Answer by Tom McFarlane:

First of all, it should be emphasized that a real number is either rational or irrational. Numbers are either one or the other, period, and botheand π are irrational.However, one can ask whether some irrational numbers are in some sense harder to approximate by rational numbers than others, and in that sense "more irrational." One way to make this notion precise is the Irrationality Measure, which assigns a positive number µ(x) to each real numberx. Almost all transcendentals, and all (irrational) algebraic numbers have µ(x)=2, includinge. But some transcendentals can have µ(x)>2. Liouville numbers, for example, have infinite µ(x). They can be approximated very well by rationals, while algebraic irrationals can not. So, counter-intuitively, the irrationality measure of an irrational islargerwhen it isbetterapproximated by a rational. The irrationality measure ofeis 2. The irrationality measure of π is not known. It is most likely 2, but could be larger.

If both π and e are irrational, how can we logically know if they are equally irrational?

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