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Can nth root of 2 be a rational number for any natural number n > 1?
0
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We know square and cube root of 2 are irrational numbers, but is it possible for any a rational number to be multiplied by itself finite times and it results in 2.
irrational-numbers rational-numbers
asked Jan 20 at 11:07
PranshuKhandalPranshuKhandal
1348
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0
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We know square and cube root of 2 are irrational numbers, but is it possible for any a rational number to be multiplied by itself finite times and it results in 2.
irrational-numbers rational-numbers
asked Jan 20 at 11:07
PranshuKhandalPranshuKhandal
1348
$endgroup$
4
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No, with the same proof used for proving that $sqrt{2}$ is irrational.
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– Jean Marie
Jan 20 at 11:10
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@Jean Marie so this is true for any prime number??
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– PranshuKhandal
Jan 20 at 11:12
3
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Exactly, due to unicity of primary factor decomposition.
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– Jean Marie
Jan 20 at 11:14
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thnx @Jean Marie for help
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– PranshuKhandal
Jan 20 at 11:15
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Um, does $1$ count as a natural number? (Maybe edit the title to say $n>1$.)
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– timtfj
Jan 20 at 11:32
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show 1 more comment
0
0
0
1
$begingroup$
We know square and cube root of 2 are irrational numbers, but is it possible for any a rational number to be multiplied by itself finite times and it results in 2.
irrational-numbers rational-numbers
asked Jan 20 at 11:07
PranshuKhandalPranshuKhandal
1348
$endgroup$
We know square and cube root of 2 are irrational numbers, but is it possible for any a rational number to be multiplied by itself finite times and it results in 2.
irrational-numbers rational-numbers
irrational-numbers rational-numbers
asked Jan 20 at 11:07
PranshuKhandalPranshuKhandal
1348
asked Jan 20 at 11:07
PranshuKhandalPranshuKhandal
1348
edited Jan 20 at 11:36
PranshuKhandal
asked Jan 20 at 11:07
PranshuKhandalPranshuKhandal
1348
asked Jan 20 at 11:07
PranshuKhandalPranshuKhandal
1348
asked Jan 20 at 11:07
PranshuKhandalPranshuKhandal
1348
1348
4
$begingroup$
No, with the same proof used for proving that $sqrt{2}$ is irrational.
$endgroup$
– Jean Marie
Jan 20 at 11:10
$begingroup$
@Jean Marie so this is true for any prime number??
$endgroup$
– PranshuKhandal
Jan 20 at 11:12
3
$begingroup$
Exactly, due to unicity of primary factor decomposition.
$endgroup$
– Jean Marie
Jan 20 at 11:14
$begingroup$
thnx @Jean Marie for help
$endgroup$
– PranshuKhandal
Jan 20 at 11:15
$begingroup$
Um, does $1$ count as a natural number? (Maybe edit the title to say $n>1$.)
$endgroup$
– timtfj
Jan 20 at 11:32
|
show 1 more comment
4
$begingroup$
No, with the same proof used for proving that $sqrt{2}$ is irrational.
$endgroup$
– Jean Marie
Jan 20 at 11:10
$begingroup$
@Jean Marie so this is true for any prime number??
$endgroup$
– PranshuKhandal
Jan 20 at 11:12
3
$begingroup$
Exactly, due to unicity of primary factor decomposition.
$endgroup$
– Jean Marie
Jan 20 at 11:14
$begingroup$
thnx @Jean Marie for help
$endgroup$
– PranshuKhandal
Jan 20 at 11:15
$begingroup$
Um, does $1$ count as a natural number? (Maybe edit the title to say $n>1$.)
$endgroup$
– timtfj
Jan 20 at 11:32
4
4
$begingroup$
No, with the same proof used for proving that $sqrt{2}$ is irrational.
$endgroup$
– Jean Marie
Jan 20 at 11:10
$begingroup$
No, with the same proof used for proving that $sqrt{2}$ is irrational.
$endgroup$
– Jean Marie
Jan 20 at 11:10
$begingroup$
@Jean Marie so this is true for any prime number??
$endgroup$
– PranshuKhandal
Jan 20 at 11:12
$begingroup$
@Jean Marie so this is true for any prime number??
$endgroup$
– PranshuKhandal
Jan 20 at 11:12
3
3
$begingroup$
Exactly, due to unicity of primary factor decomposition.
$endgroup$
– Jean Marie
Jan 20 at 11:14
$begingroup$
Exactly, due to unicity of primary factor decomposition.
$endgroup$
– Jean Marie
Jan 20 at 11:14
$begingroup$
thnx @Jean Marie for help
$endgroup$
– PranshuKhandal
Jan 20 at 11:15
$begingroup$
thnx @Jean Marie for help
$endgroup$
– PranshuKhandal
Jan 20 at 11:15
$begingroup$
Um, does $1$ count as a natural number? (Maybe edit the title to say $n>1$.)
$endgroup$
– timtfj
Jan 20 at 11:32
$begingroup$
Um, does $1$ count as a natural number? (Maybe edit the title to say $n>1$.)
$endgroup$
– timtfj
Jan 20 at 11:32
|
show 1 more comment
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4
$begingroup$
No, with the same proof used for proving that $sqrt{2}$ is irrational.
$endgroup$
– Jean Marie
Jan 20 at 11:10
$begingroup$
@Jean Marie so this is true for any prime number??
$endgroup$
– PranshuKhandal
Jan 20 at 11:12
3
$begingroup$
Exactly, due to unicity of primary factor decomposition.
$endgroup$
– Jean Marie
Jan 20 at 11:14
$begingroup$
thnx @Jean Marie for help
$endgroup$
– PranshuKhandal
Jan 20 at 11:15
$begingroup$
Um, does $1$ count as a natural number? (Maybe edit the title to say $n>1$.)
$endgroup$
– timtfj
Jan 20 at 11:32