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Byju's Answer
Standard IX
Mathematics
Natural Numbers
Prove that th...
Question
Prove that the sum , difference , product , and quotient of two irrational numbers need not be an irrational number .
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Solution
It
is
required
to
show
that
the
sum
,
difference
,
product
and
quotient
of
two
irrational
numbers
need
not
be
an
irrational
number
.
The
irrational
numbers
are
those
numbers
which
cannot
be
expressed
as
the
ratio
of
integers
.
Take
the
irrational
numbers
2
and
-
2
Note
that
their
sum
is
,
2
+
-
2
=
2
-
2
=
0
which
is
a
rational
number
Now
take
the
irrational
number
2
+
2
and
1
+
2
The
difference
of
these
numbers
are
,
2
+
2
-
1
+
2
=
2
+
2
-
1
-
2
=
1
which
is
a
rational
number
.
Take
the
irrational
numbers
12
and
3
The
product
of
these
irrational
numbers
are
,
12
×
3
=
12
×
3
=
36
=
6
2
=
6
which
is
a
rational
number
Take
two
irrational
numbers
2
27
and
3
Divide
the
above
two
to
get
,
2
27
3
=
2
27
3
=
2
9
=
2
×
3
=
6
which
is
a
rational
number
Hence
it
is
proved
that
the
sum
,
difference
,
product
and
quotient
of
two
irrational
numbers
need
not
be
an
irrational
number
.
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Similar questions
Q.
Give two irrational numbers so that their.
(i) sum is an irrational number.
(ii) sum is not an irrational number.
(iii) difference is an irrational number.
(iv) difference is not an irrational number.
(v) product is an irrational number.
(vi) product is not an irrational number.
(vii) quotient is an irrational number.
(viii) quotient is not an irrational number.
Q.
Give an example of each, of two irrational numbers whose:
(i) difference is a rational number.
(ii) difference is an irrational number.
(iii) sum is a rational number.
(iv) sum is an irrational number.
(v) product is an rational number.
(vi) product is an irrational number.
(vii) quotient is a rational number.
(viii) quotient is an irrational number.
