multiplying an dividing irrational numbers and quantifiers

Question

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.

Answer 1

For any real number $x$, rational or irrational, $\frac{x}2=\frac12x$ is a real number, and $2\cdot\frac{x}2=x$. If $x$ is rational, $\frac{x}2$ is also rational; if $x$ is irrational, $\frac{x}2$ is also irrational; but in both cases it exists.

Your difficulty seems to arise from trying to apply the notion evenly divisible to real numbers in general, when in fact it applies only to integers. It is perfectly true that if $n$ is an integer, then there is an integer $m$ such that $n=2m$ if and only if $n$ is even, so that some integers — the odd ones — are not two times some integer; and it’s often useful to distinguish the even integers from the odd integers. The odd integers are, however, still two times some rational number, albeit not an integer. E.g., $5=2\cdot\frac52$.

When we move from the integers to the rational numbers, matters change: if $p$ is rational, so is $\frac{p}2$, so every rational number is twice some rational numbers. And when we move on to the real numbers, we retain this property: if $x$ is a real number, so is $\frac{x}2$, so every real number is twice a real number.