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I know that we can obtain any rational $r\in\mathbb{R}\setminus\{0\}$ by the multiplication of two irrational numbers. There are many beautiful answers to this here.

But I want to know that if there exists a theorem/result which can exactly point out that when does (a shot of classification that) the multiplication of given two irrational numbers is a rational number?

If the above thing is much more to ask for, then so can we expect that we have a finite list of the product of two irrational numbers that are unsettled and others flow some general pattern or we have some theorems?

David G. Stork
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Kumar
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  • Why would the product of any two irrational numbers give a rational number?? $\sqrt{7} \cdot \sqrt{11}$, for example, or $\sqrt{3} \cdot \sqrt{3}$. In short: some pairs do, some pairs don't. – David G. Stork May 02 '20 at 05:25
  • @DavidG.Stork I know that any two irrational won't give a rational number. I am asking do we have some short of classification to get a rational from given two irrationals? – Kumar May 02 '20 at 05:28
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    https://en.wikipedia.org/wiki/Transcendental_number#Possible_transcendental_numbers and https://math.stackexchange.com/questions/28243/is-there-a-proof-that-pi-times-e-is-irrational may interest you. – anonymous67 May 02 '20 at 05:36
  • @DavidG.Stork I think I have made it more concise. Check if it's clear? and can you answer it? – Kumar May 02 '20 at 05:36
  • It's indicated that there are many unsettled products of two irrational numbers, to answer one of your question. :) – anonymous67 May 02 '20 at 05:41
  • @Learning I see. But is that list finite? – Kumar May 02 '20 at 05:42
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    What sort of theorem. Sometimes two irrationals multiplied together is rational-- such as $\sqrt{\frac {32}7}$ and $\sqrt{ 14}$ (which multiplied together is $4$)... or $\frac \pi {19}$ and $\frac 7{\pi}$ (which multiplied together is $\frac 7{19}$.--- and sometimes they don't-- such as $\sqrt 7$ and $\sqrt 3$ or $\pi$ and $\sqrt 2$. So what would you expect theorem to say? – fleablood May 02 '20 at 05:45
  • My guess is that the list is infinite. – anonymous67 May 02 '20 at 05:48
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    I really don't think there can be anything more that if $x$ and $y$ are irrational then $xy$ will be rational if if $x = r\frac 1y$ for some rational $r$. But that's of course a very stupid theorem. But I really don't think we can do any better. – fleablood May 02 '20 at 05:48
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    @fleablood I am expecting that the theorem should give me a give check method for a pair of irrational number that when they are multiplied will give me a rational number. I think you got some bits of the question that I am asking.:) I have already mentioned that my question might not be concise. :D – Kumar May 02 '20 at 05:51
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    But what possible language and description would help you. The only way we have to describe a rational number is that it is a ratio of two integers and we have no way of describing an irrational number unless it has some "magic" geometric or a algebraic property. So how on earth would we be able to describe what sort of irrational number multiple to a rational number. It really seems we'd have to get circular and have something like the product of two irrational numbers is rational if and only if one is a rational multiple of the inverse of the other. – fleablood May 02 '20 at 05:57
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    Try it.... say something false... The product of $x$ and $y$ is rational if ... what ... about $x$ and $y$. Don't worry about saying something true. Just try to say something that's coherent. – fleablood May 02 '20 at 05:59
  • @fleablood The method of contradiction might not be powerful, otherwise we should have been able to settle the question of $\pi\times e$ as in the links by Learning. – Kumar May 02 '20 at 06:10
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    What? I wasn't asking you to prove anything by contradiction (prove what by contradiction). I was asking you to describe and come up with language that can possibly describe what such a theory might state. "Theorem: If $x$ and $y$ are irrational then $xy$ is rational if purple monkey dishwasher". What possible words about irrational numbers could we have that purple monkey dishwasher would even make sense? What aspects of irrational numbers are there that we can even use? We can describe what they don't do, but not what they do. – fleablood May 02 '20 at 07:17

1 Answers1

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You ask:

I want to know that if there exists a theorem/result which can exactly point out that when does (a shot of classification that) the multiplication of given two irrational numbers is a rational number?

Ummm... just multiply them and find out. Works most of the time!

David G. Stork
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    Does it? `:~( If it works, can you please tell me is $\pi\times e$ rational? – Kumar May 02 '20 at 19:50
  • @Kumar :Touché. Although, for ostensibly non-known products of two presumably transcendental (not merely irrational, i.e. non-algebraic) numbers such as pi times e, the likliehood of the result being irrational (or even again transcendental, I would wager) is infinitely more likely than it being rational. I believe there are infinitely more irrationals than rationals, and indeed higher-magnitude infinity more transcendentals than algebraics. As for products of algebraic irrational numbers, I'm not sure of an algorithm or the like (of rationality of the result); I too am curious about this. – user946772 Aug 25 '21 at 16:54