prove that negative 3 underoot 17 is an irrational number
Answers
Answer:
√
17
is irrational essentially as a consequence of
17
being prime - that is having no positive factors apart from
1
and itself.
Here's a sketch of a proof:
Suppose
√
17
=
p
q
for some integers
p
,
q
, with
q
≠
0
.
Without loss of generality,
p
,
q
>
0
and
p
and
q
have no common factor greater than
1
.
[[ If they did have a common factor, then you could divide both by that common factor to get a smaller
p
1
and
q
1
with
√
17
=
p
1
q
1
]]
Then
p
2
=
17
q
2
and since
p
2
is a multiple of
17
and
17
is prime,
p
must be a multiple of
17
.
Let
k
=
p
17
Then
17
q
2
=
p
2
=
(
17
k
)
2
=
17
⋅
17
k
2
Divide both ends by
17
to find:
q
2
=
17
k
2
hence
q
is a multiple of
17
.
So both
p
and
q
are divisible by
17
, contradicting our assumption that
p
and
q
have no common factor greater than
1
.
So there is no such pair of integers
p
and
q
.
Answer:
the number 17 is prime.
therefore the root if 17 is irrational number.
|| ANISHA || ❤✨