I'm doing a homework assignment, and I am pretty sure that I know which answer they expected, but the problem made me start thinking more deeply about numbers.
So if you have an an arbitrary irrational number, is there a number which is equal to that number / 2? Is there a number which is equal to that number times two?
For the latter, I would say yes. For the first question, I've come to think no. The reason being that I do not think that an irrational number can be divided evenly. If irrational numbers cannot be divided evenly, then 2 times any number, no matter how close, will always be different than a rational number by some amount. But then, if this is the case, then also that means that there was no such number that could have been multiplied by 2 to make the irrational number. And to my knowledge, there is no irrational number which when multiplied by a rational number is not irrational. So 2 times an irrational number, is an irrational number, but that would then not be evenly divisible by 2. This is a contradiction, so it would seem to follow that all irrational numbers must be evenly divisible.
Can anyone make sense of this for me.