number that can

be expressed as the quotient or

fraction p/q of two integers, with

the denominator q not equal to

zero. Since q may be equal to 1,

every integer is a rational

number.

The decimal expansion of a rational

number always either terminates

after a finite number of digits or

begins to repeat the same

finite sequence of digits over and

over. Moreover, any repeating or

terminating decimal represents a

rational number

Rational Numbers are

divided into three main

parts :-

Integers

Whole Numbers

Natural Numbers

Rational

Integers Whole Natural

1. INTEGERS

An integer is a number that can be

written without a fractional or

decimal component. For example,

21, 4, and −2048 are integers; 9.75,

5½, and √2 are not integers. The set

of integers is a subset of the real

numbers, and consists of the natural

numbers (0, 1, 2, 3, ...) and

the negatives of the non-zero

natural numbers (−1, −2, −3, ...).

2. WHOLE NUMBERS

Whole number is collection of positive

numbers and zero. Whole number also

called as integer. The whole number is

represented as {0, 1, 2, 3, 4, 5, 6, 7, 8, 9

….}. The set of whole numbers may be

finite or infinite. The finite defines the

numbers in the set are countable. Infinite

set means the numbers are uncountable. .

Zero is neither a fraction nor a decimal, so

zero is an whole number.

3. NATURAL NUMBERS

In mathematics, the natural numbers are

those used for counting and ordering .

Properties of the natural numbers related

to divisibility, such as the distribution

of prime numbers, are studied in number

theory. The natural numbers had their

origins in the words used to count things,

beginning with the number 1.

The addition (+) and multiplication

(×) operations on natural numbers

have several algebraic properties:

Closure under addition and

multiplication: for all natural

numbers a and b,

both a + b and a × b are natural

numbers.

Associativity: for all natural

numbers a, b, and c, a + (b + c) =

(a + b) + c and a × (b × c) = (a × b)

× c.

Commutativity: for all natural

numbers a and b, a + b = b + a and a

 × b = b × a.

Existence of identity elements: for

every natural number a, a + 0

= a and a × 1 = a.

Distributivity of multiplication over

addition for all natural numbers a, b,

and c, a × (b + c) = (a × b) + (a × c)

No zero divisors: if a and b are

natural numbers such that a × b = 0,

then a = 0 or b = 0.

IRRATIONAL

NUMBERS

INTRODUCTION

In mathematics, an irrational

number is any real number that

cannot be expressed as a ratio a/b,

where a and b are integers and b is

non-zero. Informally, this means that

an irrational number cannot be

represented as a simple fraction.

Irrational numbers are those real

numbers that cannot be represented

as terminating or repeating

decimals.

It has been suggested that the

concept of irrationality was

implicitly accepted by Indian

mathematicians since the 7th

century BC, when Manava (c. 750 –

690 BC) believed that the square

roots of numbers such as 2 and 61

could not be exactly

determined. However, historian Carl

Benjamin Boyer states that "...such

claims are not well substantiated

and unlikely to be true.

THANKING YOU

Name :- jay solanki

Class :- IX

Roll No. :- 9