6. 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).
7. 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.
9. 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
10. 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.
11. 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.
12. It is divided into
two parts :-
Rational And
Irrational
16. 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.
17. 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
21. 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, ...).
23. 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.
25. 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.
26. 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.
27. 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)
28. No zero divisors: if a and b are natural
numbers such that a × b = 0, then a =
0 or b = 0.
31. 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
32. 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
