What is the quotient of two irrational numbers?

Can you divide one irrational number by another? What would be the quotient of, say, Pi divided by 2^(1/2)?What is the quotient of two irrational numbers?
yes, you can devide them, although the quotient would also be irrational. Just devide them as usual using the decimal form and take it to as many decimal places as you would likeWhat is the quotient of two irrational numbers?
The quotient of two irrational number is not always irrational.


For instance, the square root of 2 is irrational.


The sqrt(2) divided by the sqrt(2) is 1 which is a rational number. Report Abuse

Note: Fighting the battle against misinformation.


Pi is irrational. It is also transcendental.





If you can express it as a fraction with integers for the numerator and denominator, it's rational. If it's a real number, and it's not rational, it's irrational. There is no third type of real number.





And as was noted above if you divide two irrationals, the result may be irrational, it may be rational.





2e/e = 2 rational


sqrt(6)/sqrt(3) = sqrt(2) irrational
Whoops, my previous answer used a screwed up (read ';wrong';) definition of irrational...





hayharbr has essentially made my point, though. Odd that I missed that before.
Euler did this with Pi and e and showed it was a rational number. I can't find my book now sorry.


Normally it is just another irrational number. Probably nothing meaningful from a calculator, but maybe using them symbolically. One could simplify Pi/2^1/2 by squaring then you have Pi^2 / 2. Pi seems to mean a lot of things. Perhpas if you go back to a basic definition. Maybe a circle the diameter of a hypopotenuse with sides = 1,1 ,2^1/2 ? Now is the circumference have something to do with 2^1/2 ?


The right angle of the triangle also is a point on the circle.


Also M. Gardener has a proof of the area of this combinatin of circles and squares. {good question sorry maybe better left as an exercise}
Sometimes the quotient of two irrational numbers is rational. pi divided by pi is a trivial example. Also, sqrt(8) divided by sqrt(2) = sqrt(4) which is 2.
Pi is not irrational but trancedent


Pi divided by 2^(1/2) is trancedent


root(7) / root(5) is irrational


root(45) / root(20) = 3/2 is rational


root(36) / root(4) = 3 is natural
You can always divide two irrational numbers. You can do the calculations by hand; however, a calculator would be much easier.
Just a followup to combat misinformation. The quotient of pi and e is not known to be rational. I'd bet most mathematicians would be it is transcendental also, but there is no proof of that. Also, all transcendental numbers are irrational, but not the other way around.
Of course. pi/root(2) can be rationalized to avoid dividing by a long (endless) decimal as pi * root(2) / 2.





But if you want a decimal answer you have to decide how much accuracy you're going to use for pi and root(2). And the result is still irrational.

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