# Property of Irrational

• July 2nd 2010, 12:23 AM
theon
Property of Irrational
Can the product of an irrational and any real number be a rational?
• July 2nd 2010, 12:36 AM
melese
Quote:

Originally Posted by theon
Can the product of an irrational and any real number be a rational?

Simply take $\sqrt{2}$ (irrational) and take $\sqrt{2}$ (real number). Multiply $\sqrt{2}\cdot\sqrt{2}=2$(rational).
• July 2nd 2010, 12:36 AM
pickslides
What you are asking and what you really want to know might be two different things.

Remember $\mathbb{Q} \subset \mathbb{R}$

I can think of an irrational number that is multiplied by a real that gives a rational. But this does not work for all cases. Is this what you want to know?
• July 2nd 2010, 01:13 AM
theon
You're right. Based on your post, I'd appreciate greater clarity.

Are the instances where it works seperable, as one or more groups, from cases where it doesn't? If so, how might we described each group?
• July 2nd 2010, 02:47 AM
CaptainBlack
Quote:

Originally Posted by theon
You're right. Based on your post, I'd appreciate greater clarity.

Are the instances where it works seperable, as one or more groups, from cases where it doesn't? If so, how might we described each group?

I think we want you to rephrase your question so that it is unambiguous.

CB
• July 2nd 2010, 05:25 AM
theon
OK. Hope the following is better.

Pickslides had said: "I can think of an irrational number that is multiplied by a real that gives a rational. But this does not work for all cases."

So, since the product of an irrational and a real is a rational in some cases but not in other cases, are there rules by which we can know in advance whether the outcome would be an irrational or a rational?

(e.g. irrational x rational = irrational ... or... irrational x irrational = rational,... or... irrational x itself = rational... etc.)
• July 2nd 2010, 11:54 PM
CaptainBlack
Quote:

Originally Posted by theon
OK. Hope the following is better.

Pickslides had said: "I can think of an irrational number that is multiplied by a real that gives a rational. But this does not work for all cases."

So, since the product of an irrational and a real is a rational in some cases but not in other cases, are there rules by which we can know in advance whether the outcome would be an irrational or a rational?

(e.g. irrational x rational = irrational ... or... irrational x irrational = rational,... or... irrational x itself = rational... etc.)

For the product of an irrational number and some other number to be rational the other number must be a rational multiple of the reciprocal of the irrational number (and so is itself irrational).

But the above is so obvious to hardly need stating.

CB
• July 3rd 2010, 01:05 AM
theon
Quote:

Originally Posted by CaptainBlack
For the product of an irrational number and some other number to be rational the other number must be a rational multiple of the reciprocal of the irrational number (and so is itself irrational).

But the above is so obvious to hardly need stating.

CB

Perfect.

I assume this is the only rule that applies?