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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
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