ππ isn't exact if you use rational numbers — Agent Smith

Again, what is the definition of 'exact'?

Pi is an irrational number. That doesn't make it not exact.

Agreement on this point seems to be breaking out! I would only add that a true half is equally impossible to create (physically) - or, if created, impossible to know that it has been created. — Cuthbert

The application of division, fractions, ratios, is generally very problematic, and the way that conventional mathematics treats division in general is very inadequate. In physical reality, the way that a thing can be divided is governed by the type of thing which is to be divided. In reality, the type of thing to be divided actually determines the principles of division which can be employed in dividing the thing. But mathematicians appear to pay no respect to this fact, and produce principles which allow any object to be divided in any way. The mathematician's simplicity, every object is infinitely divisible.

The problem with this approach to division becomes very evident when things like sound waves are being divided. Of course physical reality actually prevents infinite frequencies and infinite wavelengths. But if we ignore the physical reality of waves, and adhere to mathematical principles of division instead, we end up with the Fourier uncertainty. This type of uncertainty is the direct consequence of a failure to determine the correct way to divide space and time, according to physical reality.

We await Metaphysician Undercover's rigorously presented new mathematics that realizes what he considers "the correct way to divide space and time according to physical reality" and avoids the many problems he imagines in current mathematics.

The problem with understanding what is meant by an "exact" number in math is tied to the concept of number systems.

In a standard positional number system, the base is a positive counting number greater than 1. The standard number system we are most familiar with is base10, but computer scientists also use base2 (binary), base8 (octal), and base16 (hexadecimal).

Consider a baseN number system where N is a positive counting number greater than 1. Let set A be the primes that divide N, and the number 1. For base10, A = {1, 2, 5}. Let set B be all the counting numbers which are products of numbers in set A only. For base10, B = {1, 2, 4, 5, 8, 10, 16, 20, 25, ...}. Then any fraction whose numerator and denominator are integers, and whose denominator comes from set B, can be expressed as a terminating decimal number in baseN. For base10, 3/8 = .375, 17/25 = .68, etc. But in base10, 1/3 = .33333333333...

So what does it mean to say π is exact? It means the same thing as saying that 1/3 is exact. 1/3 and π are simply symbols (or names) for well-defined numbers that cannot be expressed as terminating decimals in base10. But they are, nonetheless, well-defined : 1/3 is the ratio of 1 to 3, π is the ratio of circumference to diameter.

Where 1/3 and π differ, is that 1/3 can be expressed as a terminating decimal if we choose a different base, so long as that base is a positive integer greater than 1. In base3 for example, 1/3 = .1. However, π cannot be expressed as a terminating decimal in any standard number system (i.e., a number system with a positive integer base greater than 1). This is because π is irrational.

You might ask : Why can't we use a basePi number system? Then 10 = π. Seems reasonable on its face, but the problem is that basePi is what we call a non-standard number system, and weird stuff happens in these number systems. For example, 10 basePi = 3.14159... basePi.

You can look it up if you really want to know more (it might make your head hurt).

You can look it up if you really want to know more (it might make your head hurt). — Real Gone Cat

Thanks. That's enough.