Mixing
Sum of two irrational numbers being rational or irrational
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I am currently doing a project on irrational and transcendental numbers and part of this project requires me to look at sums and products of irrational numbers.
I am aware that the sum of 2 irrational numbers can be rational or irrational but was wondering if anyone knew of a definite way to look at the numbers and say if their sum/product will be rational/irrational. Is there some sort of theorem than can be applied or is the only way of knowing just working it out?
Thanks in advance for any help.
irrational-numbers rational-numbers
edited Jan 30 at 15:38
José Carlos Santos
172k22132239
$endgroup$
add a comment |
$begingroup$
I am currently doing a project on irrational and transcendental numbers and part of this project requires me to look at sums and products of irrational numbers.
I am aware that the sum of 2 irrational numbers can be rational or irrational but was wondering if anyone knew of a definite way to look at the numbers and say if their sum/product will be rational/irrational. Is there some sort of theorem than can be applied or is the only way of knowing just working it out?
Thanks in advance for any help.
irrational-numbers rational-numbers
edited Jan 30 at 15:38
José Carlos Santos
172k22132239
$endgroup$
add a comment |
$begingroup$
I am currently doing a project on irrational and transcendental numbers and part of this project requires me to look at sums and products of irrational numbers.
I am aware that the sum of 2 irrational numbers can be rational or irrational but was wondering if anyone knew of a definite way to look at the numbers and say if their sum/product will be rational/irrational. Is there some sort of theorem than can be applied or is the only way of knowing just working it out?
Thanks in advance for any help.
irrational-numbers rational-numbers
edited Jan 30 at 15:38
José Carlos Santos
172k22132239
$endgroup$
I am currently doing a project on irrational and transcendental numbers and part of this project requires me to look at sums and products of irrational numbers.
I am aware that the sum of 2 irrational numbers can be rational or irrational but was wondering if anyone knew of a definite way to look at the numbers and say if their sum/product will be rational/irrational. Is there some sort of theorem than can be applied or is the only way of knowing just working it out?
Thanks in advance for any help.
irrational-numbers rational-numbers
irrational-numbers rational-numbers
edited Jan 30 at 15:38
José Carlos Santos
172k22132239
edited Jan 30 at 15:38
José Carlos Santos
172k22132239
edited Jan 30 at 15:38
José Carlos Santos
172k22132239
edited Jan 30 at 15:38
José Carlos Santos
172k22132239
edited Jan 30 at 15:38
José Carlos Santos
172k22132239
172k22132239
asked Jan 30 at 15:35
user610274
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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No, there is not. If there was, we would know whether $e+pi$ is rational or not. But, in fact, that's an open problem.
answered Jan 30 at 15:36
José Carlos SantosJosé Carlos Santos
172k22132239
$endgroup$
$begingroup$
Oh yeah, that seems so obvious now. Thank you!
$endgroup$
– user610274
Jan 30 at 15:43
add a comment |
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"requires me to look at sums and products of irrational numbers."
so you don't ask for transcendental?
The sum or product of two irrational algebraic numbers is not necessarily irrational. Counterexamples:
$$
(2 +sqrt 2)+(2 -sqrt 2)= 4\
(2 +sqrt 2)(2 -sqrt 2)= 2
$$
answered Jan 30 at 15:43
AndreasAndreas
8,4161137
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$begingroup$
If you have any input on the sum/product of two transcendental numbers then that will be very much appreciated too !!
$endgroup$
– user610274
Jan 30 at 15:55
1
$begingroup$
For the sum/product of two transcendental numbers there are no known rules. See Joses answer.
$endgroup$
– Andreas
Jan 30 at 16:28
$begingroup$
I thought that would be the case, any idea on how to show that though?
$endgroup$
– user610274
Jan 30 at 16:29
1
$begingroup$
Well, there are examples with transcendental numbers in both ways, e.g. for multiplication of two transcendental numbers: $pi cdot frac{2}{pi} = 2$ and $pi cdot pi = pi^2$ where the first result is non-transcendental and the second one is transcendental. So there are no clear rules.
$endgroup$
– Andreas
Jan 30 at 16:41
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No, there is not. If there was, we would know whether $e+pi$ is rational or not. But, in fact, that's an open problem.
answered Jan 30 at 15:36
José Carlos SantosJosé Carlos Santos
172k22132239
$endgroup$
$begingroup$
Oh yeah, that seems so obvious now. Thank you!
$endgroup$
– user610274
Jan 30 at 15:43
add a comment |
$begingroup$
No, there is not. If there was, we would know whether $e+pi$ is rational or not. But, in fact, that's an open problem.
answered Jan 30 at 15:36
José Carlos SantosJosé Carlos Santos
172k22132239
$endgroup$
$begingroup$
Oh yeah, that seems so obvious now. Thank you!
$endgroup$
– user610274
Jan 30 at 15:43
add a comment |
$begingroup$
No, there is not. If there was, we would know whether $e+pi$ is rational or not. But, in fact, that's an open problem.
answered Jan 30 at 15:36
José Carlos SantosJosé Carlos Santos
172k22132239
$endgroup$
No, there is not. If there was, we would know whether $e+pi$ is rational or not. But, in fact, that's an open problem.
answered Jan 30 at 15:36
José Carlos SantosJosé Carlos Santos
172k22132239
answered Jan 30 at 15:36
José Carlos SantosJosé Carlos Santos
172k22132239
answered Jan 30 at 15:36
José Carlos SantosJosé Carlos Santos
172k22132239
answered Jan 30 at 15:36
José Carlos SantosJosé Carlos Santos
172k22132239
172k22132239
$begingroup$
Oh yeah, that seems so obvious now. Thank you!
$endgroup$
– user610274
Jan 30 at 15:43
add a comment |
$begingroup$
Oh yeah, that seems so obvious now. Thank you!
$endgroup$
– user610274
Jan 30 at 15:43
$begingroup$
Oh yeah, that seems so obvious now. Thank you!
$endgroup$
– user610274
Jan 30 at 15:43
$begingroup$
Oh yeah, that seems so obvious now. Thank you!
$endgroup$
– user610274
Jan 30 at 15:43
add a comment |
$begingroup$
"requires me to look at sums and products of irrational numbers."
so you don't ask for transcendental?
The sum or product of two irrational algebraic numbers is not necessarily irrational. Counterexamples:
$$
(2 +sqrt 2)+(2 -sqrt 2)= 4\
(2 +sqrt 2)(2 -sqrt 2)= 2
$$
answered Jan 30 at 15:43
AndreasAndreas
8,4161137
$endgroup$
$begingroup$
If you have any input on the sum/product of two transcendental numbers then that will be very much appreciated too !!
$endgroup$
– user610274
Jan 30 at 15:55
1
$begingroup$
For the sum/product of two transcendental numbers there are no known rules. See Joses answer.
$endgroup$
– Andreas
Jan 30 at 16:28
$begingroup$
I thought that would be the case, any idea on how to show that though?
$endgroup$
– user610274
Jan 30 at 16:29
1
$begingroup$
Well, there are examples with transcendental numbers in both ways, e.g. for multiplication of two transcendental numbers: $pi cdot frac{2}{pi} = 2$ and $pi cdot pi = pi^2$ where the first result is non-transcendental and the second one is transcendental. So there are no clear rules.
$endgroup$
– Andreas
Jan 30 at 16:41
add a comment |
$begingroup$
"requires me to look at sums and products of irrational numbers."
so you don't ask for transcendental?
The sum or product of two irrational algebraic numbers is not necessarily irrational. Counterexamples:
$$
(2 +sqrt 2)+(2 -sqrt 2)= 4\
(2 +sqrt 2)(2 -sqrt 2)= 2
$$
answered Jan 30 at 15:43
AndreasAndreas
8,4161137
$endgroup$
$begingroup$
If you have any input on the sum/product of two transcendental numbers then that will be very much appreciated too !!
$endgroup$
– user610274
Jan 30 at 15:55
1
$begingroup$
For the sum/product of two transcendental numbers there are no known rules. See Joses answer.
$endgroup$
– Andreas
Jan 30 at 16:28
$begingroup$
I thought that would be the case, any idea on how to show that though?
$endgroup$
– user610274
Jan 30 at 16:29
1
$begingroup$
Well, there are examples with transcendental numbers in both ways, e.g. for multiplication of two transcendental numbers: $pi cdot frac{2}{pi} = 2$ and $pi cdot pi = pi^2$ where the first result is non-transcendental and the second one is transcendental. So there are no clear rules.
$endgroup$
– Andreas
Jan 30 at 16:41
add a comment |
$begingroup$
"requires me to look at sums and products of irrational numbers."
so you don't ask for transcendental?
The sum or product of two irrational algebraic numbers is not necessarily irrational. Counterexamples:
$$
(2 +sqrt 2)+(2 -sqrt 2)= 4\
(2 +sqrt 2)(2 -sqrt 2)= 2
$$
answered Jan 30 at 15:43
AndreasAndreas
8,4161137
$endgroup$
"requires me to look at sums and products of irrational numbers."
so you don't ask for transcendental?
The sum or product of two irrational algebraic numbers is not necessarily irrational. Counterexamples:
$$
(2 +sqrt 2)+(2 -sqrt 2)= 4\
(2 +sqrt 2)(2 -sqrt 2)= 2
$$
answered Jan 30 at 15:43
AndreasAndreas
8,4161137
edited Jan 30 at 16:30
answered Jan 30 at 15:43
AndreasAndreas
8,4161137
answered Jan 30 at 15:43
AndreasAndreas
8,4161137
answered Jan 30 at 15:43
AndreasAndreas
8,4161137
8,4161137
$begingroup$
If you have any input on the sum/product of two transcendental numbers then that will be very much appreciated too !!
$endgroup$
– user610274
Jan 30 at 15:55
1
$begingroup$
For the sum/product of two transcendental numbers there are no known rules. See Joses answer.
$endgroup$
– Andreas
Jan 30 at 16:28
$begingroup$
I thought that would be the case, any idea on how to show that though?
$endgroup$
– user610274
Jan 30 at 16:29
1
$begingroup$
Well, there are examples with transcendental numbers in both ways, e.g. for multiplication of two transcendental numbers: $pi cdot frac{2}{pi} = 2$ and $pi cdot pi = pi^2$ where the first result is non-transcendental and the second one is transcendental. So there are no clear rules.
$endgroup$
– Andreas
Jan 30 at 16:41
add a comment |
$begingroup$
If you have any input on the sum/product of two transcendental numbers then that will be very much appreciated too !!
$endgroup$
– user610274
Jan 30 at 15:55
1
$begingroup$
For the sum/product of two transcendental numbers there are no known rules. See Joses answer.
$endgroup$
– Andreas
Jan 30 at 16:28
$begingroup$
I thought that would be the case, any idea on how to show that though?
$endgroup$
– user610274
Jan 30 at 16:29
1
$begingroup$
Well, there are examples with transcendental numbers in both ways, e.g. for multiplication of two transcendental numbers: $pi cdot frac{2}{pi} = 2$ and $pi cdot pi = pi^2$ where the first result is non-transcendental and the second one is transcendental. So there are no clear rules.
$endgroup$
– Andreas
Jan 30 at 16:41
$begingroup$
If you have any input on the sum/product of two transcendental numbers then that will be very much appreciated too !!
$endgroup$
– user610274
Jan 30 at 15:55
$begingroup$
If you have any input on the sum/product of two transcendental numbers then that will be very much appreciated too !!
$endgroup$
– user610274
Jan 30 at 15:55
1
1
$begingroup$
For the sum/product of two transcendental numbers there are no known rules. See Joses answer.
$endgroup$
– Andreas
Jan 30 at 16:28
$begingroup$
For the sum/product of two transcendental numbers there are no known rules. See Joses answer.
$endgroup$
– Andreas
Jan 30 at 16:28
$begingroup$
I thought that would be the case, any idea on how to show that though?
$endgroup$
– user610274
Jan 30 at 16:29
$begingroup$
I thought that would be the case, any idea on how to show that though?
$endgroup$
– user610274
Jan 30 at 16:29
1
1
$begingroup$
Well, there are examples with transcendental numbers in both ways, e.g. for multiplication of two transcendental numbers: $pi cdot frac{2}{pi} = 2$ and $pi cdot pi = pi^2$ where the first result is non-transcendental and the second one is transcendental. So there are no clear rules.
$endgroup$
– Andreas
Jan 30 at 16:41
$begingroup$
Well, there are examples with transcendental numbers in both ways, e.g. for multiplication of two transcendental numbers: $pi cdot frac{2}{pi} = 2$ and $pi cdot pi = pi^2$ where the first result is non-transcendental and the second one is transcendental. So there are no clear rules.
$endgroup$
– Andreas
Jan 30 at 16:41
add a comment |
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