Mixing
How can an convergent series of rational numbers result in a irrational number?
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In mathematics, nearly all significant irrational numbers can be expressed as a sum of an infinite convergent series, but according to law of addition of rational numbers, adding any to rational numbers can not result in an irrational number.
So, why is this contradiction?
sequences-and-series irrational-numbers rational-numbers
asked Jan 20 at 10:32
PranshuKhandalPranshuKhandal
1348
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|
show 5 more comments
$begingroup$
In mathematics, nearly all significant irrational numbers can be expressed as a sum of an infinite convergent series, but according to law of addition of rational numbers, adding any to rational numbers can not result in an irrational number.
So, why is this contradiction?
sequences-and-series irrational-numbers rational-numbers
asked Jan 20 at 10:32
PranshuKhandalPranshuKhandal
1348
$endgroup$
4
$begingroup$
As long as finite many rational numbers are added, the result must , of course , be rational. But for infinite sums this is no longer true.
$endgroup$
– Peter
Jan 20 at 10:40
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
@Peter plz help me out, i forgot to include you
$endgroup$
– PranshuKhandal
Jan 20 at 10:44
2
$begingroup$
Even an infinite converging sum of real numbers has a real value. If your number is not real, it cannot be represented that way.
$endgroup$
– Peter
Jan 20 at 10:45
1
$begingroup$
Also note that irrationality-proofs are , in general , extremely difficult. The sum $$pi=4-4/3+4/5-4/7pm cdots $$ does not allow an easy proof that $pi$ is irrational. In fact, $pi+e$ is not known , but conjectured to be irrational.
$endgroup$
– Peter
Jan 20 at 10:48
|
show 5 more comments
$begingroup$
In mathematics, nearly all significant irrational numbers can be expressed as a sum of an infinite convergent series, but according to law of addition of rational numbers, adding any to rational numbers can not result in an irrational number.
So, why is this contradiction?
sequences-and-series irrational-numbers rational-numbers
asked Jan 20 at 10:32
PranshuKhandalPranshuKhandal
1348
$endgroup$
In mathematics, nearly all significant irrational numbers can be expressed as a sum of an infinite convergent series, but according to law of addition of rational numbers, adding any to rational numbers can not result in an irrational number.
So, why is this contradiction?
sequences-and-series irrational-numbers rational-numbers
sequences-and-series irrational-numbers rational-numbers
asked Jan 20 at 10:32
PranshuKhandalPranshuKhandal
1348
asked Jan 20 at 10:32
PranshuKhandalPranshuKhandal
1348
asked Jan 20 at 10:32
PranshuKhandalPranshuKhandal
1348
asked Jan 20 at 10:32
PranshuKhandalPranshuKhandal
1348
asked Jan 20 at 10:32
PranshuKhandalPranshuKhandal
1348
1348
4
$begingroup$
As long as finite many rational numbers are added, the result must , of course , be rational. But for infinite sums this is no longer true.
$endgroup$
– Peter
Jan 20 at 10:40
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
@Peter plz help me out, i forgot to include you
$endgroup$
– PranshuKhandal
Jan 20 at 10:44
2
$begingroup$
Even an infinite converging sum of real numbers has a real value. If your number is not real, it cannot be represented that way.
$endgroup$
– Peter
Jan 20 at 10:45
1
$begingroup$
Also note that irrationality-proofs are , in general , extremely difficult. The sum $$pi=4-4/3+4/5-4/7pm cdots $$ does not allow an easy proof that $pi$ is irrational. In fact, $pi+e$ is not known , but conjectured to be irrational.
$endgroup$
– Peter
Jan 20 at 10:48
|
show 5 more comments
4
$begingroup$
As long as finite many rational numbers are added, the result must , of course , be rational. But for infinite sums this is no longer true.
$endgroup$
– Peter
Jan 20 at 10:40
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
@Peter plz help me out, i forgot to include you
$endgroup$
– PranshuKhandal
Jan 20 at 10:44
2
$begingroup$
Even an infinite converging sum of real numbers has a real value. If your number is not real, it cannot be represented that way.
$endgroup$
– Peter
Jan 20 at 10:45
1
$begingroup$
Also note that irrationality-proofs are , in general , extremely difficult. The sum $$pi=4-4/3+4/5-4/7pm cdots $$ does not allow an easy proof that $pi$ is irrational. In fact, $pi+e$ is not known , but conjectured to be irrational.
$endgroup$
– Peter
Jan 20 at 10:48
4
4
$begingroup$
As long as finite many rational numbers are added, the result must , of course , be rational. But for infinite sums this is no longer true.
$endgroup$
– Peter
Jan 20 at 10:40
$begingroup$
As long as finite many rational numbers are added, the result must , of course , be rational. But for infinite sums this is no longer true.
$endgroup$
– Peter
Jan 20 at 10:40
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
@Peter plz help me out, i forgot to include you
$endgroup$
– PranshuKhandal
Jan 20 at 10:44
$begingroup$
@Peter plz help me out, i forgot to include you
$endgroup$
– PranshuKhandal
Jan 20 at 10:44
2
2
$begingroup$
Even an infinite converging sum of real numbers has a real value. If your number is not real, it cannot be represented that way.
$endgroup$
– Peter
Jan 20 at 10:45
$begingroup$
Even an infinite converging sum of real numbers has a real value. If your number is not real, it cannot be represented that way.
$endgroup$
– Peter
Jan 20 at 10:45
1
1
$begingroup$
Also note that irrationality-proofs are , in general , extremely difficult. The sum $$pi=4-4/3+4/5-4/7pm cdots $$ does not allow an easy proof that $pi$ is irrational. In fact, $pi+e$ is not known , but conjectured to be irrational.
$endgroup$
– Peter
Jan 20 at 10:48
$begingroup$
Also note that irrationality-proofs are , in general , extremely difficult. The sum $$pi=4-4/3+4/5-4/7pm cdots $$ does not allow an easy proof that $pi$ is irrational. In fact, $pi+e$ is not known , but conjectured to be irrational.
$endgroup$
– Peter
Jan 20 at 10:48
|
show 5 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Adding any two rational numbers results in a rational number. By induction, adding any finite number of rational numbers together results in a rational number.
Adding together infinitely many rational numbers has no such guarantee, in exactly the same way that there is no guarantee that such a sum is finite.
answered Jan 20 at 10:37
ArthurArthur
116k7116199
$endgroup$
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
thnx @Arthur for help
$endgroup$
– PranshuKhandal
Jan 20 at 10:48
$begingroup$
No, that isn't possible. An infinite sum of real numbers which converges is guaranteed to be real.
$endgroup$
– Arthur
Jan 20 at 10:56
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
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votes
$begingroup$
Adding any two rational numbers results in a rational number. By induction, adding any finite number of rational numbers together results in a rational number.
Adding together infinitely many rational numbers has no such guarantee, in exactly the same way that there is no guarantee that such a sum is finite.
answered Jan 20 at 10:37
ArthurArthur
116k7116199
$endgroup$
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
thnx @Arthur for help
$endgroup$
– PranshuKhandal
Jan 20 at 10:48
$begingroup$
No, that isn't possible. An infinite sum of real numbers which converges is guaranteed to be real.
$endgroup$
– Arthur
Jan 20 at 10:56
add a comment |
$begingroup$
Adding any two rational numbers results in a rational number. By induction, adding any finite number of rational numbers together results in a rational number.
Adding together infinitely many rational numbers has no such guarantee, in exactly the same way that there is no guarantee that such a sum is finite.
answered Jan 20 at 10:37
ArthurArthur
116k7116199
$endgroup$
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
thnx @Arthur for help
$endgroup$
– PranshuKhandal
Jan 20 at 10:48
$begingroup$
No, that isn't possible. An infinite sum of real numbers which converges is guaranteed to be real.
$endgroup$
– Arthur
Jan 20 at 10:56
add a comment |
$begingroup$
Adding any two rational numbers results in a rational number. By induction, adding any finite number of rational numbers together results in a rational number.
Adding together infinitely many rational numbers has no such guarantee, in exactly the same way that there is no guarantee that such a sum is finite.
answered Jan 20 at 10:37
ArthurArthur
116k7116199
$endgroup$
Adding any two rational numbers results in a rational number. By induction, adding any finite number of rational numbers together results in a rational number.
Adding together infinitely many rational numbers has no such guarantee, in exactly the same way that there is no guarantee that such a sum is finite.
answered Jan 20 at 10:37
ArthurArthur
116k7116199
answered Jan 20 at 10:37
ArthurArthur
116k7116199
answered Jan 20 at 10:37
ArthurArthur
116k7116199
answered Jan 20 at 10:37
ArthurArthur
116k7116199
116k7116199
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
thnx @Arthur for help
$endgroup$
– PranshuKhandal
Jan 20 at 10:48
$begingroup$
No, that isn't possible. An infinite sum of real numbers which converges is guaranteed to be real.
$endgroup$
– Arthur
Jan 20 at 10:56
add a comment |
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
thnx @Arthur for help
$endgroup$
– PranshuKhandal
Jan 20 at 10:48
$begingroup$
No, that isn't possible. An infinite sum of real numbers which converges is guaranteed to be real.
$endgroup$
– Arthur
Jan 20 at 10:56
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
thnx @Arthur for help
$endgroup$
– PranshuKhandal
Jan 20 at 10:48
$begingroup$
thnx @Arthur for help
$endgroup$
– PranshuKhandal
Jan 20 at 10:48
$begingroup$
No, that isn't possible. An infinite sum of real numbers which converges is guaranteed to be real.
$endgroup$
– Arthur
Jan 20 at 10:56
$begingroup$
No, that isn't possible. An infinite sum of real numbers which converges is guaranteed to be real.
$endgroup$
– Arthur
Jan 20 at 10:56
add a comment |
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4
$begingroup$
As long as finite many rational numbers are added, the result must , of course , be rational. But for infinite sums this is no longer true.
$endgroup$
– Peter
Jan 20 at 10:40
$begingroup$
so, is it possible to represent iota(√-1) as a sum of infinite real numbers??
$endgroup$
– PranshuKhandal
Jan 20 at 10:43
$begingroup$
@Peter plz help me out, i forgot to include you
$endgroup$
– PranshuKhandal
Jan 20 at 10:44
2
$begingroup$
Even an infinite converging sum of real numbers has a real value. If your number is not real, it cannot be represented that way.
$endgroup$
– Peter
Jan 20 at 10:45
1
$begingroup$
Also note that irrationality-proofs are , in general , extremely difficult. The sum $$pi=4-4/3+4/5-4/7pm cdots $$ does not allow an easy proof that $pi$ is irrational. In fact, $pi+e$ is not known , but conjectured to be irrational.
$endgroup$
– Peter
Jan 20 at 10:48