Are Irrational Numbers Possible?

π. It’s value is 3.14. Well, actually, it’s 3.14159. Well, actually, it’s a 3 followed by an infinite number of digits. It is an irrational number; it cannot be expressed as a ratio of two integers and therefore must have an infinite number of digits.

Now let’s bring to bear a realization that I had a long time ago and that physicists are coming around to: the universe is defined in terms of information, not mass, energy, momentum, or any of those other things we’ve spent so much money studying. (see Information Physics, Information is the Fundamental Constituent of the Universe, and Second Thermo) The giveaway for this is the Uncertainty Principle, as well as its cognate, the Second Law of Thermodynamics. Now, we normally think of the Uncertainty Principle as something that applies to a single measurement. If we want to measure the position and momentum of a particle, we have to use a photon that unavoidably disturbs the particle we’re trying to measure. This principle applies to every measurement we make. 

But consider what happens when we think about the universe as a whole. There’s a cumulative limit on how much information we can obtain from the universe. At the very least, the fact that there is a finite amount of everything — mass, energy, space, time, angular momentum, charge — strongly suggests that there is also a finite amount of information in the universe.

This means that we can never express the value of π properly. If we use all the matter, energy, charge, and other “stuff” in the universe to express the value of π, there will STILL be a place where our number trails off into an ellipsis. 

The same thing applies to ANY measurement in the universe: the universal gravitational constant, the charge on an electron, the fine structure constant, even the Heisenberg uncertainty constant. Every one of these numbers, even if measured to perfection, would still end by trailing off into an ellipsis representing the limits of information in the universe. 

So what would happen if we attempt to directly measure π with extremely high precision? What if we were to draw a circle centered on the center of the universe, with radius equal to the radius the known universe, and then walk along that circle with measuring tape, measuring the circumference of that circle with a precision of nanometers? That result would take a mere 35 digits to express in full precision. Hmm...