Mixing
Sums and Products of Algebraic and Transcendental numbers
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I am doing a project about irrational and transcendental numbers and part of this project involves looking at sums and products of various combinations of rational, irrational, algebraic and transcendental numbers. I am quite happy that I have done what I need to with rational and irrational numbers however I need help with algebraic and transcendental numbers
I need to find out and prove what the solutions of the following are:
1) algebraic + algebraic
2) algebraic + transcendental
3) transcendental + transcendental
4) algebraic x algebraic
5) algebraic x transcendental
6) transcendental x transcendental
Also if there are any interesting results from sums and products of (ir)rationals with algebraic or transcendental numbers then these would be of great help too.
Any help is greatly appreciated, and thanks in advance.
number-theory algebraic-number-theory irrational-numbers rational-numbers transcendental-numbers
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add a comment |
$begingroup$
I am doing a project about irrational and transcendental numbers and part of this project involves looking at sums and products of various combinations of rational, irrational, algebraic and transcendental numbers. I am quite happy that I have done what I need to with rational and irrational numbers however I need help with algebraic and transcendental numbers
I need to find out and prove what the solutions of the following are:
1) algebraic + algebraic
2) algebraic + transcendental
3) transcendental + transcendental
4) algebraic x algebraic
5) algebraic x transcendental
6) transcendental x transcendental
Also if there are any interesting results from sums and products of (ir)rationals with algebraic or transcendental numbers then these would be of great help too.
Any help is greatly appreciated, and thanks in advance.
number-theory algebraic-number-theory irrational-numbers rational-numbers transcendental-numbers
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1
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algebraic + algebraic and algebraic x algebraic are slightly messy, but easy to find in books (e.g. Niven's "Irrational and Irrational Numbers"), and the rest you can do in straightforward ways. For example, let * be a fixed choice of + or x, let $a$ be algebraic and let $t$ be transcendental. Then $a*t$ equal to an algebraic number implies $t$ is the difference or quotient of algebraic numbers, which is a contradiction. And for transcendental x transcendental, note that for $t$ transcendental we have both $t$ and $frac{t}{2}$ transcendental.
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– Dave L. Renfro
Jan 27 at 16:59
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Thanks, very much appreciated
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– user610274
Jan 27 at 17:01
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I just noticed that the end of my previous comment should be "both $t$ and $frac{2}{t}$ transcendental". (To show that $frac{2}{t}$ is transcendental, assume for later contradiction that it's algebraic, write down what this means in the form of a specific equation that's satisfied when it's plugged in, and then rewrite the equation in a way that shows $t$ is algebraic.)
$endgroup$
– Dave L. Renfro
Jan 27 at 17:57
add a comment |
$begingroup$
I am doing a project about irrational and transcendental numbers and part of this project involves looking at sums and products of various combinations of rational, irrational, algebraic and transcendental numbers. I am quite happy that I have done what I need to with rational and irrational numbers however I need help with algebraic and transcendental numbers
I need to find out and prove what the solutions of the following are:
1) algebraic + algebraic
2) algebraic + transcendental
3) transcendental + transcendental
4) algebraic x algebraic
5) algebraic x transcendental
6) transcendental x transcendental
Also if there are any interesting results from sums and products of (ir)rationals with algebraic or transcendental numbers then these would be of great help too.
Any help is greatly appreciated, and thanks in advance.
number-theory algebraic-number-theory irrational-numbers rational-numbers transcendental-numbers
$endgroup$
I am doing a project about irrational and transcendental numbers and part of this project involves looking at sums and products of various combinations of rational, irrational, algebraic and transcendental numbers. I am quite happy that I have done what I need to with rational and irrational numbers however I need help with algebraic and transcendental numbers
I need to find out and prove what the solutions of the following are:
1) algebraic + algebraic
2) algebraic + transcendental
3) transcendental + transcendental
4) algebraic x algebraic
5) algebraic x transcendental
6) transcendental x transcendental
Also if there are any interesting results from sums and products of (ir)rationals with algebraic or transcendental numbers then these would be of great help too.
Any help is greatly appreciated, and thanks in advance.
number-theory algebraic-number-theory irrational-numbers rational-numbers transcendental-numbers
number-theory algebraic-number-theory irrational-numbers rational-numbers transcendental-numbers
asked Jan 27 at 16:46
user610274
1
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algebraic + algebraic and algebraic x algebraic are slightly messy, but easy to find in books (e.g. Niven's "Irrational and Irrational Numbers"), and the rest you can do in straightforward ways. For example, let * be a fixed choice of + or x, let $a$ be algebraic and let $t$ be transcendental. Then $a*t$ equal to an algebraic number implies $t$ is the difference or quotient of algebraic numbers, which is a contradiction. And for transcendental x transcendental, note that for $t$ transcendental we have both $t$ and $frac{t}{2}$ transcendental.
$endgroup$
– Dave L. Renfro
Jan 27 at 16:59
$begingroup$
Thanks, very much appreciated
$endgroup$
– user610274
Jan 27 at 17:01
$begingroup$
I just noticed that the end of my previous comment should be "both $t$ and $frac{2}{t}$ transcendental". (To show that $frac{2}{t}$ is transcendental, assume for later contradiction that it's algebraic, write down what this means in the form of a specific equation that's satisfied when it's plugged in, and then rewrite the equation in a way that shows $t$ is algebraic.)
$endgroup$
– Dave L. Renfro
Jan 27 at 17:57
add a comment |
1
$begingroup$
algebraic + algebraic and algebraic x algebraic are slightly messy, but easy to find in books (e.g. Niven's "Irrational and Irrational Numbers"), and the rest you can do in straightforward ways. For example, let * be a fixed choice of + or x, let $a$ be algebraic and let $t$ be transcendental. Then $a*t$ equal to an algebraic number implies $t$ is the difference or quotient of algebraic numbers, which is a contradiction. And for transcendental x transcendental, note that for $t$ transcendental we have both $t$ and $frac{t}{2}$ transcendental.
$endgroup$
– Dave L. Renfro
Jan 27 at 16:59
$begingroup$
Thanks, very much appreciated
$endgroup$
– user610274
Jan 27 at 17:01
$begingroup$
I just noticed that the end of my previous comment should be "both $t$ and $frac{2}{t}$ transcendental". (To show that $frac{2}{t}$ is transcendental, assume for later contradiction that it's algebraic, write down what this means in the form of a specific equation that's satisfied when it's plugged in, and then rewrite the equation in a way that shows $t$ is algebraic.)
$endgroup$
– Dave L. Renfro
Jan 27 at 17:57
1
1
$begingroup$
algebraic + algebraic and algebraic x algebraic are slightly messy, but easy to find in books (e.g. Niven's "Irrational and Irrational Numbers"), and the rest you can do in straightforward ways. For example, let * be a fixed choice of + or x, let $a$ be algebraic and let $t$ be transcendental. Then $a*t$ equal to an algebraic number implies $t$ is the difference or quotient of algebraic numbers, which is a contradiction. And for transcendental x transcendental, note that for $t$ transcendental we have both $t$ and $frac{t}{2}$ transcendental.
$endgroup$
– Dave L. Renfro
Jan 27 at 16:59
$begingroup$
algebraic + algebraic and algebraic x algebraic are slightly messy, but easy to find in books (e.g. Niven's "Irrational and Irrational Numbers"), and the rest you can do in straightforward ways. For example, let * be a fixed choice of + or x, let $a$ be algebraic and let $t$ be transcendental. Then $a*t$ equal to an algebraic number implies $t$ is the difference or quotient of algebraic numbers, which is a contradiction. And for transcendental x transcendental, note that for $t$ transcendental we have both $t$ and $frac{t}{2}$ transcendental.
$endgroup$
– Dave L. Renfro
Jan 27 at 16:59
$begingroup$
Thanks, very much appreciated
$endgroup$
– user610274
Jan 27 at 17:01
$begingroup$
Thanks, very much appreciated
$endgroup$
– user610274
Jan 27 at 17:01
$begingroup$
I just noticed that the end of my previous comment should be "both $t$ and $frac{2}{t}$ transcendental". (To show that $frac{2}{t}$ is transcendental, assume for later contradiction that it's algebraic, write down what this means in the form of a specific equation that's satisfied when it's plugged in, and then rewrite the equation in a way that shows $t$ is algebraic.)
$endgroup$
– Dave L. Renfro
Jan 27 at 17:57
$begingroup$
I just noticed that the end of my previous comment should be "both $t$ and $frac{2}{t}$ transcendental". (To show that $frac{2}{t}$ is transcendental, assume for later contradiction that it's algebraic, write down what this means in the form of a specific equation that's satisfied when it's plugged in, and then rewrite the equation in a way that shows $t$ is algebraic.)
$endgroup$
– Dave L. Renfro
Jan 27 at 17:57
add a comment |
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$begingroup$
algebraic + algebraic and algebraic x algebraic are slightly messy, but easy to find in books (e.g. Niven's "Irrational and Irrational Numbers"), and the rest you can do in straightforward ways. For example, let * be a fixed choice of + or x, let $a$ be algebraic and let $t$ be transcendental. Then $a*t$ equal to an algebraic number implies $t$ is the difference or quotient of algebraic numbers, which is a contradiction. And for transcendental x transcendental, note that for $t$ transcendental we have both $t$ and $frac{t}{2}$ transcendental.
$endgroup$
– Dave L. Renfro
Jan 27 at 16:59
$begingroup$
Thanks, very much appreciated
$endgroup$
– user610274
Jan 27 at 17:01
$begingroup$
I just noticed that the end of my previous comment should be "both $t$ and $frac{2}{t}$ transcendental". (To show that $frac{2}{t}$ is transcendental, assume for later contradiction that it's algebraic, write down what this means in the form of a specific equation that's satisfied when it's plugged in, and then rewrite the equation in a way that shows $t$ is algebraic.)
$endgroup$
– Dave L. Renfro
Jan 27 at 17:57